Keywords: identi cation, characterisation, and quanti cation of cosmological information Cosmology: Large-scale structure of Universe, Cosmic web, Methodology: Statistical methods, Pattern analysis, Unsupervised machine learning, Mixture models. v Structures grandes échelles de l'Univers, Toile cosmique, Méthodes: Méthodes statistiques, Reconnaissance de motifs, Apprentissage automatique non supervisé, Modèles de mélange . . . . . . . . 6.19 . . . . .

Mes premiers remerciements vont évidemment à Nabila Aghanim et Aurélien Decelle. Bien plus que de simples superviseurs, vous avez non seulement modelé le contexte scienti que de cette thèse, son contenu et sa direction, mais m'avez également o ert une liberté d'exploration et d'expression à laquelle je n'avais osé songer jusqu'alors. Au-delà du partage de vos vastes connaissances scienti ques dont j'ai béné cié durant ces trois années, je vous remercie pour votre enthousiasme, votre bienveillance et votre bonne humeur qui font que cette thèse n'a pas simplement été un travail, mais une expérience humaine unique ayant mis à rude épreuve mes tendances misanthropes. J'ai entendu qu'un chercheur était, humainement et scientiquement, façonné par ses directeurs de thèse. Si tel est le cas, sachez que je suis er de ma généalogie académique.

Bien sûr, comment survivre à une thèse sans le soutien sans faille et quotidien de collègues et amis avec lesquels se plaindre de nos superviseurs. C'est donc tout particulièrement que je remercie les thésards de l'équipe cosmo, Daniela, Thibaut et Marion avec qui j'ai eu la chance de subir partager ces trois années, mais aussi ceux partis trop tôt en post-doc, Adélie, Victor et Louis, ou Raphaël, arrivé trop tard en doctorat, sans lesquels le quotidien se serait trouvé bien moins amusant. Des remerciements spéciaux vont aux "hommes simples", Victor et Thibaut, pour ces échanges sains et cordiaux sur slack qui ont égayé nombreuses de mes journées pendant les sinistres périodes de con nement.

Que serait une thèse sans co-bureaux exceptionnels ? Céline et Marion, je ne pouvais pas rêver meilleur bureau que le 216, à mi-chemin entre le poste de travail et la salle de pause de l'équipe cosmo. Mélangez plaintes et chants, amour et haine, un peu de critique de ceux qui viennent de sortir, ajoutez-y une once tonne de travail et vous obtenez les ingrédients essentiels du "bureau des princesses". Je remercie plus généralement l'équipe de cosmologie de l'IAS pour leur accueil, à commencer par les chercheurs permanents, Marian, Mathieu, Julien, Hervé, Nabila, et désormais Laura pour avoir maintenu une ambiance à la fois chaleureuse, familiale, mais aussi propice au travail et à la re exion. Je remercie également les post-doctorants de l'équipe, passés et présents, Hideki, Giulio, Nicola, Alex et Joseph pour leur partage. D'un point de vue plus personnel, je remercie ma famille pour leur soutien tout au long de mes études, et surtout Anaïs pour m'avoir poussé à poursuivre ce souhait et avoir toujours cru en moi. La thèse est la consécration d'un long parcours académique durant lequel deux enseignants en particulier ont joué un rôle central dans mon intérêt et mon accès à la recherche. Pour cela, j'exprime ma sincère gratitude à Charlie Kahloun pour m'avoir communiqué, aux travers de ses cours et de mon expérience en stage, cette passion pour la physique mais également à Pierre-Yves Richard pour ses conseils précieux durant mes années à Supélec, et après. Merci aussi Guillaume, pour ces heures de discussions, ces questions et ces admirations communes qui ont alimenté mon intérêt pour la Science. Tu disais "faire une thèse par procuration". Je suis heureux d'aujourd'hui te décerner le titre de "Docteur par procuration".

J'adresse des remerciements à tous les membres de l'IAS, les scienti ques des autres équipes, le personnel administratif, et des encouragements aux futurs docteur(e)s de l'IAS.

En n, je souhaite remercier le jury de thèse pour le temps consacré à la lecture de mes travaux, mais aussi pour leurs commentaires sur le manuscrit et la présentation orale.

 6.4Normalised power spectra in the cosmic environments in real space . . . . . . 6.5 Partial derivatives ∂P αα (k)/∂θ i for the di erent environmental auto-spectra in real space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Correlation matrices of matter and environmental auto-spectra in real space . 6.7 1σ con dence ellipses for all the pairs of cosmological (Ω m , Ω b , h, n s , σ 8 , M ν )

and nuisance (λ th ) parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Zoomed con dence ellipses in the M ν -Ω m and M ν -σ 8 planes . . . . . . . . . 6.9 Evolution of the marginalised cosmological constraints with the maximum scale involved in the real-space analysis . . . . . . . . . . . . . . . . . . . . . . 6.10 Evolution of the SNR with the maximum scale for the power spectra used in the Fisher analysis in real space . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Monopoles in real and redshift spaces for the ducial cosmology . . . . . . . . 6.12 A closer look at the impact of σ 8 , Ω m and M ν on the matter power spectrum and those derived from cosmic environments . . . . . . . . . . . . . . . . . . . 6.13 Partial derivatives ∂P αα (k)/∂θ i for the di erent environmental auto-spectra in redshift space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.14 Correlation matrices of matter and environmental auto-spectra in redshift space 6.15 Fisher ellipses in redshift space . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction

"Accepte-la, cette thèse. "

A. D

Cosmology in the era of data science

The rst mappings of galaxies in the sky [Joeveer et al., 1978;Einasto et al., 1980;de Lapparent et al., 1987] together with rst simulations [Zel'dovich, 1970;Doroshkevich & Shandarin, 1978] show that today's matter, may it be dark or baryonic, is not lling the Universe uniformly. Instead, it is spreading over a gigantic well-organised structure shaped by billion years of gravitational forces. This pattern, commonly referred to as the cosmic web [Bond et al., 1996], exhibits massive nodes that are linked together by uni-dimensional bridges of matter, the laments, themselves found at the intersection between thin and mildly-dense planes of matter labelled walls forming the vast shells enclosing underdense volumes almost devoided from galaxies called voids. This complex spatial pattern provides a rich amount of information about the content of the Universe but also contains an imprint on its history, how it formed and evolved through time. One of the aims of modern cosmology is to understand this spatial distribution and extract all the possible information to link it with theoretical principles. To do so, the e ort of various communities is being put together that allows: (i) deeper, wider and always more complete observations of di erent matter tracers [e.g. York et al., 2000;Colless et al., 2001;Driver et al., 2009;Laureijs et al., 2011;Abbott et al., 2016]; (ii) accurate simulations of the Universe and its evolution from large to small scales enabling to model the physics of stars and galaxies to clusters and lament [e.g. Springel et al., 2005;Vogelsberger et al., 2014;Dubois et al., 2014;Nelson et al., 2019;Villaescusa-Navarro et al., 2020]; (iii) the development of state-of-the-art methods and algorithms to analyse the large amount of data (web nders [see the review of Libeskind et al., 2017, and reference therein], statistical descriptors of non-Gaussian elds [e.g. Hahn et al., 2020;Cheng et al., 2020], robust and accurate cosmological constraints etc.). All these analyses then provide paramount information to confront to theory thus validating or invalidating some models on the formation and evolution of the Universe itself and its components at all scales.

As such, data analysis is a pillar of modern observational cosmology since its beginning. The interplay between the two elds is even more emphasised by the recent and upcoming collections of always larger and more complex datasets allowing the measurement of cosmological quantities with an unprecedented accuracy. New galaxy surveys such as Euclid [Laureijs et al., 2011], the Vera Rubin Observatory [Collaboration et al., 2009] or the Dark Energy Survey Instrument [DESI, Levi et al., 2013] and upcoming observations of physical quantities like the cosmic microwave background with the Simons Observatory [Ade et al., 2019] or the 21cm neutral Hydrogen line with the Square Kilometer Array will require the development and application of state-of-the-art methods to analyse and interpret the huge amount of complex data. By providing orders of magnitudes more data than previous generations, this soon-available ood of data is a double challenge in both designing the right tools to deliver meaningful and precise cosmological analyses but also in their optimisation to obtain reasonable time complexity. Standard statistics, for instance based on the n-point correlations, are heavy to derive for such large datasets and require computational enhancements [see e.g. Philcox & Eisenstein, 2020;Philcox, 2021]. In parallel, the growth of machine learning algorithms of the past decade proposes a unique opportunity to handle such data. Their rst applications in astrophysics showed outstanding results in many tasks, ranging from the re nement of redshift estimates in photometric surveys [e.g. Carrasco Kind & Brunner, 2013] to the encoding of high-order information [e.g. Cheng et al., 2020;Allys et al., 2020] and constraining cosmological parameters [e.g. Ribli et al., 2019]. Despite these successes, it is essential that, when applied to physics, machine-learning-based algorithms, and in particular deep-learning ones, are well-controlled and understood. Beyond the fact that machine learning can be particularly opaque to the user in the way it learns a representation of a dataset, a special attention must be drawn on the inclusion of errors and uncertainties on the estimated quantities. Understanding the features of importance for the learning, the biases that can be induced when models are trained from prede ned datasets and how to exploit correctly the output of such algorithms are key questions to which several communities are jointly trying to answer.

With the motivation of enriching our knowledge on large-scale structures of the Universe for cosmological analyses, we will explore, throughout this thesis, these two linked facets of data science and cosmology. The work presented in the manuscript is in particular addressing two main questions that are:

1. How can the matter distribution at present time be used to identify and characterise cosmic web environments?

2. What is the cosmological information contained in the cosmic web environments?

In the corpus of the manuscript, the context of the rst question is made broader than the cosmological one and can be formulated, in a data science point of view, as "How can we e ciently extract meaningful representations of spatially structured point-cloud datasets?" Point-cloud data are indeed ubiquitous in many elds of science and even though cosmological datasets representing the large-scale matter distribution are the main object of study, we include the presented work in its larger context of data science when necessary.

Organisation of the manuscript

To distinguish the di erent contributions, the manuscript is divided into three parts. The rst one is dedicated to the presentation of the cosmological context of the thesis by rst focusing on the theory of structure formation in Chapter 1. Building-up on this theoretical introduction, we introduce in Chapter 2 the manifestation of cosmic structures in data and simulations, and we discuss the di culty of linking observable quantities with theoretical predictions and cosmological models. This chapter also sets up the astrophysical and cosmological interests of identifying the di erent components of the cosmic web with a particular emphasis on the central role of laments. Finally, it also discusses the di culties of carrying an e cient extraction of structures from both simulated and observed data.

The second part of the manuscript exposes the methods developed to tackle the extraction of patterns and features from generic point-cloud datasets. Chapter 3 rst analyses a statistical physics formulation of a machine learning procedure to gain a physical insight on the structure of point-cloud data. The associated developments and results led to a published article Tran. Pattern Anal. Mach. Intell. [arXiv:2106.09035]). This formulation provides the statistical formalism of the actual method allowing the extraction of a continuous one-dimensional structure from a given dataset. This method, named T-ReX, was also presented in T. Bonnaire, N. Aghanim, A. Decelle & M. Douspis (T-ReX: a graph-based lament detection method, Astron. Astrophys. 337 A18 (2020) [arXiv:1912.00732]).

In the third and last part of the manuscript, we propose to analyse the cosmic web pattern by rst exploiting the previously-mentioned principal graph formulation to represent the lamentary part of the pattern in Chapter 5. In particular, we focus on the structural properties of laments and we investigate in C. Gouin, T. Bonnaire & N. Aghanim (Shape and connectivity of groups and clusters: Impact of dynamical state and accretion history, Accepted for publication in Astron. Astrophys. [arXiv:2101.04686]) the impact of the lamentary pattern on the physical properties of astrophysical objects like galaxy clusters. Finally, Chapter 6 presents a comprehensive analysis aiming at theoretically quantifying the statistical information contained in the di erent cosmic web environments in order to constrain cosmological parameters. This work will be published in a forthcoming article T. Bonnaire et al., (in prep).

Part I Emergence of large-scale structures

Chapter 1.

Structure formation in the Universe

"Ce week-end, je me suis rendu compte que la structure de l'univers est quand même vachement compliquée. " T. P This rst chapter brie y presents the theoretical context of large-scale structure formation in cosmology. It exposes the ideal setup of linear theory describing the evolution of matter perturbations at the largest scales of the Universe and the mathematical representations of common cosmological observables like the density eld. Finally, it introduces the current favoured theoretical model of the Universe, the Lambda cold dark matter cosmological model, review its successes and main parameters. Of course, the chapter only introduces the headlines of these rich topics and do not constitute a thorough theoretical introduction. For a more detailed presentation of the cosmological context, the interested reader is referred to seminal text books such as Peebles [1980] or to articles such as Bernardeau et al. [2002].

The homogeneous universe 1.Distances in an expanding universe

As observed locally around us, the Universe appears highly inhomogeneous with small-scales structures made of stars and galaxies. However, when averaged over large scales, its structure is relatively "simple" and nearly homogeneous. Said di erently, by smoothing the matter distribution at large enough scales, it appears at with no strong dominant structures. This statement is partly at the foundation of theoretical cosmology and is expressed as the cosmo-logical principle describing the Universe as statistically homogeneous and isotropic at large scales. This principle was initially introduced as a philosophical concept by arguing that our position in the Universe has nothing special and that there should not be preferred directions for the distribution of matter. The incredible advances in observing the sky, and particularly the tiny anisotropies detected in the cosmic microwave background [CMB, Smoot et al., 1992], however brought evidences supporting this principle.

Under these assumptions, the mathematical framework o ered by general relativity, allowing to describe the evolution of the matter, energy and geometry of the Universe, lead to solutions of Einstein's eld equations known as the Friedmann-Lemaître-Robertson-Walker metric [FLRW, Friedmann, 1922;Lemaitre, 1931;Robertson, 1935;Walker, 1937]. This metric describes the way to compute distances in a Universe that is statistically isotropic and homogeneous, and leads to

ds 2 = c 2 dt 2 -a(t) 2 dχ 2 , (1.1)
where c is the speed of light in vacuum, dχ the comoving distance independent of how the Universe expands or contracts, and a(t) is the scale factor1 normalised such that today, a(t 0 ) = a 0 = 1. In our expanding universe [Hubble, 1929], it is indeed convenient to rely on distances between objects that are independent of the Universe global ow. The proper distance d(t) of a source and its comoving counterpart are linked by the scale factor as d(t) = a(t) dχ. In Eq.

(1.1), the squared distance element dχ 2 can take di erent forms depending on the geometry and curvature of the Universe. For instance, in Cartesian coordinates and assuming a at universe, it simply reduces to dx 2 + dy 2 + dz 2 . More generally expressed in terms of spherical polar coordinates (noting r the radial distance and Ω the solid angle) and with a curvature term, we have dχ 2 = dr 2 / (1 -kr 2 ) + r 2 dΩ 2 , with k = {-1, 0, 1} representing the sign of the spatial curvature. Considering the light emitted by a given source at time t em following a null geodesic with ds 2 = 0, we have from Eq. (1.1)

χ(t em ) = tem 0 c dt a(t) .
(1.2)

Due to the expansion, the observed wavelength λ obs appears however larger than the emitted one λ em , which de nes the redshift z as

λ obs λ em = 1 a(t em ) = 1 + z, (1.3)
which also allows us to exhibit the relation between the redshift and the scale factor as a = (1 + z) -1 .

The dynamics of the homogeneous Universe

Now equipped with a means of computing distances by the metric (1.1), the dynamics of the Universe can be described by the Friedmann equations linking its expansion with its energy content. By assuming that the Universe is lled with perfect uid, the rst Friedmann equation reads

H 2 = 8πG 3 ρ + Λc 2 3 - kc 2 a 2 , (1.4)
where G is the gravitational constant, Λ is the cosmological constant, ρ is the mass density of considered uid and H(t) := ȧ(t)/a(t) is the Hubble parameter characterising the expansion rate of the Universe at time t. Assuming energy conservation and taking the time derivative of Eq. (1.4), we obtain the second Friedmann equation

ä a = - 4πG 3 ρ + 3p c 2 + Λc 2 3 , (1.5)
where p is the pressure. The Universe is in fact assumed to lled with a mixture of perfect uids made of matter and radiation leading to ρ = ρ m + ρ r . Similarly, Λ can be interpreted as a uid with density ρ Λ = Λc2 /8πG and k as a uid of curvature with density ρ k = -3kc 2 /8πGa 2 . Assuming equations of state of the form ρ = wp for uids, we get w r = 1/3 for the radiation component, w m = 0 for the pressureless matter, w Λ = -1 for the cosmological constant and w k = -1/3 for the curvature. Introducing further the critical density de ned as ρ crit (t) = 3H(t) 2 /8πG, the uid densities can be expressed in terms of the critical one, also known as the density parameters Ω i (t) = ρ i (t)/ρ crit (t), where i designates either radiation, matter, Λ or curvature. Note that, in the rest of the thesis, when referring to an energy density Ω without specifying a precise time, we refer to the one at present time, t 0 . Under these notations, the rst Friedmann equation (1.4) reads Ω r (t) + Ω m (t) + Ω Λ (t) + Ω k (t) = 1 and explicitly expresses the conservation of the matter-energy content in the Universe. By deducing a time evolution of the matter densities for each component using the rst law of thermodynamics and coupling it back with the Friedmann equations, it yields

H(t) = H 0 Ω r,0 a(t) 4 + Ω m,0 a(t) 3 + Ω k,0 a(t) 2 + Ω Λ,0 , (1.6)
with Ω i,0 = Ω i (t 0 ) denoting the energy density of component i at the present time t 0 and H 0 the value of the Hubble parameter at present time. This simple paradigm of a homogeneous Universe already highlights the importance of the cosmological density parameters and gives an idea of the interplay between the dynamics of the Universe and its content. By knowing the energy density values at the present time, one gets crucial information on the evolution of the Universe, and on how it expanded through H(t).

The birth of large-scale structures

The homogeneous model presented in the previous section brought tremendous knowledge on the evolution the Universe but also allows the computation of distances from the redshift of a source or the notion of horizon. Despite these achievements, the assumption of a homogeneous and isotropic Universe is only valid at large scales, above roughly 100 Mpc 2 [Yadav et al., 2005] and, by de nition, does not predict the formation of structures at smaller ones. Astronomers, striving to map objects in the sky however depict a Universe made of a multitude of structures at di erent scales, with stars, galaxies, but also clusters of galaxies at larger scales. At even larger one is drawn an interconnected network of galaxies forming oriented structures in space that we call laments found at the border of gigantic quasi-empty regions called voids. This non-uniform distribution is shown in the rst mappings of galaxies for instance reported in Fig. 1.1. All these hierarchical structures suggest the presence of inhomogeneities in the matter distribution. The current leading theory of structure formation assumes that the seeds of these density perturbations have a quantum origin which led to the small inhomogeneities observed in the initial matter distribution. These small uctuations of densities evolved through time and gravity to the rich structures that are observed today. In this section, we brie y expose3 this scenario of gravitational instability.

Linear perturbation theory

In general, the introduction of uctuations in the matter distribution makes the equations governing the dynamical evolution of the uid not analytically tractable. Restricting the analysis to small amplitude uctuations is already instructive and known as the linear perturbation theory leading, as we will see, to their growth. The perturbations can be characterised by the density contrast de ned as

δ(x) = ρ(x) - ρ ρ , (1.7)
where x is a spatial location and ρ is the mean density. We focus in this section on perturbations that have small amplitudes |δ| 1. In addition, we restrict the analysis on a Universe made of matter only4 where the matter component is the dominant one. modeled as a perfect uid and making use of Newtonian dynamics. Naturally, this single-ow uid description is expected to break at small-scales where the interactions and crossings between matter particles occur, that is, at scales where cosmic environments like laments or dark matter halos are formed.

The three equations governing the motion of the uid are given by the continuity, the Euler and the Poisson equations that we can respectively write, in comoving coordinates x and linearised considering at the same time small perturbations in the density and small velocities with respect to the Hubble parameter H(t), ∂δ (x, t) ∂t

+ 1 a ∇ x • v(x, t) = 0, (1.8) ∂v(x, t) ∂t + ȧ a v(x, t) = - 1 a ∇Φ(x, t),
(1.9) ∆ x Φ(x, t) = 4π ρGδ(x, t), (1.10)

where v is the uid peculiar velocity (independent from the expansion of the Universe) and Φ is the gravitational potential. The rst equation is the conservation of matter, the second one is the Euler equation of motion stating that changes in the velocity ow are caused by the gravitational acceleration and the last one is the Poisson equation linking the gravitational potential Φ to the overdensity δ through the Laplacian ∆. The equations leads to the linear homogeneous second order di erential equation describing the time evolution of the density perturbation δ, ∂ 2 δ(x, t) (1.11) in which we note the absence of spatial derivatives with respect to x. The solution can hence be expressed as the linear combination of two independent solutions, one leading to a growing and the other a decaying perturbation, δ(x, t) = D + (t)δ + (x, 0) + D -δ -(x, 0) in which we decoupled the spatial and time evolution of the perturbation. At the considered linear scales, where interactions between matter particles can be neglected, the matter distribution at time t can thus be described by multiplying the initial distribution by the growth factor D ± (t).

∂t 2 + 2H ∂δ(x, t) ∂t - 3 2 H 2 Ω m (t)δ(x, t) = 0,

Zel'dovich formalism

We have seen that an initial small perturbation in the matter distribution can either grow or vanish at linear scales. This already powerful description of the matter distribution at any time is useful for theoretical predictions but, for scales of few tenth of Mpc, we need to resort to the non-linear versions of the continuity and Euler equations (1.8) and (1.9). Zel'dovich [1970] proposes to describe the evolution of perturbations using a Lagrangian description of the uid. This leads, in particular, to re-write Eq. (1.11) in terms of a displacement eld encoding the information of how a particle moved from its initial position. Focusing only the growing mode, Zel'dovich nds that the perturbation grows such that

1 + δ(x, t) = 1 [1 -λ 1 D + (t)] [1 -λ 2 D + (t)] [1 -λ 3 D + (t)]
,

(1.12)

where λ 1 > λ 2 > λ 3 are the eigenvalues of a deformation tensor of the displacement eld.

In contrast to the linear evolution of perturbations from Sect. 1.2.1, the overdensity in the Zel'dovich approximation is evolving depending on the local properties of the matter distribution. It hence allows to describe the evolution of a perturbation in the mildly non-linear regime and predict the collapse of matter forming the locally anisotropic structures that are observed today and that form the cosmic web. If all eigenvalues are negatives in Eq. (1.12), then the density eld is deformed to create a locally underdense region called a void, while if λ 1 is positive, then the collapse occurs in a preferential direction forming a locally 2D structure named a wall. Following this reasoning, if λ 1 and λ 2 are positive, we end up with a two-dimensional contraction creating tubular-like structures called laments while a quasi-isotropic collapse (λ 1 λ 2 λ 3 and all positive) forms the nodes of the cosmic web. This Lagrangian description is very powerful to understand how initial seeds of inhomogeneities in the matter distribution give birth to the diversity of structures that are observed in data and simulations (that we will more precisely discuss in the Chapter 2). The collapse takes place at di erent scales, along di erent directions and at di erent times as cadenced by the amplitude of the eigenvalues. Even though very powerful, this rst order Lagrangian approximation also assumes that particles are not interacting which breaks at small scales and cannot represent accurately the formation of bounded structures. Several ways to improve the model have been proposed, such as the introduction of a viscosity term to the Euler equation (1.9) which led to the adhesion model [Kofman & Shandarin, 1988;Kofman et al., 1992].

Alternatively, the study of non-linear scales can be performed through numerical simulations that we discuss in the forthcoming Sect. 2.1.

Statistical descriptions of the matter distribution

One way to treat mathematically observables like the matter distribution in cosmology is to consider the uctuations δ(x) as a random variable and the observed overdensity eld as a realisation of a spatial process that is called a random eld.

Discrete random elds

Putting ourselves in a discrete setup where the space can be regularly gridded such that a value of the studied quantity δ can be measured at a location x ∈ R D , a random eld is de ned as the collection of random variables {δ(x i )} i∈{1,...,n} and associates to a given realisation a probability to occur. Taking the simple example of an image, one can consider that an Rvalued random variable is associated to each pixel, making the entire image the set of random variables constituting one realisation of the random eld. In the nite case, the random eld is described by the joint probability density function (pdf) of the n discrete random variables p(δ(x 1 ), . . . , δ(x 1 )).

The cosmological principle, stating that the distribution of matter in the Universe is, at large scales, homogeneous and isotropic (see Sect. 1.1), has a direct implication on the mathematical properties of the considered elds. The rst is statistically expressed as the stationarity of the associated random process yielding that the joint pdf is invariant under translations (i.e. not a function of the spatial index set). The isotropy in turn induces the eld to be invariant under rotation relieving the statistics from any preferential direction. Such invariant elds, widely used in cosmology are also a key element in many mathematical formulations in other elds of physics and applied mathematics. When equipped with additional local properties, they are for instance called Markovian and are at the basis of many developments in image processing such as texture recognition, classi cation and synthesis [see e.g. Efros & Leung, 1999;Varma & Zisserman, 2009].

In statistics, a common one-point summary of a probability distribution is given by the ensemble average over many realisations. One thing that makes cosmology a special science is that the object of study, the Universe, is the only realisation 5 we have access to. By invoking ergodicity, cosmologists however are able to assimilate ensemble and volume averages to extend the statistical properties of the uniquely observed Universe. As an example, the mean matter density in the Universe that we denoted ρ in Sect. 1.2.1 can be seen as the average over a su ciently large volume of the observed density, without requiring other samples of the Universe.

Correlation functions and poly-spectra

The natural way to describe centred elds with δ = 0 is through the next non-vanishing moment, the covariance function, also called the two-point correlation function, de ned at a given time t as ξ(r, t) := δ(x, t)δ(x + r, t) , (1.13)

where r = r . The dependency of this function on the norm of the separation only comes from the assumed statistical isotropy and homogeneity of the Universe. By the Wiener-Khinchin theorem [Wiener, 1930;Khintchine, 1934], the Fourier transform of the two-point correlation function de nes the power spectrum, as P (k, t) = ξ(r, t) exp (ik • r) d 3 r,

(1.14) also de ned as the covariance of Fourier modes, which can be expressed, for a statistically homogeneous and isotropic eld, as

P (k, t)δ D (k 1 + k 2 ) = 1 (2π) 3 δ(k 1 , t) δ(k 2 , t) ,
(1.15)

where δ(k, t) denotes the Fourier transform of the eld δ(x, t) and δ D the Dirac delta distribution. In this latter expression, the assumed statistical properties of the overdensity eld δ are expressed in the dependence on the norm k only in P (k) for the isotropy and in the Dirac delta for the translation invariance. One case of particular importance is the Gaussian random eld [for a review, see Adler & Taylor, 2007] which has the particularity of being fully de ned by its two rst moments that are the average and the correlation function. Mathematically, a random eld is said Gaussian if the joint probability of any subset of eld points {x 1 , . . . , x n } follows a multivariate Gaussian distribution

p(δ | µ, Σ) = 1 2π|Σ| exp - 1 2 [δ -µ] T Σ -1 [δ -µ] , (1.16)
with δ := {δ(x i )} n i=1 , µ and Σ are the sets of the n averages and covariances respectively. For centred elds6 like the matter overdensity δ, the Gaussian random eld is characterised by the sole knowledge of the covariance function. Indeed, by Wick's theorem, all the higherorder moments can be expressed as products of two-point functions summed over all possible pairings. Therefore, all the information is indeed contained in the covariance of the eld Σ. In cosmology, the Gaussian random eld, beyond its appealing mathematical tractability, is also representing to great accuracy the density perturbations arising after the cosmic in ation as observed by the CMB [Planck Collaboration VII et al., 2020;Planck Collaboration IX et al., 2020]. The corresponding power spectrum of these primordial uctuations, noted P 0 (k) is known to be scale invariant, usually expressed as P (k, 0) = A s (k/k * ) ns-1 , with k * a pivot scale. In the context of linear theory exposed in Sect. 1.2.1, the matter power spectrum of the density uctuations can be simply expressed as the product of P (k, 0) and the growth factor D + (t) responsible for the growth of perturbations. This is particularly powerful since we are hence able, at least at large enough scales, to describe the matter power spectrum based only on the knowledge of the initial one. The gravitational evolution of the instabilities however leads to a distribution that is not well-represented by a Gaussian anymore with a pdf that is mostly skewed towards higher density contrast and better represented by a log-normal distribution [Peacock, 1998]. In such cases, there is a leak of information into higher-order moments that can be expressed similarly to Eq. (1.13). For instance, the three-point correlation function can be written ζ(r 12 , r 13 , r 23 , t) = δ(x 1 , t)δ(x 2 , t)δ(x 3 , t) , (1.17)

where r ij is the norm of the vector x j -x i . The three-point correlation function can equivalently be expressed in Fourier space and is known as the bispectrum. Such summary statistics are at the basis of statistical analyses of large-scale structure datasets. Using higher order estimators of course bring more information about the underlying cosmology but also comes with a higher computational cost that innovative works try to circumvent [see e.g. Philcox, 2021].

The ΛCDM model

The current favoured theory for modelling the Universe and its evolution is called the Lambda Cold Dark Matter (ΛCDM) model. In this section, we review its main parameters, its main successes explaining why it stands as the "standard" model and introduce in a general manner the way cosmological parameters are constrained.

Presentation of the model

In its simplest form, the ΛCDM rely on a set of six cosmological parameters among which two are related to the physical density of baryons and dark matter, obtained by multiplying the density parameters and the reduced Hubble parameter de ned as h := H 0 /100 km s -1 Mpc -1 , namely Ω b h 2 and Ω m h 2 . Two parameters are then describing the amplitude and tilt of the power spectrum of the primordial matter uctuations, A s and n s . Finally, one parameter accounts for the geometry of the Universe, and the last one is linked with the reionisation era of the Universe at which the rst stars are formed. This minimalist set of six cosmological parameters together with theoretical equations governing the dynamics of the Universe assuming the cosmological principle partly exposed in Sect. 1.1 is the most simple one agreeing remarkably well with observations that made its supremacy. This is the reason why the ΛCDM model is also sometimes referred to as the concordance model of cosmology. Among all the observational evidences of the ΛCDM paradigm, perhaps the most important is the direct detection of the CMB [Smoot et al., 1992;Komatsu et al., 2011;Planck Collaboration I et al., 2016] constituting the rst emitted light in the Universe that probes the existence of hot photons emitted at the recombination (t ∼ 3.8 × 10 5 ) that cooled down while travelling to us. The apparent isotropy of the emission argue in favour of the cosmological principle and the small temperature uctuations advocates for small inhomogoneities in the initial matter distribution at the origin of the large-scale structures, as exposed in Sect. 1.2.1.

In the ΛCDM model, the Universe is born nearly 13.8 billion years ago from a singularity and, as indicated by its name, is made of three main components that are: (i) the cold dark matter (CDM) thought to be made of unrelativistic particles interacting only by gravity; (ii) the dark energy coming from the Λ cosmological parameters and responsible for the late-time acceleration of the Universe expansion; and (iii) the baryons composing the observable part of the matter like dust, stars and galaxies. The re ned measurements provided by Planck Collaboration XIII et al. [2016] report that 25.8% of the total mass/energy inventory of the Universe is brought by the CDM and 69.4% which are actually associated to the dark energy. Even though lling most of the energy content of the Universe, the nature of the dark energy component remains unknown nowadays and is at the heart of many cosmological experiments trying to improve our understanding of this mysterious ingredient like Euclid [Laureijs et al., 2011] or the Dark Energy Survey [Abbott et al., 2016]. More importantly, this leaves us with only less than 5% of baryonic, visible matter which are distributed to form galaxies and over di erent gas phases [see for instance Fig. 8 from de Graaf et al., 2019]. Of course, several g. Milgrom, 1983;Di Valentino et al., 2012, 2017;Maeder, 2017;Capparelli et al., 2018].

Cosmological parameters and matter power spectrum

Note that the six previous cosmological parameters were the fundamental ones describing the ΛCDM but some others are xed or can be derived, including the Hubble constant H 0 and the density parameters Ω m , Ω b , Ω Λ and ρ crit already encountered in Sect. 1.1.2. Another common parameter of interest, equivalent to xing the amplitude of the primordial spectrum A s , is σ 8 which corresponds to the variance of the late-time matter uctuations smoothed at a scale of 8 Mpc/h. Among the goals of observational cosmology are to test the validity of the ΛCDM model but also to provide accurate measurements its parameters. To do so, cosmologists rely on statistical representations such as the two-point correlation function of cosmological observables like the galaxy distribution, as discussed in Sect. 1.3. The cosmological parameter values can then be estimated by tting parameters of the model to the observed statistics. By combining the information provided by di erent observables, the community was for instance able to draw a picture of the 3D matter power spectrum matching with precision the one from the linear matter prediction. This is shown in Fig. 1. Salvati et al., 2018]. In Chapter 6, we shall also see how cosmic web environments can be used to constrain cosmological parameters of the ΛCDM model.

In this chapter, we have built the theoretical context of the large-scale structures formation. Starting from weak initial matter uctuations and in the framework of rst-order approximations, we were able to draw a picture of the late-time distribution of matter in which perturbations grow in an anisotropic way. This gives rise to the cosmic web [Bond et al., 1996] made of gravitationally collapsed structures in one, two or three spatial directions forming respectively the walls, laments and nodes in the matter distribution. We have seen that such elds of uctuations around the average noted δ can be represented by means of the random eld theory and derived statistics like the n-point correlation functions are keystones that cosmologists seek to measure in order to enrich their knowledge about the Universe. Indeed, the power spectrum of the matter distribution provides a wealth of information on the cosmological model and its parameters that are themselves describing the dynamics of the observed Universe.

Chapter 2. Building upon the theoretical prescriptions of structure formation presented in Chapter 1, the current one supplies the complementary view of the large-scale structures provided by simulations and observations. We rst introduce and motivate the central place of simulations for the study of non-linear physical processes and then focus on the di culties induced by realworld observations of the cosmic web pattern. We nally emphasise through many previous works the importance of detecting cosmic structures and carry on an extended review of the current literature for doing so.

Large-scale structures manifestation

Large-scale structures in simulations

First exhibitions

Even with the limited computational resources available in the 70s, numerical simulations quickly appeared as an indispensable tool to study the evolution of the density eld in the non-linear regime where |δ| 0. By resorting to approximations allowing the inclusion of the growth of non-linear structures through a rst order approximation of perturbations, theory predicts the birth of structures depending on the eigenvalues of the local deformation tensor as exposed in Sect. 1.2.2 [Zel'dovich, 1970;Doroshkevich & Shandarin, 1978; Klypin &

Dark matter only and hydrodynamical simulations

Early developments of simulations were based on the dark matter only evolution of the density eld including solely the e ect of gravity (also called N -body simulations). Starting from a set of initial conditions at very high redshift, usually taken from a Gaussian random eld1 , the main idea is to dynamically evolve a set of N particles by solving the Vlasov-Poisson system of equations and iteratively move particles. Solving equations of motion for large N is a computationally heavy task and require N 2 operations at each timestep for which several sophisticated techniques emerged allowing a more e cient processing [Efstathiou et al., 1985;Springel, 2005]. The right panel of Fig. 2.1 shows the overdensity eld computed in 2014 from a set of N = 1820 3 particles by the Illustris collaboration 2 showing the evolution made in this domain the past decades.

One of the main achievements of these N -body simulations is to allow the analysis of the density eld at both large and small scales, reproducing accurately its statistics. Figure 2.2 shows for instance the power spectrum computed from one box of the Quijote simulation [Villaescusa-Navarro et al., 2020] and the one predicted by linear theory. We can see the deviation between the two at k ∼ 0.15 h/Mpc showing the incapacity of using such theoretical modelling at small scales. In principle, elements can be added to the linear theory to allow the description of smaller scales. Yet they are still limited, reaching the percent accuracy at a mildly-non linear scales of k ∼ 0.3 h/Mpc [Carrasco et al., 2012] which is already an achievement but is not su cient to carry out precise cosmological analyses and to understand the physical processes occurring at smaller scales (galaxy evolution, baryonic physics, etc.). Simulations are thus indispensable for the accurate assessment and analysis of gravitational dynamics at small scales, but also for the development of statistical tools used for the study of future large-volume surveys like Euclid [Laureijs et al., 2011] or Vera Rubin Observatory [Collaboration et al., 2009]. As an example, simulations are particularly interesting to assess how a statistics derived from an observable varies with cosmological parameters, or to build accurate covariance matrices, problems encountered in Chapter 6. For all these reasons, dark matter only simulations strive pushing further the scale limit by using always larger volume and ner resolutions, as for instance Millenium [Angulo et al., 2012], MultiDark [Klypin et al., 2016] or Quijote [Villaescusa-Navarro et al., 2020]. Beyond statistical analysis of the matter distribution, N -body simulations are also particularly interesting to study collapsed objects like halos identi ed by means of post-processing to re ne predictions of their number counts or density pro les [Tinker et al., 2008;More et al., 2011].

However, the dark matter is not directly observable and, with the interest of reaching smaller and smaller scales, grew that of including baryonic matter in the simulations. Modelling the complex non-linear interactions between baryons, gas, stars, black holes and dark matter happening at all scales in cosmological volumes is one of the goals of hydrodynamical simulations. For that precise purpose, the inclusion of additional equations of motion is required which increases even more the complexity of the computation, hence requiring tradeo s between mass/volume resolutions and computational time of such simulations. Even with those di culties, many large-scale hydrodynamical simulations were developed to study the role of baryonic physics in the evolution of large-scale structures, like Horizon-AGN [Dubois et al., 2014], EAGLE [Schaye et al., 2015], Illustris [Vogelsberger et al., 2014] or Illustris-TNG [Nelson et al., 2019]. As such, hydrodynamical simulations enlighten our understanding of structure formation and evolution of individual objects like galaxies or stars that can then be tested against astrophysical observations to support or refute the proposed models [Pearce et al., 2001;Dubois et al., 2014;Schaye et al., 2015;Crain et al., 2015;Nelson et al., 2019;Donnari et al., 2019].

2.2 From darkness to light: cosmic web and galaxies

Galaxy surveys

The rst observations of the large-scale distribution of galaxies were made soon after the depiction of the lamentary pattern in simulations using the rst available mappings of galaxies such as the Palomar Observatory Sky Survey [Joeveer et al., 1978] or the CfA survey show in Fig. 1.1. Since then, astronomers carried out extensive surveys to map as precisely as possible the galaxies observed in the sky and trace the lamentary pattern of the cosmic web. For that purpose, the past twenty years witnessed the succession of numerous surveys like the Sloan Digital Sky Survey [SDSS, York et al., 2000], the 2 Degree Field Galaxy Redshift Survey [2dFGRS, Colless et al., 2001], the Galaxy and Mass Assembly survey [GAMA, Driver et al., 2009] or the Dark Energy Survey [DES, Abbott et al., 2016]. This ood of spectroscopic and photometric surveys aim at both covering the largest portion of the sky and have the deepest possible observations. This race to the most accurate mapping is being still pursued nowadays and for the forthcoming years with surveys like the Vera Rubin Observatory's Large Synoptic Survey Telescope [LSST, Collaboration et al., 2009], Euclid [Laureijs et al., 2011] or the Dark

Linear scales Non-linear scales

Linear theory N-body simulation Fig. 2.2. Illustration of the crucial role of simulations for statistical analyses at small scales. The red dashed line shows the power spectrum computed from linear theory while the black one is the average computed from 7, 000 realisations of the Quijote simulation. The grey shaded area shows the 1-σ interval around the mean. Both agree at the percent level below the grey vertical dashed line at k = 0.1455 h/Mpc.

Energy Spectroscopic Instrument [DESI, Levi et al., 2013].

Observational e ects

In addition to the general e ects that one can expect from observations, like uncertainty on the estimated quantities, outliers and Poisson noise emerging from the discrete sampling of galaxies, cosmology is subject to additional e ects related to the physics of the Universe and of the observed objects. In this section, we review the main observational e ects and discuss their practical impact on the observation of large-scale structures.

Redshift-space distortions

When dealing with real-world observations, the redshift is used as a measure of the distance. However, this quantity is the linear combination of two contributions: the rst one corresponding to the motion of the source (such as a galaxy) due to its peculiar velocity and the another one due to the expanding universe. Considering x the source position in the comoving space, r its position in the redshift space, v its peculiar velocity and n a unit vector in the line of sight (LoS) direction, it holds the mapping relation

r = x + v • n aH(z) n, (2.1)
with a the scale factor and H(z) the Hubble parameter. It is worth mentioning that this e ect, even though corrupting the position of observed objects along the LoS, also carries cosmological and astrophysical information about both the dynamics of galaxies and the expansion of the Universe. In particular, it is nowadays well established that redshift-space distortions (RSDs) provide key information on the growth rate of structures, denoted f , which scales with the matter density Ω m (z) γ and can be used to study dark energy as well as alternative gravity models [Hamilton, 1998;Linder, 2005].

The fact that observations are carried out in redshift space has a direct implication on the collected data. More speci cally, the spatial distribution of matter is distorted with overdense clustered regions appearing elongated in the LoS direction due to the high velocity of sources describing the overdensity. This e ect is called "Finger-of-god" [Jackson, 1972] and is illustrated schematically in the right part of Fig. 2.3 where the peculiar velocities of the two sources having a non-zero velocity component with respect to the LoS elongate the shape of the spherical overdensity in this direction. At larger scales is observed a squashing e ect of dense regions along the LoS [Kaiser, 1987] as shown in the left part of Fig 2 .3. All these e ects can be visually appreciated in the right panel of Fig. 2.4 where RSDs have been simulated by assuming the Z axis as the LoS and moving dark matter particles using Eq. (2.1).

One of the keystone of cosmology is the assumption of statistical homogeneity and isotropy of the Universe at large scales which makes the overdensity eld δ a homogeneous random eld invariant to rotations and translations. As depicted in Sect. 1.3, cosmological elds are statistically described by quantities derived from their correlations functions, or Fourierequivalent poly-spectra. By breaking the isotropy, RSDs are breaking the rotation invariance of δ used for the establishment of some estimated statistics such as the matter power spectrum. This is more precisely described in Sect. 6.3.2.

Alcock-Paczynski e ect

When working with observed objects, the translation of the estimated redshift z s of a source into a physical distance d s is expressed as

d s = zs 0 c H(z) dz, (2.2)
and requires the assumption of a cosmological model for H(z), the evolution of the Hubble expansion with redshift. Models of H(z) vary depending on the mass Ω m , curvature Ω k and dark energy Ω Λ densities in the Universe. The di erence between the assumed model and the truth induces some geometrical e ects in the observed matter distribution known as Alcock-Paczynski distortions [Alcock & Paczynski, 1979]. In particular, it modi es the ratio between the radial extent and the angular size of an observed object (like clusters) making them either elongated or squashed along the LoS depending on the mismodelling of cosmology. As for RSDs, the Alcock-Paczynski e ect turns out to be a way to assess cosmological models, and more precisely the expansion and geometry of the Universe by measuring the induced distortions when disentangled from the dynamical ones [López-Corredoira, 2014].

Real space

Redshift space

Line of sight

Large scale (~100 Mpc/ℎ) Small scale (~10 Mpc/ℎ)

Galaxy bias

As presented in Chapter 1, the dark matter is the dominating matter component in the Universe which consequently rules the formation and evolution of gravitational potential. It is also on the dark matter overdensity eld δ that most of the theoretical predictions are based in cosmology, such as the commonly used power spectrum of today's matter. The real-life observation of the large-scale structure is however performed through the luminous matter made, for instance, of galaxies. In theory, galaxies are the result of the non-linear evolution of dark matter creating potential wells in which baryons fall to form them and consequently trace preferentially overdense regions of the dark matter [Bardeen et al., 1986;Mo & White, 1996]. When one wants to link observations made from galaxies and the extracted statistics with theory, it is hence crucial to know how this statistics is actually linked with the one that would be derived directly from dark matter. This e ect, called "bias", appears whenever a tracer di erent from the dark matter itself is used. The current literature hence provides a rich amount of information on the handling of bias induced by galaxies but also by other tracers of the matter distribution like quasars, voids or galaxy clusters, to name only a few. Accounting for this e ect requires the understanding of how the spatial distribution of tracers is linked with the one of dark matter and can be studied either theoretically or numerically, with the above-mentioned observational e ects complicating even more the picture. The simplest bias form implies a linear relation [Kaiser, 1984] between the overdensity eld of tracers δ tracer and of dark matter

δ δ tracer (k, z) = b(z) δ(k, z), (2.3)
with b(z) the linear bias factor. In the case of the power spectrum, this translates into a constant b(z) 2 factor between the two spectra [Mo & White, 1996], only impacting its amplitude and inducing no shape dependencies. This simple linear bias model has shown great concordance for large and linear scales but more sophisticated bias models were investigated, accounting, for instance, for the tidal and environmental e ects [Yang et al., 2017;Paranjape et al., 2018b]. In practical applications to real surveys, this bias is one of the biggest contributions to errors and is simply due to the lack of understanding of the tracers which appear as a set of "nuisance parameters" in the analyses. For a complete review on this rich topic, we refer the reader to Desjacques et al. [2019].

Motivations for cosmic web classi cation 2.3.1 The limitations of statistical analyses

Statistical representations of random elds based on the rst orders of poly-spectra decomposition are limited in their representation of non-Gaussian patterns. In particular, most cosmological analyses of the matter distribution are based on the evaluation of the power spectrum which is completely insensitive to the texture of the cosmic web. Since P (k) is only taking into account the modulus of the random eld in the Fourier space, it omits the information contained in the phase. This is illustrated in Fig. 2.5 where the two elds have the exact same power spectrum, yet showing very di erent structural information, easily captured by eye.

The inclusion of a sensitivity to this pattern in the analyses requires the evaluation of higher order statistics becoming already computationally hardly tractable at the order three with the bispectrum and come with a theoretical expression of the uncertainty which involves the fourth moment. The measure of such rst high-order statistics also require many datapoints to be accurate. As a consequence, many works try to identify the best way to reduce the information contained in these non-Gaussian elds, which remains one of the biggest challenge of modern cosmology. The fair representation of such elds should ideally satisfy three constraints: i) include, in some ways, higher-than-two orders information to go beyond the Gaussian representations; ii) be as compressed as possible such that only few coe cients are needed to store most of the information; iii) be physically interpretable, at least giving, in an empirical way, an idea of the information stored. To extract partial information from higher-order statistics, solutions based on topological criteria were for instance proposed back in the eighties [Gott et al., 1986;Mecke et al., 1994]. Using a set of topological invariants relying on Euler characteristics, Betti numbers and Minkowski functionals to describe non-Gaussian elds like the latetime matter distribution or weak lensing convergence maps has indeed successfully shown to encode higher-order information [Kratochvil et al., 2012;[START_REF] Shirasaki | [END_REF]. For the task of representing and analysing astronomical data, wavelet decomposition has also been extensively used. The rst surveys were for instance quantitatively analysed by means of the wavelet transform [Martinez et al., 1993] which was also used to detect voids in the rst galaxy catalogues [Slezak et al., 1993]. Later were formulated wavelets speci cally sensitive to the di erent cosmic environments that are laments and walls with the 3D ridgelets and beamlets [Starck et al., 2005;Woiselle et al., 2010]. This interest of statistically representing the matter or galaxy distributions never vanished and, still nowadays, recent developments improve the eciency of the computation and analysis of the third order statistics at non-linear scales [Philcox & Eisenstein, 2020;Philcox, 2021;Hahn et al., 2020;Hahn & Villaescusa-Navarro, 2021] or exploit the additional information contained in velocities [Kuruvilla & Aghanim, 2021]. The rise of data science also led some older descriptions of non-Gaussian elds to be redesigned based on machine-learning derived tools. In that sense, the wavelet scattering transform [Mallat, 2012] mixes elements from convolutional neural networks [LeCun et al., 1999] and wavelet decomposition to compute a set of representative coe cients exploiting the invariance under rotation and translation of the eld at hand. The resulting summary is shown to be sensitive to higher-than-two points features in 2D cosmological elds in Cheng et al. [2020] and Allys et al. [2020]. Similarly, topological data analysis and persistent homology [see Edelsbrunner & Harer, 2008;Bubenik, 2015;Chazal & Michel, 2017] are extending the previous topological de nitions by studying the properties of the topological features over a range of di erent spatial scales (which can be seen as a continuous analogous to the discrete scale-space analysis). The analysis of such topological features is also a promising way to encode non-Gaussian cosmic information, as shown in Wilding et al. [2020]. Even though interesting for the building of summary statistics taking into account the textural information, most of these statistical representation do not directly allow the possibility of a spatial identi cation nor extraction of cosmic structures.

The cosmological sensitivity of environments

Since the resulting pattern of the cosmic web is mainly driven by gravitational dynamics, the extraction of quantitative information from the observed structures provides key insights on the underlying cosmological model and enlighten our understanding of dark matter and dark energy. The rst extensive and quantitative analyses of the multi-scale cosmic web in simulations suggests that each individual environment span a broad range of densities [see the right panel from Fig. 2.6, reproduced from Cautun et al., 2014] which in turn advocates for di erent cosmological histories. One could hence expect that individual environments inherit from di erent imprints and may show dissimilar behaviours with respect to cosmological models and parameters. As an example, voids are believed to be pristine environments, only little deformed by gravity and free from complex multi-streaming thus providing a perfect playground for the study of dark energy [Lee & Park, 2009;Lavaux & Wandelt, 2012;Hamaus et al., 2014Hamaus et al., , 2015;;Pisani et al., 2015] or for constraining neutrino mass [Massara et al., 2015]. In the opposite way, clusters are highly non-linear objects with high over-density enclosing a large fraction of the mass for a small part of the volume. The statistics of these peaks (number, distribution with redshift) in the density eld have been shown to be particularly sensitive to some cosmological parameters like the normalisation of the matter power spectrum or the matter density [Bahcall et al., 1997;Bahcall & Fan, 1998;[START_REF] Holder | [END_REF]. They are also unique laboratories to constrain the baryon gas fraction [White & Frenk, 1991;White et al., 1993] and to study the evolution of galaxies [Butcher & Oemler, A., 1984;Baldry et al., 2006]. This relationship between the di erent environments of the cosmic web and the cosmological parameters of the ΛCDM model is an aspect that we will develop in Chapter 6 using the two-point statistics of the di erent environments.

The role of the environment in shaping galaxies and clusters

At the astrophysics level, detecting cosmic structures may also help in proposing scenarios for the formation and evolution of galaxies. The rst hints of environmental e ects on galaxies were reported in [Oemler, 1974] showing that the densest regions of the Universe were more likely hosting elliptical than spiral galaxies. These observations were then re ned with the recrudescence and availability of web nder algorithms. In particular, the most prominent structure, also traced in observations, is the lamentary part of the pattern. These massive bridges act like highways in the cosmic web, allowing the transport of the matter. In this picture, galaxies escape low-density regions and travel along the network being carried by the ow of matter in laments towards the most massive parts of the web, the nodes [Aragon Calvo et al., 2019;Cadiou et al., 2020]. This journey leaves an imprint on galaxy properties and revealing the lamentary pattern of the cosmic web in data and simulations hence o ers the possibility to study the in uence of the environment on the formation and evolution of galaxies. This topic has received a considerable interest these past years showing many correlations between the physical properties of galaxies (e.g. their mass, shape, luminosity, orientation or ability to form stars) or halos and those of the underlying web or related tidal anisotropies [Kau mann et al., 2004;Hahn et al., 2007;Martinez et al., 2016;Kuutma et al., 2017;Malavasi et al., 2017;Laigle et al., 2018;Ganeshaiah Veena et al., 2018;Sarron et al., 2019]. For instance, it has been shown that galaxies closer to the spine of lament are more likely to be red and massive while it gets bluer and lighter when the radial distance is larger [e.g. Bonjean et al., 2020]. Some studies also draw a correlation between the orientation of galaxies and the direction of the lament they are hosted in [Ganeshaiah Veena et al., 2018, 2019;Kraljic et al., 2020].

The insightful analysis of the cosmic web in large-scale simulations carried out by Cautun et al. [2014] teaches us that the lamentary structure contains half of the dark matter mass of the Universe at the present time for only few percents of the total volume. By also hosting halos of various masses, typically from 10 10 M /h to roughly 10 13 M /h, laments are the ideal place to study collapsed objects like galaxies. Numerous works, based on simulations or observations, additionally show that a considerable fraction of baryons are hidden in the form of hot gas in laments [Cen & Ostriker, 2006;Martizzi et al., 2019;Tanimura et al., 2020a;Galárraga-Espinosa et al., 2021] hence emphasising the crucial role this particular environment is playing in baryonic processes shaping the formation and evolution of galaxies. These ndings highlight the importance of detecting the lamentary pattern both to improve the quality of the predictions in simulations and to discover new correlations with hosted tracers.

Galaxy clusters are massive biased tracers of the underlying matter observed both in simulations and surveys. It is now well-established that studying their properties like shapes, masses and redshift is a wealthy source of information on how they structure and evolve with time and on the underlying cosmological model [Yoshida et al., 2000;Peter et al., 2013;Sereno et al., 2018]. These properties have also been shown to be in uenced by the local environment of halos and clusters and how they are locally embedded in the cosmic web [Poudel et al., 2017;Darragh Ford et al., 2019;Gouin et al., 2020]. In particular, Musso et al. [2018] expect that lowmass halos are more likely lying inside laments while massive halos are found to be closer to nodes. cosmic web anisotropies are hence indicators of halo assembly bias and therefore strongly correlated with halo properties [Paranjape et al., 2018a;Ramakrishnan et al., 2019]. At a topological level, the number of lament a node, or massive cluster, is connected to, a quantity called the connectivity, is expected to depend on the growth factor hence allowing to put constraints on dark energy [Gay et al., 2012;Codis et al., 2018]. These relations between nodes and their local environments of the cosmic web will be investigated in more details in Chapter 5, Sect. 5.5.

Challenges in detecting cosmic laments

Several di culties make the detection of cosmic structures like laments a challenging task. In this section, we discuss the main obstacles and review the several de nitions adopted for cosmic laments in the past and current literature.

Structural complexity of the pattern

Cautun et al.

[2014] draw a multi-scale picture of laments probing several order of magnitudes in densities, from tenuous laments of δ ∼ 0.1 to dense bridges of matter with overdensities of few hundreds, as shown in the right panel of Fig. 2.6. This large range implies overlapping in densities between the several environments and suggests the use of more re ned methods than simple density thresholds. It is also shown that laments are spreading over a wide range of widths with densities above the background from few to tenth of megaparsecs around the spine [Cautun et al., 2014;Galárraga-Espinosa et al., 2020], as illustrated in the lamentary structure depicted by the NEXUS+ algorithm [Cautun et al., 2013] in the left panel of Fig. 2.6. Designing methods applicable for data surveys also makes the picture more complex. Indeed, the incompleteness of the galaxy samples at all masses shows a pattern traced by the most luminous objects only and prevents an accurate sampling (illustrated in Fig. 2.7). Additionally, the redshift-space distortions induced by the peculiar velocities of galaxies alter the pattern drawn by the tracers. In particular, we have seen in Sect. 2.2.2, and illustrated by Fig. 2.4, that it makes clustered regions appear elongated along the line-of-sight and may hence create "fake laments" [Malavasi et al., 2020a]. Finally, galaxy surveys are usually subject to holes in the observations with complex spatial selection functions creating absence of data in large portions of the sky. All these e ects are, in addition to the natural complexity of the cosmic web pattern, creating further di culties that the developed methods should take into account, either intrinsically or using appropriate pre-or post-processing.

Non-unicity of the de nition

Section 2.3 emphasise how essential is the detection of cosmic environments for both improving our cosmological and astrophysical understanding of the Universe. However, there is no consensus on the de nition of web elements and, these past decades have seen the emergence of many algorithms to identify structures with their own mathematical de nition. Since laments are of particular interest for astrophysics purposes and because these have been one of the main topics of this thesis, we will discuss in detail the several de nitions provided for this precise environment. A non-exhaustive list is reported in Table 2.1 together with the de nition adopted for laments in each case. For a more detailed view of classi cation schemes, we refer the interested reader to Libeskind et al. [2017] or to individual references mentioned in Table 2.1. Broadly speaking, there are six families of methods proposed for detecting cosmic web laments:

Graph-based. The rst representations of the lamentary pattern by Barrow et al. [1985] were carried out using a tool coming from graph theory called "the minimum spanning tree" that links galaxies together with the minimum total Euclidean distance to do so (discussed in more detailed in Sect. 4.2). In an attempt to extract quantitative information from it, Colberg [2007] and Alpaslan et al. [2014b,a] propose post-processing operations of the graph by cutting short edges or removing spurious ones to de ne laments as branches of the tree structure. Thanks to its ability to easily capture the pattern with no parameter and by relying directly on the position of galaxies, the minimum spanning tree is still used today and extended to obtain smooth laments, passing in the middle of the distribution of galaxies they host [Bonnaire et al., 2020[Bonnaire et al., , 2021b;;Pereyra et al., 2020b,a]. At the interface between graph-based methods and point-based geometric ones, Chapters 4 and 5 will present ways to incorporate ideas of smoothness and robustness while learning a graph structure into a single formulation under the ag of principal graphs.

Point-based geometric. Also addressing the detection from the observational point of view by directly relying on discrete input, Stoica et al. [2005aStoica et al. [ , 2007] ] introduce a stochastic formalism based on the point-process theory de ning laments as a set of random connected and aligned cylinders paving the galaxy distribution and ful lling some criteria based on sizes and local densities. The application of this latter, named the Bisous model in current surveys were later carried out by Tempel et al. [2014Tempel et al. [ , 2016]]. Alternatively, the FINE algorithm proposed in González & Padilla [2010], by considering physical principles based on the mass of halos or galaxies, is extracting laments between two node tracers as the shortest line following the local highest density. In the same spirit, Genovese et al. [2014] andChen et al. [2014] use a principal curve formulation to de ne laments as the set of projected points standing on the ridge passing in the middle of galaxies in 2D datasets in the Subspace Constraint Mean-Shift algorithm [SCMS, Ozertem & Erdogmus, 2011]. Duque et al. [2021] recently proposed a way to overcome the hand-tuning of the xed scale in the SCMS algorithm by using machine learning based combination of multiple scales.

Hessian-based. With the evolution of simulation accuracy and availability, a class of method based on a smooth estimate of the density eld traced by dark matter particles emerged using the Hessian matrix of either the density [Aragon-Calvo et al., 2007;Cautun et al., 2013] or tidal [Hahn et al., 2007;Forero-Romero et al., 2009;Kitaura, 2013] eld to classify cells according to their level of local anisotropy measured by the number of eigenvalues below a given threshold. The Multi-scale Morphology Filter [MMF, Aragon-Calvo et al., 2007] and NEXUS+ [Cautun et al., 2013] methods are carrying their analysis using the joint analysis of the curvature within a range of various Gaussian smoothing scales thus highlighting features of di erent sizes in the smooth eld. Note that these classi ers provide a segmentation at the cell level but do not allow the identi cation of single objects like a lament without post-processing of the output.

Topological. In parallel to hessian-based methods were proposed descriptions of the density eld invoking topological criteria [Sousbie et al., 2008;Pogosyan et al., 2009]. Topology is at the heart of cosmological analyses of the large-scale galaxy distribution since its rst observation [Gott et al., 1986;Mecke et al., 1994]. By describing the density eld through the Discrete Morse Theory, the Discrete Persistent Structures Extractor [DisPerSE, Sousbie, 2011] proposes a mathematically elegant formulation of the cosmic network using prescriptions from computational geometry and persistent homology as the set of constant gradient lines connecting saddles and maxima. Such topological arguments coupled with watershed transform based methods, originally used for the detection of voids [Platen et al., 2007], led Aragón-Calvo et al.

[2010] to de ne laments as the intersections between three or more watershed basins in the SpineWeb algorithm.

Phase-space. Speci cally dedicated to simulations, phase-space methods rely on the additional information of velocities to describe the structures using the full 6D information.

Conclusions and perspectives for the thesis

The comparison of this landscape of methods is not easy and heavily depends on the targeted application and underlying goals. Eventually, there is no best de nition for cosmic environments and each method come with its own caveats which point the user to a particular algorithm depending on the application. As a matter of fact, some algorithms, mainly those from the two rst categories have been developed targeting point-based real-data applications by relying on matter tracers like galaxies. Other methods, usually requiring continuous input, have been mainly designed to be applied on simulations with dark matter particles as input to obtain an accurate estimate of the density eld δ. These di erent inputs are illustrated in Fig. 2.7 showing in the bottom panel the galaxy distribution, visible counterpart of the dark matter density eld tracing with much more details the cosmic web in the top panel. Even in the absence of other observational e ects, the sparse sampling of the distribution complicates the task of extracting accurately the web-like pattern. Several applications of procedures based on continuous inputs on survey data have however been carried out with the DisPerSE algorithm [e.g. Malavasi et al., 2017;Laigle et al., 2018;Kraljic et al., 2020], and imply the computation of δ directly from the observed galaxy distribution. The accurate estimate of δ is hence an additional di culty that needs to be tackled and for which several algorithms have been proposed, like the Delaunay Tessellation Field Estimator [DTFE, Schaap & Weygaert, 2000] or B-spline interpolation schemes [START_REF] Hockney | [END_REF]Jing, 2005]. To alleviate the requirement of methods based on smooth eld inputs, one could use jointly sophisticated reconstruction algorithms like BORG [START_REF] Jasche | [END_REF]Leclercq et al., 2013] or Barcocde [Bos et al., 2014] providing a Bayesian estimate of the full underlying dark matter density eld from its visible counterpart only (galaxies or clusters respectively). However, in direct applications, e ects like masking or RSDs, are usually easier to handle in formalisms relying directly on discrete inputs like the MST [Alpaslan et al., 2014a]. They also generally allow the de nition of laments as continuous one-dimensional objects standing in the cloud of galaxies, which is not the case for other classi ers like NEXUS+ or T-web that need post-processing to obtain individual object detection. Despite all these e ects and the di erent de nitions for the cosmic web that do not always agree one with each other [Libeskind et al., 2017], it is remarkable to see the many results obtained both in simulations and observations. In addition to the achievements listed in Sect. 2.3.3 on the interplay between the physical properties of laments and galaxies, the detection of large-scale laments in other wavelengths have been made possible with the recent advances in observational astronomy. This usually involves stacking methods to increase the signal after their detection via the distribution of galaxies. The observation of the lamentary pattern is hence currently performed and used in di erent observables like X-ray [Dietrich et al., 2012;Eckert et al., 2015;Nicastro et al., 2018;Tanimura et al., 2020b], weak lensing [Gouin et al., 2017;Epps & Hudson, 2017;Tanimura et al., 2020c] or through the Sunyaev-Zel'dovich e ect [Bonjean et al., 2018;Tanimura et al., 2019;de Graaf et al., 2019;Tanimura et al., 2020c]. Yet, the observable that traces best the web-like pattern is the galaxy distribution from which laments are rst extracted to then study their properties in other wavelengths and this is why numerous works are providing catalogues of laments to the community [e.g. Tempel et al., 2014;Chen et al., 2016;Malavasi et al., 2020b].

One of the aim of this manuscript is to propose ways to analyse such complex spatial patterns such as the one depicted by the spatial matter distribution. In particular, in Chapter 4, we present a method for the learning of a principal graph that extends some algorithms of the graph-based category. We then discuss the interest of the method for cosmological purposes and apply it to related datasets in Chapter 5.

Part II Statistical methods for pattern extraction

Chapter 3. This chapter is based on the results from Bonnaire et al. [2021a]. As the rst chapter of the second part of the manuscript, it sets up the general context of machine learning and statistical methods of the thesis and discusses some of its caveats for applications in science, and more precisely in physics. It also introduces a generic formulation of the unsupervised optimisation problems encountered throughout the manuscript and the required knowledge about mixture models and the expectation-maximisation algorithm, two concepts exploited in the present and forthcoming chapters. Finally, the chapter exposes how a statistical physics formulation of an unsupervised machine learning task, the clustering, can be utilised to obtain an insight on the learning dynamics of the classes during the iterative procedure providing information about the structure of the data.

Statistical physics for clustering

Context and motivations

Machine learning and physics

The incredible advances of Machine Learning (ML) we witnessed the past decade made it an indispensable tool embedded in our daily life through our smartphones, the software we use and even our cars. Beyond these industrial successes and thanks to its ability to extract information from huge amount of data, ML also sparked the interest of researchers in a large variety of domains ranging from physics to biology. Perhaps one of the most common applications to various elds is the image processing in which ML algorithms, and more precisely deep learning ones, have shown an unprecedented power in the learning of relevant features from large labelled databases [LeCun et al., 2015]. Astrophysics and cosmology were no exception and even showed themselves to be particularly interesting for the application of ML algorithms. The recent availability of data in cosmology, such as larger and larger simulations with various cosmologies (see Sect. 2.1) for instance paved the way for the estimation of cosmological parameters [Ravanbakhsh et al., 2017;Ribli et al., 2019;Cheng et al., 2020;Allys et al., 2020] or for the mapping of the dark matter and its visible counterpart [Zhang et al., 2019;Hong et al., 2021] using deep-learning-based architectures. The use of such methods also led to promising results in providing fast and accurate alternatives to the computationally costly N -body simulations [Rodríguez et al., 2018;He et al., 2019;Ullmo et al., 2021]. For observational tasks, ML-based algorithms also allowed the automatic classi cation of galaxies morphology based on optical data [see Barchi et al., 2020, for a review], the improvement of redshift estimation from photometric surveys [Carrasco Kind & Brunner, 2013] or the prediction of unknown galaxy properties like their masses, star formation rates or full spectra based on their photometry [Bonjean et al., 2019;Mucesh et al., 2021]. All these successes rely on di erent learning schemes that fall into three categories:

Supervised Learning in which is learnt a function mapping an input to an output based on paired examples. This goal can be achieved using various methods such as the random forest [Ho, 1995] or derived from neural-network architectures [Rosenblatt, 1959;Rumelhart et al., 1986], to cite only the most popular ones. Unsupervised Learning methods that aim at identifying hidden structures or patterns in a given dataset without requiring labels. Usually based on the learning of the probability distribution (de ned explicitly or not) that most probably generated a given dataset, famous algorithms of this category are generative models like Generative-Adversarial Networks [Goodfellow et al., 2014] and derived models [Mirza & Osindero, 2014;Radford et al., 2016;Arjovsky et al., 2017], Restricted Boltzmann Machines [Smolensky, 1986] or Variational Auto-Encoder [Kingma & Welling, 2013]. All the studied mathematical problems we will encounter in this part of the manuscript fall in this family, may it be the clustering or the principal graph learning presented in Chapter 4.

Reinforcement Learning [see Sutton & Barto, 2018, for an extensive introduction] is the task of learning multiple functions, called agents, to predict an output based on environmental information (for instance from sensors) in order to maximise a reward function. A famous example of algorithm in this category is the AlphaGo program [Silver et al., 2016] that learnt how to play Go against itself. Despite their increasing popularity in physics applications and the performance they can achieve, ML approaches, in their fundamental idea of automatically "learning" a model from the sole information of the data remain very opaque. Particularly true in the case of deep learning, the learnt models and the features that were used to build them are generally not exploitable to understand the underlying phenomena, which is precisely what physics aims at. This opacity has led researchers to address questions like "Can we ourselves interpret the models learnt by the machine?" or "What are the important features in the data?". To an-swer these questions but also more fundamental ones on the unusual generalisation abilities of deep learning models [Zhang et al., 2016;Hastie et al., 2019;Advani et al., 2020;D'Ascoli et al., 2020], the joint e ort of various communities from computer science, mathematics and physics is being put together. Such answers are essential to provide prescriptions for the developed algorithms, their range of optimality but also increase their interpretability. For that purpose, leads can be found in the application of theoretical physics to ML algorithm. Back in the eighties already, the learning dynamics of the rst neural networks [Little, 1974;Little & Shaw, 1978;Hop eld, 1982] were analysed by means of spin-glass models from statistical physics [Amit et al., 1985]. In a shameful attempt to give, in a few lines, an intuition on the connections between the two elds (and leave to Nishimori [2001] the task of giving the reader a more rigorous exposition), let us consider the case of the famous Convolutional Neural Networks [CNN, LeCun et al., 1999]. Taken individually, each component of the procedure is "simple", with linear operations (here, convolutions) interwined with somehow simplistic non-linear activation functions (like sigmoids). Parameters of the full resulting model ( lter weights and biases) are estimated iteratively using gradient-descent-based algorithms minimising a quadratic cost functions. This composition of basic mathematical operations allows the modelling of complex functions and led to the best known results in numerous applications, as exposed above. On the other hand, statistical physics, in its most fundamental de nition, is aiming at describing the microscopic interactions between simple elementary components of a system and understanding how complex behaviours can macroscopically emerge from it. In physical systems, statistical physics usually deals with a large number of particles and hence omits the individual characteristics to rather focus on the average properties of the entire system. In this analogy, the overall model learnt by the CNN is the macroscopic system while the many parameters are the particles composing it. Global characteristics of the model like its ability to generalise are analogous to macroscopic properties of the system. By setting themselves in idealised setups where the learning is "controlled", statistical physicists shown numerous analogies with ML -or related -algorithms allowing new theoretical insights and prescriptions [see Zdeborova & Krzakala, 2016, for a review]. In particular, many optimisation and inference problems have shown an equivalent formulation in statistical physics [Mezard & Montanari, 2009] that allowed a brand-new look at some long-standing problems and improved the understanding of complex algorithms [Mézard & Mora, 2009]. As an example, the identi cation of the phase diagram and phase transitions of a model can bring interesting insights such as knowing under what conditions a given algorithm is optimal or if a given information can be retrieved from the model and dataset at hand [such as Decelle et al., 2011;Lesieur et al., 2016;Tubiana & Monasson, 2017;Decelle & Furtlehner, 2021]. Such insights on the developed algorithms are precious, both in supervised and unsupervised contexts, since they provide key information on what matters during the learning process at di erent levels, from the topology of the cost function to be optimised [Choromanska et al., 2015;Spigler et al., 2019;D'Ascoli et al., 2020] to the impact of the data structure itself [Goldt et al., 2019].

In what follows, we give a general formulation of the optimisation problems that we will be focusing on in this part of the manuscript and introduce the concept of regularisation right before drawing the purpose and drawbacks of clustering algorithms this chapter is analysing.

Optimisation problems and regularisation

Inference (or learning) problems often come as the minimisation of an energy function (also called cost or loss function depending on the community) [see for instance Vapnik, 1998]. To keep the formulation general, let us consider two sets of points, a dataset {x i } N i=1 with

x i ∈ X and {y k } K k=1 with y k ∈ Y such that {f (y k )} builds a representation of the data, with f : Y → X a mapping function between the two spaces. Calling F the set of all such mappings, many mathematical problems in machine learning and inference contexts come formulated as

argmin f ∈F N i=1 min y E(x i , f (y)), (3.1)
where E(x i , f (y)) is the data delity term quantifying the energy cost of representing x i by f (y). As an example, one can express the empirical mean as the one-point representation of the dataset resulting from Eq. ( 3.1) under a quadratic energy cost with K = 1 and F the set of constant functions, leading to nd y minimising N i=1 x i -y2 2 . In the special context of statistics, one can see equivalently the maximisation of the log-likelihood function as the minimisation of an energy cost between the observed data and a model. The widely known and studied problematic of supervised machine learning can also be expressed in a similar manner. Supervised machine learning aims at nding the mapping f drawn from a set of tuples (y i , x i ) encoding a signal and its label. A popular form of the cost function in these contexts is a quadratic energy cost

E(x i , f (y i )) = x i -f (y i ) 2 2 .
Note that this is also the cost emerging when assuming Gaussian additive noise in the error of the representation of x i by f (y i ), i.e. a Gaussian likelihood in a probabilistic setting.

When the mathematical problem is inherently ill-posed and solutions exist only for a subset of F, one can constrain the form of the solution, may it be to penalise its complexity or because of prior knowledge, by adding a regularisation term leading to

argmin f ∈F N i=1 min y E(x i , f (y)) + λR(f ) , (3.2)
where R is a functional penalising the solution and λ a parameter acting like a trade-o between the two terms. The introduction of such a penalty in the optimisation scheme is very popular in the machine learning community to avoid the over-tting of the learnt models [Kukačka et al., 2017], but also in signal processing to relax some reconstruction problems [Jalalzai, 2012]. One of the common choices for R is an L 2 norm1 hence penalising unsmooth functions leading to Tikhonov-like estimators [Tikhonov & Arsenin, 1977]. Under these assumptions of Gaussian distributed errors and quadratic regularisation, equivalent to a bounded L 2 norm constrained optimisation, the typical problem one has to solve is thus

argmin f ∈F 1 2 N i=1 min y x i -f (y) 2 2 + λ 2 f 2 , (3.3)
which will be the generic form of most formulations we shall encounter in this part of the manuscript, like the clustering studied in this chapter or the principal graph in Chapter 4. Note that this choice of quadratic regularisation is not innocuous. From a Bayesian point of view, this is the form arising when imposing a Gaussian prior 2 distribution. Equation (3.3) thus corresponds to the maximisation of a log-posterior with a Gaussian log-likelihood and a Gaussian prior distribution. Thereby, λ can be seen as the ratio of variances between the Fig. 3.1. Illustration of the purpose of clustering. Given a set of collected points in an arbitrary space (here the two-dimensional dataset in the left panel), one wants to identify K = 3 clusters (shown as coloured points in the right panel). Note that the choice of K is application-and user-dependent and that we could have chosen to identify only two clusters as well.

two Gaussian laws. Note also that, when maximising Eq. ( 3.3) with λ = 0, the variance of the likelihood does not play any role and is coupled with the variance of the prior distribution when λ > 0.

Clustering and its drawbacks

Clustering is a one of the most ancient unsupervised tasks of ML aiming at identifying a partition of a given dataset into multiple groups called "clusters". Figure 3.1 illustrates this objective in a 2D case where the aim is to go from the left panel to the right one, by attributing a class to each input datapoint. The common ground to many methods that have been proposed these past decades [MacKay, 2002] is that they all embed, in some ways, a measure of the "similarity" between datapoints living inside the same cluster. The simplest way of thinking this is a measure based on the Euclidean distance leading to representations in which "close" datapoints more likely fall into the same cluster while distant ones should reside in di erent clusters. The prototypical method to partition a D-dimensional dataset into K clusters is the K-means algorithm [MacQueen, 1967]. Taking back the general formulation of optimisation problems of Sect 3.1.2, we can write the K-means goal as nding the set of points minimising the sum of squared distances between datapoints and cluster centres. We end up in a setting seeking to nd the set of points µ = {µ 1 , . . . ,

µ k } T minimising N i=1 min k∈{1,...,K} x i -µ k 2 2 .
(3.4)

The K-means algorithm solves this optimisation problem in a simple way. Starting from a set of initial positions µ (0) , it assigns to each datapoint x i the closest (in the sense of the Euclidean distance) centroid among µ, noted µ(x i ), and moves µ k accordingly to the average of all datapoints projecting on it, namely µ

(t+1) k = E(x i | µ(x i ) = µ (t)
k ). As we can see, this formulation assigns a single centroid to a datapoint without any level of uncertainty, making it a "hard" version of clustering. In case of overlapping clusters, such as the two on the left of Fig. 3.1, it is however natural to think that datapoints living at the border could be either in the blue or the orange cluster. Also, because the association energy appearing in Eq. (3.4) is based on the Euclidean distance, it tends to generate spherical groups. These restrictions of the K-means algorithm later led to the Gaussian mixture model formulation of the clustering that we will present in this chapter. This latter introduces a fuzziness by assigning a probability to each datapoint of being generated by one the K clusters but also extends the de nition to anisotropic clusters of various densities.

Thanks to their ability to describe clustered patterns by grouping datapoints without prior information, clustering algorithms naturally found many applications in science [see Saxena et al., 2017, for a review]. In the particular context of astrophysics and cosmology, they found applications in the identi cation of astronomical sources (stars, quasars, galaxies, etc.) together with their morphological properties to complete and improve survey catalogues [see e.g. Aghanim et al., 2015;Barchi et al., 2016;Logan & Fotopoulou, 2020]. Despite their attractive data-driven and unsupervised foundations3 , clustering algorithms are not free of aws and, in addition to speci c drawbacks proper to each method, most of them come with the requirement of inputting a number of cluster K to identify in the dataset. They also usually act as opaque boxes and the clustering procedure only outputs a partition of the dataset without any information on what has been "learnt" from their separation. For a successful application of most clustering procedures, one hence needs to have strong priors on the wanted clusters, such as their number, and a broad idea of the family of datapoints that need to be grouped together, even if not labelled, to assess the result of the clustering.

The aim of the rest of the chapter is to expose precepts originating from statistical physics to obtain a physical insight on what happens during the clustering procedure when carried out using the Gaussian mixture model and the Expectation-Maximisation procedure to optimise the log-likelihood. After having introduced both concepts, we will show that, in the context of deterministic simulated annealing, we are witnessing a cascade of phase transitions and that we can study the linear stability of the iterative scheme of the Expectation-Maximisation algorithm to derive theoretical thresholds at which these transitions occur. More importantly, we show that, by tracking empirically some physically relevant quantities related to the size of the clusters being learnt by the model, we end up with crucial information on the number of structures at a given scale, their hierarchy (i.e., how they are embedded in space with respect to each other), and their size. This is done without relying on prior knowledge, such as the number of components usually required by clustering methods. All the physical transitions are visible in a two-dimensional diagram which allows the extraction of the structural information, even in high dimensions when visualisation is not possible, hence opening the path for the exploration of complex high-dimensional datasets.

Mixture models

General formalism

Real-life data often come as drawn from probability distributions with complex shapes and multiple modes that cannot be satisfactorily described by a single well-known probability distribution. Mixture models [see McLachlan & Krishnan, 1997;Bishop, 2006] propose to model this complexity by using a linear combination of K known laws. need on a multi-modal 2D distribution of points. Although easy to study, a single Gaussian component (grey ellipse) do not explain the non-Gaussian dataset while a linear combination of three Gaussian (coloured ellipses) leads to a better representation. This idea of combination of known laws, despite its conceptual simplicity, can lead to accurate representations of highly complex density distributions. This is in particular why mixture distributions are nowadays at the basis of many mathematical tools like kernel density estimation [Parzen, 1962;Li & Barron, 2000], clustering [Jain et al., 2000] or mixture density networks [Bishop, 1994] in machine learning.

Keeping the formulation as general as possible, a mixture distribution is the linear combination of K probability distributions {f k } with parameters θ k . The probability of a datapoint x being generated by the model is hence

p(x | Θ) = K k=1 π k f k (x, θ k ), (3.5)
with Θ = {π 1 , . . . , π K , θ 1 , . . . , θ K } the set of model parameters, π k the mixing coe cient (also called amplitude) of component k. Note that, given the properties of probability distributions, amplitudes are normalised and positive by de nition, namely K k=1 π k = 1 and ∀k ∈ {1, . . . , K}, π k ≥ 0.

The Gaussian case

A particular class of mixture model is the Gaussian case, where ∀k ∈ {1, . . . , K},

f k (x, θ k ) = N (x, θ k ), with N (x, θ k ) = exp -1 2 (x -µ k ) T Σ -1 k (x -µ k ) (2π) D/2 |Σ k | 1/2 , ( 3.6)
the Gaussian probability distribution with parameter θ k = {µ k , Σ k }, respectively corresponding to the average of the k th Gaussian component and its covariance matrix. The particular Gaussian formulation of mixture models has been extensively used in machine learning, mainly for density estimation and clustering purpose.

Expectation-Maximisation algorithm 3.3.1 Introduction through Mixture Models

In a parametric setup, one can estimate the optimal set of parameters Θ of a model by maximising the log-likelihood function log p(X | Θ). Assuming a mixture model as de ned in Eq. ( 3.5), we get

log p(X | Θ) = N i=1 log K k=1 π k f k (x i , θ k ) . (3.7)
This function cannot be analytically maximised because of the sum inside the logarithm function. To circumvent this issue, the Expectation-Maximisation algorithm [EM, Dempster et al., 1977] allows the optimisation of the log-likelihood of models with latent variables. To reformulate the mixture model in terms of latent variables, we rst introduce a set of random variables Z = {z i } N i=1 with z i ∈ {1, . . . , K}. Z represents the partition of the dataset and z i denotes by which of the K component the datapoint x i has been generated from. By doing so, the log-likelihood from Eq. (3.7) can be written as the marginal distribution

log p(X | Θ) = log Z p(X, Z | Θ), (3.8)
where the joint probability distribution log p(X, Z | Θ) is often referred to as the "completed log-likelihood" of the model in the sense that the joint knowledge of datapoints and latent variables {X, Z} is the "completed dataset". It can be expressed as

log p(X, Z | Θ) = N i=1 log (π z i f (x i , θ z i )) . (3.9)
The key idea of EM is that the log-likelihood of {X, Z} is easier to maximise than the one of Eq. (3.7): if we knew the vector Z, we could easily maximise Eq. (3.9) and nd parameters Θ of the model. Unfortunately, Z are unknown hidden variables that we need to estimate, in addition to Θ. For this precise purpose, EM provides a procedure with two alternating maximisation steps to iteratively estimate both quantities. The presented formalism is more general than the context of mixture models and Eq. (3.8) remains true for any model with latent variables. In this section, if not clearly stated, we consider a general log-likelihood, without referring to the particular one of Eq. (3.7).

Iterative scheme

From the observation that the completed log-likelihood log p(X, Z | Θ) is easier to maximise, the goal is thus to obtain an estimate of the probability distribution over the latent variables.

Jensen's inequality4 , from the concavity of the logarithm function, together with Eq. (3.8) allow us to write, for any normalised distribution q(Z) over the latent variables,

log p(X | Θ) ≥ L(q, Θ) := Z q(Z) log p(X, Z | Θ) q(Z) . (3.10)
Hence, the log-likelihood of the model is bounded below by L(q, Θ). If easier to handle, maximising this quantity could help maximising log p(X | Θ). We can re-write this lower-bound, using the normalisation of q(Z) and the decomposition p(X,

Z | Θ) = p(Z | X, Θ)p(X | Θ), as L(q, Θ) = -D KL (q, p(Z | X, Θ)) + log p(X | Θ), (3.11)
where D KL (q, p) := q log q/p ≥ 0 is the Kullback-Leibler divergence [Kullback & Leibler, 1951]. To maximise the lower-bound L(q, Θ), the Kullback-Leibler term, at the origin of the inequality in Eq. ( 3.10), has to cancel out, hence leading to estimate the probability distribution of the latent variables. To this end, the current values of the parameters, Θ (t) , are used to compute q(Z) = p(Z | X, Θ (t) ) that is then exploited to obtain the lower-bound L(q, Θ (t) ). This step is called the "E-step" as a reference to "Expectation" because the lower-bound can be written, using Eq. (3.10), as

L(q, Θ) = E Z | X,Θ (t) {log p(X, Z | Θ)} - Z q(Z) log q(Z), := Q(Θ, Θ (t) ) + H(Θ (t) ), (3.12)
where Q is the expectation of the complete log-likelihood over the latent variables and H is the negative log-entropy of the latent variables distribution q(Z).

In a second step, the values of the parameters are updated by maximising the lower-bound over Θ,

Θ (t+1) = argmax Θ L(q, Θ), = argmax Θ Q(Θ, Θ (t) ). (3.13)
This step is the M-step, standing for "Maximisation". The full EM algorithm, by alternating between E and M steps, iteratively maximises the log-likelihood of the model through the double maximisation of a lower-bound, rst over the distribution of latent variables and then over the parameters of the model, Θ.

One of the key advantages of EM, beyond its tractability, is its guaranteed convergence towards a local maximum of the log-likelihood through successive monotonic increases [Wu, 1983;McLachlan & Krishnan, 1997]. To give an intuition of that, it is straightforward to write L(q, Θ (t) ) = log p(X | Θ (t) ) from Eq. (3.11). Then, Θ (t+1) is chosen such that maximising L(q, Θ) which yields L(q, Θ (t+1) ) ≥ L(q, Θ (t) ). Since L(q, Θ) is a lower-bound of the loglikelihood, increasing it guarantees the increase of the log-likelihood, which in fact increases by an amount of D KL (q, p(Z | X, Θ)), as suggested by Eq. (3.11). To better visualise this property, Fig. Despite these mathematically appealing characteristics, some of the main drawbacks of the EM algorithm are found in its convergence properties, mainly because of its slow convergence rate [Wu, 1983] and its dependency on the initialisation [Kloppenburg & Tavan, 1997;Ueda & Nakano, 1998]. Indeed, the direct application of the procedure is known to be easily trapped in local maxima of multi-modal likelihoods leading to variability in the provided results depending on the initialisation of the algorithm. This can be intuited from Fig. 3.3 where starting from di erent points (grey or red) will irremediably lead to one of the two modes of the loglikelihood. When used in practice, it is hence natural to start from several random locations and keep the realisation with the highest value of the nal log-likelihood (as implemented in Pedregosa et al. [2011] for instance).

The particular case of Gaussian mixtures

When considering a mixture model (see Sect. 3.2), the log-likelihood and the completed loglikelihood are respectively given by Eq. (3.7) and Eq. (3.9). In this scenario, we can derive, in the E-step, the probability of the latent variables given the observed data and the current parameters of the model, also called responsibilities, .14) This estimation is then used in the M-step to evaluate the rst term of the lower-bound through Eq. (3.12) (the other one being independant of Θ) as

p ik := p(z i = k | x i , Θ (t) ) using Bayes' theorem, p ik = π k f (x i , θ (t) k ) K j=1 π j f (x i , θ (t) j ) . ( 3
Q(Θ, Θ (t) ) = N i=1 K k=1 p ik log [π k f (x i , θ k )] , (3.15)
and update the parameters through Eq. ( 3.13) by maximising over Θ. Note that, in this equation, parameter values referring to time t are hidden in the responsibilities p ik of Eq. (3.14).

In the case of a Gaussian mixture model (GMM, see Sect. 3.2.2), we have

f k (x i , θ k ) = N (x i , θ k ), with θ k = {µ k , Σ k }. The M-step consists then in solving argmax Θ N i=1 K k=1 p ik log π k - 1 2 log Σ k - 1 2 (x i -µ k ) T Σ k (x i -µ k ) , (3.16)
where we can recognise a relaxed version of the general form of Eq. ( 3.2) with λ = 0 (no prior). Indeed, compared to the rigid K-means formulation of Eq. (3.4) in which a datapoint is associated to a unique cluster, the GMM method not only includes a parameter for the shape of the cluster through Σ k but also quanti es the probability of a datapoint x i to be represented by a given cluster k through the responsibility p ik . From Eq. ( 3.16), it is possible to derive an update equation for each parameter of Θ (t+1) as

π (t+1) k = 1 N N i=1 p ik , µ (t+1) k = N i=1 x i p ik N i=1 p ik , Σ (t+1) k
= N i=1 p ik (x i -µ k )(x i -µ k ) T N i=1 p ik .
(3.17)

Note that, if we consider spherical Gaussian clusters only, meaning that ∀k ∈ {1, . . . , K}, Σ k = σ 2 k I D , the sum of weighted covariances to update Σ k simply reduces to the one of

L 2 norms x i -µ k 2 2 .

Phase transitions in Gaussian mixtures

Statistical physics formulation of clustering

In the context of clustering, it has been demonstrated that the GMM can be formulated as a statistical mechanics problem [Rose et al., 1990;Akaho & Kappen, 2000] where the negative log-likelihood can be interpreted as a free energy. We review here the main steps establishing this analogy that we will exploit in the next sections. Assuming that the clustering aims at identifying a partition Z of the data into K clusters with centres {µ k }, the average energy cost of a given con guration can be written

E(Z) = N i=1 K k=1 p(z i = k | x i , µ k )E k (x i , µ k ), (3.18)
where E k (x i , θ k ) is the energy cost of the association of x i to the cluster k. Under a quadratic energy cost

E k (x i , µ k ) = x i -µ k 2 2
, the direct minimisation of the total energy E(Z) leads to the K-means clustering of Eq. ( 3.4) and associates hard probabilities for the associations, i.e. 0 or 1 by simply attributing to datapoints the closest cluster in the sense of the distance measurement provided by E k (x i , µ k ). Instead, we can introduce a level of uncertainty through a temperature in the associations by maximising the free energy F = H -βE(Z) where H is the entropy of the system [Shannon, 1948] and β acts like the inverse temperature of the analogue physical system. For low temperatures β → ∞, we retrieve the hard solution while nite values lead to a fuzzy association of datapoints to clusters. The distribution p(Z | X, µ) maximising the entropy under the constraint of Eq. ( 3.18) is, by invoking the principle of maximum entropy [Jaynes & Rosenkrantz, 1983], the Boltzmann distribution, hence reading

p(z i = k | x i , µ k ) = exp{-βE k (x i , θ k )} ξ(x i ) , (3.19)
where ξ(x i ) = K k=1 exp{-βE k (x i , µ k )} is the individual partition function. For non-interacting systems, the partition function is hence Ξ = N i=1 ξ(x i ) (usually denoted Z but that we modi ed to avoid confusion with the partition of the dataset). The free energy can hence be equivalently written

F = - 1 β log Ξ, = - 1 β N i=1 log K k=1 exp{-βE k (x i , µ k )}. (3.20)
Assuming a quadratic energy cost for E k (x i , µ k ), we recognise the negative log-likelihood from Eq. (3.7) under the spherical and equal variances and uniform amplitudes assumptions, meaning that ∀k ∈ {1, . . . , K}, Σ k = σ 2 I D and π k = 1/K. As such, this formulation of the clustering is equivalent to the one of a mixture model with a set of parameters reduced to Θ = {µ 1 , . . . , µ K }. In the model, the clusters variance is thus playing the role of the temperature of the system 1/β = 2σ 2 . EM update equations, thanks to this analogy, can be used to minimise the free energy F . The Boltzmann distribution of Eq. (3.19) is exactly the responsibility obtained in the E-step

p ik = p(z i | x i , θ k ) = exp{-x i -µ k 2 2 /2σ 2 } K j=1 exp{-x i -µ j 2 2 /2σ 2 } . (3.21)
The M-step then re nes the position of cluster's centres based on the current values of p ik to minimise F as

µ (t+1) k = N i=1 p ik x i N k (3.22)
where N k = N i=1 p ik stands for the number of datapoints associated with the cluster k and with parameters not indexed by iteration are relative to time (t). By applying iteratively Eq. (3.21) and Eq. ( 3.22), one gets the set of positions maximising the log-likelihood, or negative free energy, of the model. In the context of clustering, it means that one hence gets a set of K positions {µ k } and variances {σ k } corresponding to the center and extensions of the K clusters. If a new datapoint is given and one wants to predict its belonging, Eq. ( 3.21) provides exactly the probability of it having been generated by the several detected clusters under the assumptions of the model.

From paramagnetic to condensation phase

At very high values of σ 2 (high temperatures), the responsibilities (3.21) are uniformly distributed with ∀i ∈ {1, . . . , N }, p ik = 1/K. A given datapoint x i has hence an equal probability to be attributed to any of the K clusters. Injecting this result in the update equation of cluster positions (3.22) yields µ k = N i=1 x i /N meaning that they are all collapsed at the center of mass of the dataset. This phase is called the paramagnetic phase as an analogy to the Ising model phases, and is, for instance, illustrated in the top left panel of Fig. 3.4.

We are now interested in deriving the transition value of the temperature σ 2 c at which the centres split into several subgroups. Without any loss of generality, the dataset is considered centred, with N i=1 x i = 0 D where 0 D is the D-dimensional zero vector. Linearly Taylor expanding the expression of p ik given by Eq. ( 3.21) for small perturbations µ k ≈ 0 D gives

p ik = 1 K 1 + 1 σ 2 k x T i µ k + o(µ k ). (3.23)
Re-writing the update equation with this expansion leads to study the stability of a dynamical linear system whose evolution is governed by the matrix [Rose et al., 1990]

M = 1 σ 2 c 1 N N i=1 x i x T i , (3.24)
where we recognise C = 1/N N i=1 x i x T i , the data covariance matrix. Consequently, the system leaves stability and enters the condensation phase when the spectral radius of M is strictly greater than 1, meaning that the stability is governed by the maximum eigenvalue of the M . In that phase the position of the means starts to be correlated with the position of the clusters. More speci cally, {µ k } K k=1 move away from the centre of mass when σ 2 c < Γ, where Γ is the largest eigenvalue of C.

Since we are focusing on datasets supposed to be clustered, with multiple groups spanning a D-dimensional space, we can expect to see successive transitions when a Gaussian component of the model is describing the local size of a subgroup in the data. This idea is the one explored in the forthcoming sections in the context of simulated annealing where the temperature is decreased progressively.

Hard annealing

Cascade of phase transitions

The combination of the previously exposed drawback of the EM algorithm, namely the trapping in local maxima of multi-modal likelihoods and the one of clustering with the choice of the number of components to model the data, makes parametric mixture models very sensitive to the initialisation and the choice of hyper-parameters [Ueda & Nakano, 1998]. In that regard, the statistical physics formulation of clustering established in Sect. 3.4.1 helped to overcome these issues by making use of deterministic simulated annealing. In particular, it allows the relaxation of the non-convex optimisation problem by solving it iteratively while the temperature, or equivalently in our our case, the variance of all components σ 2 , is controlled and slowly reduced [Kirkpatrick et al., 1983]. The idea behind these approaches is to smooth out the likelihood by starting with a very high variance leading to a concave function. Decreasing it slowly leads to a ner and ner description of the dataset hence resulting in a more complex likelihood function with multiple modes appearing. In cosmological analyses, simulated annealing is for instance used successfully for the optimisation of the energy function in the Bisous model used to extract cosmic laments from galaxies in Stoica et al. [2004Stoica et al. [ , 2005a,b] ,b] and Tempel et al. [2016].

In Sect. 3.4.2, we focused on the range of temperatures σ 2 for which the likelihood is concave and hence for which all of the K components are collapsed into a single location centred at 1/N N i=1 x i . We shown that the critical quantity above which this behaviour is observed, that we note T hard c , is the maximum eigenvalue Γ of the data covariance matrix C [see also Rose et al., 1990]. Figure 3.4 displays the centres position for an arti cial dataset with ve Gaussian clusters at di erent temperatures. When σ 2 > T hard c (top left panel), even though K = 25 components are used in the model, they are all collapsed as K r = 1 physical cluster at the center of mass of the dataset (here chosen to be 0). When σ 2 becomes slightly smaller than T hard c (top right panel), the likelihood is deformed [see Ueda & Nakano, 1998, for illustrations and discussions about the likelihood point of view] and centres get aligned with the rst principal direction given by the data covariance matrix C. When σ 2 continues to decrease, the dataset description becomes more and more detailed and K r takes increasing values (bottom panels of left part of Fig. 3.4).

These transitions are physically relevant of the underlying structure of the data by occurring at variances related to the local size of the sub-system being represented. We hence argue that it is possible to extract information on the structure of a dataset from the clustering during the annealing procedure. This can be achieved by tracking the evolution of the physical size represented by a given component k through the maximum eigenvalue Γ k of its weighted covariance matrix, namely

Σ k = 1/N k N i=1 p ik (x i -µ k )(x i -µ k )
, that is the quantity governing the successive transitions.

The right panel of Fig. 3.5 illustrates the evolution of the ratio Γ k /σ 2 during the annealing for the same arti cial dataset as Fig. 3.4 coloured by their end-point cluster. In the left panel of Fig. 3.5, we see the cascade of transitions and successive splitting of centres when σ 2 decreases. When two or more centres collapse, they share similar values of µ k and p ik leading to similar evaluations of Γ k . This is why all lines are superimposed for σ 2 > T hard c in the right panel. Each time a curve reaches the horizontal unit line, one of the K r sub-systems made of collapsed 3.25) that we identify as an order parameter. This quantity is not represented for σ ≤ 1 since the number of physical clusters K r begins to be higher than the number of generated clusters q.

components reached the temperature of the sub-dataset it represents, namely σ 2 Γ k . From there, we observe either a bounce or a cross of the line. When bouncing, there is a split between two populations of centres that were representing the same part of the dataset but that will, from now on, take di erent paths. Centres thus move towards a smaller cluster and the value of Γ k decreases. Crossing the line occurs when centres split inside an individual cluster due to its inner random structure. In that case, the imposed variance gets smaller than the physical one. Since the transitions are ruled by the maximum eigenvalues of the empirical covariance matrices, it is this quantity that is used to measure the size of spherical clusters and plotted as vertical lines on the gures. Note however that if we use instead the empirical variance σ2 k as a measure of the cluster sizes, transitions would be shifted since σ2 k = Tr{Σ k }/D is the average of all eigenvalues of Σ k . In this case, transitions would thereby occur at a di erent temperature in the annealing than the estimated variances of the cluster because they are driven by Γ k .

Following a posteriori the several curves and the successive transitions in the right panel of Fig. 3.5 provides an informative insight on the structure of the dataset. In particular, it allows to visualise the evolution of the local size representation of the data and the interactions between centres. For instance, we clearly see that the {purple, red, green} sets of centres represent the same information when σ 2 > 9 and then split into {purple} and {red, green}. This indicates the presence of a sub-system made of two clusters. Later, we observe a crossing of the horizontal line for the {purple} centres before splitting again after crossing. This indicates that the e ective variance of the cluster is larger than the one xed by the annealing and, therefore, that these centres now describe uctuations within a "true" cluster. The {red, green} sets of centres split at lower σ 2 followed as well by a crossing of the line at the scale of individual cluster sizes (indicated as coloured dashed vertical lines on the gure).

Successive transitions can be computed in two steps: rst by identifying the K r macrocomponents resulting from the collapse of centres based on their positions and then by assigning to each datapoint (black dots in the left panel of Fig. 3.5) the label of the macro-component that most probably generated it. We are thus assuming, at a given iteration, a GMM with K r components to compute responsibilities from Eq. (3.21). Thereby, we can group together 0.0 0.2 0.4 0.6 0.8 1.0 0.9

1.0 1.1 1.2 1.3 1.4 2 / 2
true Fig. 3.6. Ratio between the estimated variances σ2 obtained when freezing the K = 25 centres when Γ k /σ 2 1 and the empirical ones σ 2 true from data of Fig. 3.5 for several values of ρ, the fraction of datapoints kept for the computation.

datapoints with identical labels and compute the next transition as the maximum eigenvalue of the covariance matrix for each sub-system. These quantities, basically corresponding to successive evaluations of the critical temperature in sub-systems, can be approximated during the annealing and are shown as black stars in the right panel of Fig. 3.5.

We identify the overlap Q, de ned as the quality of the data classi cation at each temperature, as an order parameter whose value changes throughout the several phases during the annealing. Formally,

Q ({ẑ i }, {z i }) = max π∈Π i δ K ẑi ,π(z i ) /N -1/q 1 -1/q , (3.25)
where δ K is the Kronecker delta, Π denotes all the possible permutations of the set {1, . . . , q} with q the true number of clusters used to generate the data and ẑi = argmax k p ik the estimated latent variable for the a liation of the datapoint x i . Q is based on the responsibilities p ik , all being 1/K for a random assignation giving Q = 0 and taking value 1 when ∀i, ẑi = z i . By doing so, Q is zero when σ 2 > T hard c and undergoes successive transitions as the system is cooled down, as illustrated by the black dashed curve of the left panel in Fig. 3.5. During the annealing, Q remains at 1 for the range of σ 2 between the last split of centres between two true clusters and before the rst split due to the inner random structure of the largest of them. Note that this metric is meaningless when K r > q since the dataset is partitioned into more clusters than actually used for the generation, and this is why the curve is not shown for σ 2 ≤ 1.

To further assess the robustness and accuracy of the transitions with the density of input points, we use the dataset from Fig. 3.5 where only a fraction ρ of the datapoints is randomly kept for the computation. During the annealing, we freeze all the K = 25 centres at the last split before reaching Γ k /σ 2 = 1 and then let variances evolve freely hence providing an estimate for each detected cluster that we note σ2 . Figure 3.6 shows that the retrieved variances are, even in highly sparse sampling settings, with ρ ≤ 30%, close to the true ones of the clusters. It is worth emphasising that all ve clusters are always correctly identi ed and that the value of Q is always close to 1 at the end of the process, showing the ability of the method to highlight structures, even in sparse con gurations.

Higher dimensions and nested representations

As exposed in Sect. 3.1.3, usual applications of GMMs for clustering are performed blindly by inputting the desired K to obtain a classi cation making use of all components. Some criteria, based on information theory [Akaike, 1974;Oliver et al., 1996] or Bayesian approaches [Schwarz, 1978;Roeder & Wasserman, 1997], were proposed to overcome this major drawback of unsupervised clustering. Here, we propose an approach to avoid such a K-dependent unique solution. A key aspect of the annealing is the collapse of {µ k } K k=1 at the center of mass of successive sub-datasets providing a hierarchical view of clustering with an increasing number of physical clusters. The proposed 2D diagram enables to capture this set of nested representations by catching the several transitions, even if the input space is of high dimensions. Figure 3.7 represents an application for a 5D arti cial dataset made of ten Gaussian clusters, spatially appearing as three clusters at larger scale. Transitions occurring at large scales in each panel (grey vertical dashed lines) clearly indicate that the dataset is described as three di erent physical clusters. Pursuing the decrease of variances leads to a ner description where each of the three macro-clusters splits into smaller ones that still have physical interpretations5 . Such information on the spatial organisation of the dataset are of crucial importance when having no prior idea of its structure nor the number of underlying components. The method hence allows a hierarchical view of clustering at di erent scales as proposed by hierarchical clustering methods [Murtagh & Contreras, 2012]. Because sometimes there is not a unique solution for the number of clusters depending on what they physically represent for the application at hand, the proposed way to explore the data can be of interest before running any kind of blind clustering algorithm.

Soft annealing

When the dataset is more complex with nested structures or overlapping clusters of di erent sizes, the previously presented analysis is not suitable. Multiple scales cannot be represented at the same time, hence biasing those of embedded structures towards higher values. To overcome this, we relax the equal variance assumption considered until now for the GMM, the parameter set including thereby both the positions and variances of components

Θ = {θ 1 , . . . , θ K }, with θ k = (µ k , σ k ). Under these circumstances, EM update equations are µ (t+1) k = N i=1 p ik x i N k , (3.26) σ (t+1) k = N i=1 p ik x i -µ k 2 2 DN k 1/2 . (3.27)
Keeping the idea of a temperature being slowly reduced, we propose a modi ed annealing acting on the mode of an a priori distribution on each σ 2 k allowing the representation of multiple size in the same time. In particular and to obtain a closed-form expression of the posterior distribution, we use the conjugate prior for variances, namely an inverse-Gamma distribution, with shape parameter 1 + λ σ and scale parameter λ σ σ 2 so that the distribution has a mode at the position σ 2 . Formally, it reads

ln p(σ 2 k ) = -λ σ ln σ 2 k + σ 2 σ 2 k + cst, (3.28)
where the constant comes from the normalisation of the probability distribution. Introducing such a prior modi es the update Eq. (3.27) for variances as

σ 2 k = N i=1 p ik x i -µ k 2 2 + 4λ σ σ 2 D N i=1 p ik + 4λ σ . (3.29)
Consequently, when λ σ → 0, the prior, and hence the annealing, has no e ect and components update their variances as Eq. (3.27). Inversely, for a large enough value of λ σ , σ 2 k will be close to σ 2 resulting in the classical hard annealing procedure discussed in Sect. 3.4.3. Choosing intermediate values for λ σ imposes a broad trend for all components but lets each of them correct the prior by the actual value of the neighbouring covariance. In what follows, we refer to this procedure as "soft annealing" that we distinguish from the "hard annealing" to describe the classical procedure acting directly on the variance parameter. There is no general rule to

x the hyper-parameter λ σ and, in this work, we adopt6 λ σ = 2.

Similarly as in the hard annealing case (see Sect. 3.4.2), we can compute the threshold value T soft c such that all components are collapsed at the centre of mass when σ 2 > T soft c . In the context of soft annealing, we however deal with two sets of parameters that are the position of Gaussian components and their associated variances which leads us to study the stability of the joint perturbations over the two sets. Still considering a centred dataset, we propose in a rst place to derive the xed-point variance σ 2 0 of all centres when σ 2 T soft c assuming that

σ 2 > T soft c =⇒ ∀k ∈ {1, . . . , K}, σ 2 k = σ 2 0 .
Injecting the linearly Taylor expanded expression of responsibilities (3.23) in the update equation (3.29) gives

σ 2 0 = 4λ σ Kσ 2 + N i=1 x T i x i N D + 4λ σ K . (3.30)
This equation links σ 2 0 , the actual variance attributed to all Gaussian components, to σ 2 , the inverse of the annealing temperature and is valid in the large σ 2 limit. Taking now into account both parameters hence considering perturbations k and δ k , respectively around the xed points µ k = 0 D and σ 2 k = σ 2 0 , we can derive the set of equations for the vectorised perturbations = T 0 , . . . ,

T K T ∈ R KD×1 and δ = (δ 0 , . . . , δ K ) ∈ R K×1 .
Responsibilities can be Taylor expanded as

p ik 1 K 1 + 1 σ 2 0 x T i k + δ k 2σ 4 0 x i 2 2 - 1 Kσ 2 0 x T i l l - x i 2 2 2Kσ 4 0 l δ l , (3.31)
which, when combined with the update equations (3.26) and (3.27) lead to the system of perturbations in positions and variances

     (t+1) = 1 σ 2 0 (U ⊗ C) (t) + (U ⊗ a) T δ (t) , δ (t+1) = (U ⊗ b) (t) + c U δ (t) , (3.32)
where ⊗ denotes the Kronecker product, U = I K -1 K J K with I K is the K × K identity matrix, J K the K × K all-ones matrix, and

a = i x i 2 2 x T i 2N σ 4 0 , (3.33) b = i x i 2 2 x T i mσ 2 0 , (3.34) c = 1 2σ 2 0 m i x i 4 2 σ 2 0 -D i x i 2 2 , (3.35) with m = N D + 4λ σ K.
Putting it all together leads to the matrix representation of the system perturbations η

= , δ ∈ R K(D+1)×1 η (t+1) = (U ⊗ M ) η (t) , (3.36)
with M the squared block matrix of order D + 1 where C is the data covariance matrix. Since the eigenvalues of the Kronecker product are given by the product of all individual eigenvalues of the two matrices involved and that U has eigenvalues 0 or 1, we can only restrict the analysis to those of M . Therefore, the value of σ 2 at which the rst transition occurs, namely T soft c , can be derived as the value of σ 2 such that the spectral radius of M is 1, leading to instabilities in the dynamic of the system (3.32).

M =        C/σ 2 0 a T b c        , ( 3
The left panel of Fig. 3.8 illustrates the result of the soft annealing procedure on an arti cial dataset made of six clusters similar to Fig. 3.5 but with more complexity such as overlapping (the two clusters on the left) and nested clusters (the two bottom right clusters). The right panel focuses on the evolution of the ratio between the size of the represented sub-system by a given component and its actual variance, namely Γ k /σ 2 k . This is the same physical quantity as in the hard annealing case, except that we relax the constraint on σ 2 k which is now varying for each component. In this soft con guration, all the µ k are collapsed for σ 2 > T soft c followed by steep transitions when σ 2 decreases. This relaxed annealing is especially useful for the representation of the two nested clusters. Even though we would learn that those structures are encapsulated, reaching an accurate size description for the smallest nested component would not be possible in hard annealing because its variance would be boosted by neighbouring datapoints of the surrounding cluster. The left panel of Fig. 3.8 illustrates positions (red crosses) and variances (grey dashed circles) of all K = 25 components when xing parameters a posteriori at the value they had during the annealing at their last transition point just before crossing the line Γ k /σ 2 k = 1. Although K = 25 components are used, we correctly identify K r = 7 physical clusters with their variances and means.

To assess the robustness of the soft annealing procedure in clustering complex datasets, we focus on a setup restricted to the two nested clusters of Fig. 3.8 in the bottom right part of the left panel. We dilute the small one by varying its number of sampling points N . This translates into a decreasing contrast between the signal to noise ratios σ/ √ N of the two clusters. The inset of the right panel of Fig. 3.8 shows the evolution of the ratio between the maximum overlap value Q obtained during the annealing and Q th , the theoretical overlap computed us-ing the ground truth parameters as a function of the contrast. It can be seen that both clusters are recovered when the contrast is su ciently high (above 1.5 in practice) while below, there is not the necessary information for the model to retrieve it. The ratio between the estimated variances is also shown to and we observe an overestimated variance at lower and lower contrast which explains the decreasing max Q/Q th ratio. This e ect can partly be explained by the uniform weights hypothesis being less and less true when the contrast decreases.

Graph-regularised mixture model

In Chapter 4, we shall introduce a method to learn a smooth graph representation from pointcloud distributions. For the purpose of studying the transitions of the model, we set ourselves in a simpler context than the one presented in Chapter 4. We hence consider that the learning of the graph structure is done through a regularised mixture model with xed-variance. The regularisation term acts on component averages and constrains the graph smoothness through the Laplacian de ned as

µ 2 G = K i=1 K j=1 a ij µ i -µ k 2 2 , (3.38)
where a ij is an element of A, the adjacency matrix taking value 1 when centres i and j are linked and 0 otherwise. This added term on the log-likelihood acts as an attractive quadratic interactions of centres connected on the graph A. As for the soft annealing case, the introduction of a prior in EM equations only impacts the M-step update of center positions of Eq. (3.26) as

µ (t+1) k = N i=1 x i p ik /σ 2 + 2λ µ K j=1 a kj µ (t+1) j N i=1 p ik /σ 2 + 2λ µ K j=1 a kj .
(3.39)

As done for previous iterative learning models, it is possible to compute the value of σ 2 for which the high-temperature system becomes unstable, noted T graph c considering perturbations around the xed point µ k = 0 D . Analogous derivations as in Sect. 3.4.4 shows that the system is unstable when the maximum eigenvalue of M is greater than 1, with

M = I K - 1 K J K ⊗ C σ 2 I KD + 2λ µ Kσ 4 N L ⊗ I D -1 , (3.40)
where L the Laplacian matrix de ned as

L = D -A with D the diagonal K × K degree matrix with d kk = i a ik .
The graph prior in Eq. (3.40) is expressed through L and, in a rst place, we propose to solve analytically the simple case of a complete graph prior in which a node is connected to all other nodes. For more general cases, the threshold depends on the form of the Laplacian and is computed numerically depending on the graph and the data at hand. Using a complete graph prior, the temperature can be computed with A = J K and D = (K -1) I K , leading to solve a second degree equation in T graph c with a unique non-negative solution for λ µ > 0 given by .41) This simple case shows how the critical temperature varies with λ µ . The limit λ µ → 0 and Eq. as a function of λ µ in case of a complete graph prior given by Eq. (3.41). The grey dashed horizontal line represents Γ, the maximum eigenvalue of the data covariance matrix C.

T graph c = -1 + 1 + 8Γλ µ K 2 /N 4λ µ K 2 /N . ( 3
as illustrated in Fig. 3.9 showing the evolution of T graph c given by the relation (3.41). Physically, since the prior is basically pulling adjacent centres, increasing λ µ increases the strength of the bonds between nodes of the graph.

In Fig. 3.10 is shown the result of an annealing procedure for K = 100 components, λ µ = 300 and using a graph prior given by the minimum spanning tree construction [Borůvka, 1926] assuming that centres are all linked together with the minimum total length (see Sect. 4.2.3 for a more detailed presentation of graph constructions). In this case, the graph is computed from the random set of initial nodes and is updated at each iteration of the EM procedure when the temperature of the annealing is below the critical one, σ 2 < T graph c . When paving a continuously structured data distribution with Gaussian clusters standing on a prior graph structure as proposed by the regularised mixture model, the scale of interest is the local width of the elongated structure. This size is given locally, in our 2D case, by the minimum eigenvalue γ k of the weighted covariance Σ k that we now follow during the transitions As predicted by linear stability, centres are rst aligned with the principal axis of the dataset at the beginning of the annealing and then spread over the structure to pave it more precisely, as shown on the top left panel of Fig. 3.10. By tracking the evolution of the ratio γ k /σ 2 in the bottom panel, we clearly distinguish four types of behaviours, signature of four distinct scales for structures in the dataset. It is also interesting to observe the absence of sharp phase transitions or splits within this continuous dataset tending to smooth out the evolution of the energy when the temperature is decreasing. By imposing, for each component, the variance σ 2 during

Summary and prospects

In this chapter, we introduced the rich context of machine learning and discussed its applicability and interpretability for physical sciences. We saw that various communities were addressing such questions and that statistical physics was one of the promising lead to do so. After exhibiting some connections between the two elds, we introduced the clustering aim-ing at identifying, in an unsupervised way, some subgroups in the data. For that purpose, we saw that mixture models can be used together with the Expectation-Maximisation algorithm to estimate parameters of the model with latent variables.

We then established a statistical physics formulation of the clustering procedure and showed that it was equivalent to a particular case of mixture models which can be solved by the EM algorithm. We used this physical analogy to study the learning dynamics of the clustering scheme in the context of deterministic simulated annealing. In particular, we observed and characterised the cascade of phase transitions occurring when tracking the evolution of eigenvalues of the successive covariance matrices of mixture components in multiple cases. We showed that the thresholds at which the rst transition occurs can be computed analytically by studying the linear stability of the xed-point iterative scheme and that we can approximate the next ones during the annealing based on the current estimate of the responsibilities. The proposed way to use these transitions to explore the data and build a hierarchical description independent from K provides a qualitative and quantitative insight on the structure of the dataset at di erent scales and without requiring any prior knowledge. The 2D diagram representing the evolution of the physical size of the represented clusters with the annealing temperature allows an analysis independent from the data dimensionality and highlights characteristic scales at which physical transitions occur. This diagram carries information about the number of components, their scale and hierarchy that can be used a posteriori for data exploration before running blind clustering methods.

Interestingly, we saw that the latent variables of the GMM are directly related to the values taken by the order parameter in Eq. (3.25). Since the GMM can be recast as a particular case of Restricted Boltzmann Machines [RBM, Smolensky, 1986] using a soft-max prior on the hidden nodes, making it a particular type of auto-encoder [Bourlard & Kamp, 1998], the presented work can be seen as the learning of a latent representation of the phase transitions, as is discussed in Van Nieuwenburg et al. [2017] and Wetzel [2017]. Intriguingly, we also witnessed a complex dynamical behaviour in the learning of a simple algorithm of clustering with a restricted number of parameters and putting ourselves in a simplistic setup with uniform and spherical Gaussian mixtures. This says long on the di culty of handling overparametrised deep neural models in such setups, a vast current topic of research [Bahri et al., 2020]. One of the aspects I personally would be interested to explore is the relation between the data structure and the learning dynamics in more generic distributional learning approaches. As a starting point, the RBM is the perfect candidate since it extends the GMM formulation but it is also a simple version of a neural network with a single hidden layer that aims at learning a statistical representation of the data. The statistical physics analysis of the RBM through spin-glass analogies and mean-eld approximations already led to the characterisation of the phase diagram of the model in many cases [see Decelle & Furtlehner, 2021, for a review]. Mixing this more general version of the GMM and the hidden manifold model proposed in Goldt et al. [2020], meant to study the learning of two-layer networks when high-dimensional data are embedded onto a submanifold of smaller dimension, is a promising way to build a framework enabling the understanding of relationship between the data structure and the learning dynamics in unsupervised setups.

Among the di erent and numerous applications of clustering, the task of segregating datapoints in di erent classes is a long-standing problem in astrophysics with for instance the galaxy morphological classi cation (elliptic, spiral, etc.) based on optical data [Weinmann et al., 2006;Shamir, 2009] or the classi cation of gas phases based on gas properties (temperature, density, etc.) [see e.g. Martizzi et al., 2019;Galárraga-Espinosa et al., 2021]. By including the identi cation of hierarchical structures, it may help in de ning the classes in a physically motivated way. It could also tackle the problem of detecting subhalos in simulations based on discrete particle positions as an alternative to the SubFind algorithm [Springel et al., 2001;Dolag et al., 2009]. By proposing a non-unique solution to clustering, it also o ers the possibility to explore the di erent relevant substructures. Applications of such a clustering procedure are hence of interest in these contexts to study the link between the di erent possible physical and spatial structural information of a cluster (may it be through its dark or baryonic matter composition) and how it correlates with those of the neighbouring lamentary structure.

Chapter 4.

Principal graph learning

"Graph theory is a natural language to describe the Cosmic Web. " M. A C 4 [2020]. In Chapter 3, we mainly studied some datasets appearing as split into multiple groups that we wished to identify. With the same interest for the spatial structure of datasets, we aim at learning non-linear representations of continuously structured data assuming as standing on one-dimensional underlying manifolds. The goal of this chapter is to expose an alternative formulation of the principal curve problem that overcome some of the drawbacks of the existing de nitions. In particular, we focus on establishing a procedure that allows the handling of outliers and variations in the local size of the sampling noise in view of applying it to the detection of the lamentary pattern depicted by matter tracers (halos or galaxies) in Chapter 5. To do so, we rst review some basic elements of graph theory and mix them with the mixture model (see Sect. 3.2) framework to propose a formulation in which the graph is modelling the underlying 1D manifold. After a comparison to other ridge nders, we discuss several technical aspects of the algorithm, like its convergence properties and time complexity before using the detection of roads from noisy GPS measurements as a showcase.

Context

Spatially structured point-cloud data

Many datasets come as a set of discrete D-dimensional datapoints X = {x i } i=0,...,N with x i ∈ X ⊂ R D sampled from an unknown probability distribution. These datapoints are usually not spreading uniformly over the entire R D space but often result from the sampling of a lower dimensional manifold whose topology and characteristics are linked with the process that generated the data. Taking the example of the famous MNIST dataset, in which datapoints correspond to images of 28 × 28 pixels digits, making it a 784 dimensional dataset, there is only a very restricted volume in this space that would generate images that actually look like a digit [Hein & Audibert, 2005;Facco et al., 2017]. Capturing this information, may it be for visual, geometrical or topological analyses of the dataset requires the application of non-linear methods that are parts of the manifold learning eld [Roweis & Saul, 2000;Belkin & Niyogi, 2003;van der Maaten & Geo rey, 2008;Van Der Maaten et al., 2009]. In some applications, data even appear as standing on a continuous one-dimensional structure. It is for instance the case for GPS measurements collected by vehicles standing on the road network [Ahmed et al., 2015], vessel networks transporting blood through the human body [Moccia et al., 2018] but also for the large-scale matter distribution describing the lamentary structure of the cosmic web, as seen in Chapter 2 illustrated for instance by the left panel of Fig. 2.6. As part of unsupervised machine learning methods (see Sect. 3.1.1 for a discussion on categories of machine learning approaches), one of the key aspects of pattern analysis is to extract from such inputs, with the least prior knowledge, the su cient information to build a meaningful representation of the data to understand the underlying structure that generated it, build models and make predictions.

Principal curves

The problem of estimating an one-dimensional manifold approximating the underlying distribution of X is a particular case of dimensionality reduction also known as ridge detection or principal curve extraction. The seminal work of Hastie & Stuetzle [1989] provides an intuitive de nition of a principal curve as the line passing in "the middle" of the point cloud distribution hence providing a non-linear generalisation of principal components. More formally, in this early formulation, a principal curve is a self-consistent smooth non-intersecting line with nite length. Self-consistency is maybe the most important property and implies that each point of the curve corresponds to the average of the datapoints projecting right on it, i.e. f (y) = E(X | y f (X) = y), with y f (x) the projection index of a datapoint x on the curve f (see Fig. 4.1 for an illustration). The self-consistency property is also shared by principal components to which principal curves are a relaxation of the straight line condition and can be seen as what local regression is to linear regression. Exactly as principal components, principal curves can be written as the minimisation of the quadratic sum of projected distances.

In that regard, it can be formulated in the general form (3.1)

argmin f ∈F N i=1 x i -f (y f (x i )) 2 2 . (4.1)
which, under the constraint that F is the set of linear relations f (y) = f 0 y, is solved by the straight line in the direction of the rst principal component. If F is more generally the set of continuous functions, optimisation problem (4.1) de nes the principal curve formulation proposed by Hastie & Stuetzle [1989]. Figure 4.1 illustrates the rst principal component of a 2D non-linear dataset and a principal curve. Although well-de ned for the principal component case, the minimisation of problem (4.1) in its most general form do not lead to "principal curves". A simple way to see that is to imagine the curve passing exactly on every datapoints, hence minimising the total cost but which is not a principal curve. One way to take into account that prior knowledge is to include regularisation terms (see Sect. 3.1.2) to penalise the set F and constrain the solution f . In Kegl et al. [2000] is for instance proposed an algorithm xing the overall length of the curve while Smola et al. [2001] propose more general smoothness constraints. Since the work of Hastie & Stuetzle [1989], several studies followed to extend these de nitions to higher dimensions (principal surface or volume) or to give more suitable de nitions of principal curves as ridges of a probability density [e.g. Tibshirani, 1992;Ozertem & Erdogmus, 2011].

To allow the description of more exible structures than those imposed by curves and to bypass their inability to represent self-intersecting or cycling topologies, a formulation relying on graph theory to model the one-dimensional structure was introduced in Gorban & Zinovyev [2005] and extended in Gorban & Zinovyev [2009]. This latter is based on prede ned rules for growing the graph and includes regularization terms to limit its complexity. However, this model come with a large number of parameters and with no guaranteed convergence to which the double optimization scheme of Mao et al. [2015] o ers an alternative for the learning of a tree structure. In this landscape of methods, only a few address the problem of estimating a principal graph with a proper handling of outliers, which considerably complicates the learning of the graph structure. A built-in robustness to outliers is proposed in Gorban et al. [2016] and Albergante et al. [2020] by discarding from the update of a node position all datapoints beyond a robustness radius R 0 . However, the choice of R 0 is not trivial, scale-dependent and require a careful tuning to deal with uniform background noise [Albergante et al., 2020].

The aim of the chapter is to combine mixture models presented in Sect. 3.2 to approximate the underlying data distribution and regularise it over a graph structure to constrain Gaussian centroids to pave the approximation of the manifold given by the graph. The method extends the original presentation of Tibshirani [1992] which is making use of smooth di erentiable curves to a more general representation given by a graph structure that acts like a topological prior in the Bayesian model turning the problem into a maximum a posterior estimation. We will see that the proposed formulation through mixture models naturally allows the learning of the local width of the represented one-dimensional structure with robustness to outliers hence freeing it from heavy pre-processing, as opposed to the previous principal curve or graph algorithms. It also comes with guaranteed convergence to a local maximum of the log-posterior inherited from the Expectation-Maximisation algorithm.

1st principal component Principal curve data

x i f(y f (x i ))
Fig. 4.1. Illustration of a principal curve for a non-linear dataset. The principal curve (in red), minimising the total sum of squared projection distances of all datapoints under some smoothness constraints generalise the straight line principal component analysis (dashed blue line).

Elements of graph theory

Graph theory is a branch of mathematics that became popular in many elds of science such as social science [Borgatti et al., 2009], biology [Koutrouli et al., 2020] and physics [Estrada, 2013]. For this latter use, it has been exploited in elds ranging from quantum physics for the propagation of waves [Kuchment, 2008;Berkolaiko, 2017] to cosmology for the study of the galaxy distribution [Barrow et al., 1985;Colberg, 2007;Coutinho et al., 2016]. Widely known for their great representative power of relationships between objects, graphs became also a keystone of computational geometry by allowing the representation of non-Euclidean geometries. This section hence aims at providing the required formal background of graph theory that is of importance to establish and understand the method proposed in this manuscript. For a more complete introduction to graph theory, we refer the reader to the book of Bondy & Murty [2008].

Introduction and de nitions

In simple terms, a graph is a collection of nodes with edges linking them together. Nodes can represent physical observables, like galaxies in astrophysics, towns and cities in the road network, human beings in a social network or URL addresses in the internet. Edges indicate a physical or conceptual relation between two vertices, linking them if they share some similarities. In the case of a social network, they can be thought as a connection between two individuals if they know each other while, in the road network, two cities can be linked if there is a route to travel from one to the other. Edges usually come with a weight that encodes the similarity measurement or the cost of associating two nodes and depend on the application. Taking back the example of roads, they could be a measure of the geodesic distance between two linked cities. Mathematically, we write G = (V, E, W), where V is the set of K := |V| nodes, E ⊂ {1, . . . , K} × {1, . . . , K} is a set of tuples of two vertices that are linked and W = {w ij } (i,j)∈V 2 is the set of edge weights such that ∀(i, j) ∈ V 2 , w ij ≥ 0 being non-zero when (i, j) ∈ E. If E is an unordered (resp. ordered) set, the graph is said undirected (resp. directed), meaning that the link between two nodes i and j is reciprocal and w ij = w ji (resp. not reciprocal and, in general, w ij = w ji ). In the example of the internet network, a URL address can refer to another one while it might not be the case in the other way, making it a directed network.

All of the graphs discussed in this manuscript are parts of the undirected and simple graphs family, meaning that they exhibit no self-loop (i.e., no node i such that (i, i) ∈ E) and no multiple edges (i.e., each association (i, j) ∈ E is unique in E). Although one can de ne many characteristics for nodes in a graph, we only restrict ourselves to the introduction of the degree deg(i), which is the number of individual edges incident with the node indexed i.

Moreover, in all the applications presented in this thesis, graphs are considered as objects embedded in R D , whose nodes have a spatial position, making them "spatial graphs". As for any other graphs, the weights {w ij } associated to nodes pairs are still measuring a similarity but based on their spatial proximity. More precisely, for a graph with K nodes with positions {µ k }, we will consider weights corresponding to the Euclidean distance, w ij = µ i -µ j 2 if nodes indexed i and j are linked.

Linear algebra representations

To any graph G is associated a K × K matrix A fully encoding the graph information by embedding the relations between nodes. Such a matrix representation is called the adjacency matrix and is particularly useful in linear algebra contexts to have tractable representation of the graph. Elements of A are de ned as

a ij = 1 if (i, j) ∈ E, 0 otherwise. (4.2)
One can note that summing over the i th row gives the degree of the node i, meaning that deg(i) = K j=1 a ij and that i j a ij = 2|E|. In the particular case of simple graphs, A has the additional properties of being symmetric and has a zero-valued diagonal from the absence of loops. If weights are incorporated into the adjacency matrix then W fully characterises the graph G including node connections and the associated weights.

A second possible matrix representation is provided by the discrete graph Laplacian de ned as the symmetric and semi-positive de nite matrix

L := D -A, (4.3

Some graph constructions

There are multiple ways to build a graph from a given set of spatial nodes µ = {µ i } K i=1 . We review below some remarkable graph topologies that are being used throughout this manuscript.

Complete graph. The complete graph is represented by a plain adjacency matrix, ∀(i, j) ∈ {1, . . . , K} 2 , a ij = 1, meaning that all possible pairs of nodes are linked together. Even though weights can be chosen to stress more local relations based on the proximity of nodes, this graph encodes a lot of redundant paths and do not emphasise a particular one in the data. Moreover, even if dense graphs have interesting properties (like robustness), this construction induces a plain adjacency matrix which, for applications including a large number of nodes (i.e. galaxies for instance), becomes quickly hardly tractable in both time and memory.

-neighbourhood. Two nodes i and j are linked together if µ i -µ j 2 < (see illustration in the left panel of Fig. 4.2). Despite its natural de nition, where close nodes are linked together, one of the main drawbacks of this graph construction is that the represented pattern highly depends on the choice of that can be arbitrary and scale-dependant.

k-nearest neighbours. The k-neighbourhood (k-nn) of a node i is de ned as the set of k nodes that are the closest in terms of the Euclidean norm. In this construction, two nodes i and j are hence linked together if i is in the k-neighbourhood of j or reciprocally. The choice of k is easier than in the -neighbourhood case but the k-nn has the disadvantage of creating long-range links for datapoints standing in almost-empty areas (as illustrated on the middle panel of Fig. 4.

2).

Minimum Spanning Tree. Let us rst de ne a subgraph H = (V H , E H , W H ) of an arbitrary graph G = (V, E, W) as the graph such that V H ⊂ V and E H ⊂ E. We call a spanning tree a particular type of subgraph in which the set of vertices is the same as the original one but with the minimum number of edges. That is, the graph G admits a spanning tree T = (V, E T , W T ) with |E T | = |V| -1 linking all nodes of V together. Building on that knowledge, a minimum spanning tree [MST, Borůvka, 1926] is a spanning tree with the minimum total weighted length i j w ij . The MST is, by de nition, a simple graph and is unique only if all weights are di erent. This de nition can be rephrased in terms of an integer programming problem aiming at nding the adjacency matrix A MST such that

A MST = argmin A K i=1 K j=1 a ij w ij (4.4) s.t.                      A = A T , a ij ∈ {0, 1}, K i=1 K j=1 a ij = 2 (K -1) , ∀k ∈ {1, . . . , K}, a kk = 0, ∀S ⊆ V, i∈S j∈S a ij ≤ |S| -1,
The rst two constraints encode the undirected graph de nition, the third one denotes a graph with K -1 edges, the fourth the absence of self loops in the structure and the last one requires that any subset of |S| vertices has at most |S|-1 edges. By relaxing the integer constraint to a ij > 0, the problem can be solved in quasi-linear time with the number of edges using Kruskal's algorithm [Kruskal, 1956].

As pointed out in its de nition, the MST is a subgraph, meaning that it is built from the set of nodes, edges and weights of another graph. When considering a complete spatial graph embedded in R D and weights {w ij } based on the Euclidean distance, the MST is called Euclidean and is linking all the input points together with the minimum total Euclidean distance to do so. Because in applications including a large number of graph nodes K, the use of a complete graph can make the computation of the MST costly, we rely on a graph built from the Delaunay Triangulation [Cavendish, 1974] from which the MST is theoretically a subgraph [Aurenhammer et al., 2013]. In particular, since the Kruskal algorithm has a time complexity of O(|E| log|E|), reducing the number of edges to explore from the complete graph with K 2 edges to the O(K) of the Delaunay Triangulation allow to maintain an overall complexity of O(K log K) for the computation of the Euclidean MST.

The Euclidean MST (that we simply call MST in the rest of the manuscript) graph construction has the advantage of being scale-independant and parameter-free. These properties made it an especially well-suited tool to study the lamentary part of the cosmic web in the past decades [Barrow et al., 1985;Colberg, 2007;Alpaslan et al., 2014b;Bonnaire et al., 2020;Pereyra et al., 2020a]. The right panel of Fig. 4.2 allows to appreciate the preferred path obtained for the "S" shape and gives a good intuition that, by post-processing it with simple operations (like pruning), the main underlying structure can be obtained, even though locally spiky. The main focus of the next section is precisely to obtain a locally smooth representation of the graph structure by embedding it into the probabilistic framework of mixture models (see Sect. 3.2). The limited tree topology of the MST for the representation of cycles will be the topic of Sect. 4.4 in which we present how to extend it by combining several MSTs obtained from random sub-samplings.

Graph regularised mixture models 4.3.1 Full model and formalism

We assume in this section that we are given a dataset X ∈ R N ×D , resulting from the noisy sampling of a continuous unknown one-dimensional manifold. As previously exposed, the -neighbourhood k-nn, k = 6 MST direct application of a graph construction may not represent the topological and geometrical properties of the underlying pattern. To build a coherent-with-data graph representation, we seek to combine the mixture model formalism (Sect. 3.2) with a graph representation of the manifold, hence assuming a given topology, to extract a principal graph from the data (as illustrated in Fig. 4. 3).

The data generation model

We rst model the data using a mixture model with a set of K Gaussian clusters with their own centres µ k and covariances Σ k . Figure 4.3 illustrates the proposed model of the data distribution with spherical Gaussian clusters Σ k = σ 2 k I D hence assuming that the sampling noise around the 1D manifold is Gaussian and isotropic. The isotropy is characteristic of the tubular structures observed in many datasets like blood vessels or laments and this spherical assumption will hold for the rest of the dissertation.

One of the drawbacks of the principal curves formulation presented in Sect. 4.1.2 is their sensitivity to outliers tending to bias the curve estimation. Real-world datasets, however, often come with outliers or noisy measurements that are not part of the underlying pattern that generated the data we aim to extract. In the mixture model formalism, we can tackle this problem by directly assuming an explicit distribution for the outliers. The choice of this distribution can be speci c to the problem at hand and adapted depending on the knowledge one has on their generation. In the present work, we consider a uniform distribution of outliers over the data support volume. Hence, in addition to the K Gaussian clusters paving the distribution, we make use of a uniform background component to take into account observations that should not be part of the pattern of interest. The full mixture model can hence be written

p(x|Θ) = K k=1 π k N (x, θ k ) + αρ(x), (4.5)
with Θ = {π 1 , . . . , π K , α, θ 1 , . . . , θ K } the set of all parameters of the model with normalised and non-negative amplitudes the only information required to handle the outliers is the inverse of the data support volume, its amplitude being part of the parameters Θ, adjusted during the learning. In practice, ρ(x) is estimated as the inverse volume of the convex hull of the input dataset. One of the key advantage of the proposed formulation is that outliers do not need to be removed in a preprocessing step as it is for instance the case in Chen et al. [2014] in which an arbitrary density threshold over a kernel-density estimate of the probability distribution is used. The proposed idea of built-in robustness is also explored in Gorban et al. [2016] and Albergante et al. [2020], who propose to discard from the update of a node position all datapoints beyond a robustness radius R 0 . Still, the choice of R 0 is not trivial, scale-dependant and require a careful tuning to deal with uniform background noise [Albergante et al., 2020]. In our formulation, the parameters related to outliers, α and ρ(x), are fully part of the model and do not require any heavy prior information nor ne-tuning to deal with uniformly distributed outliers.

K k=1 π k + α = 1 and ρ(x) = R D 1 X (x) -1 . Consequently, data { k }

Graphs as approximations of manifolds

As of now, we simply modelled the data generation process through a mixture model and wrote its likelihood. However, we additionally assume that the D-dimensional dataset X is lying on an one-dimensional Riemannian manifold M. In simple words, manifolds extend the notion of curves, surfaces and volumes to any dimensions. When locally Euclidean and equipped with a metric, they are called Riemannian which allows for the de nition of common geometric notions such as lengths, areas, volumes or curvatures. To avoid over-tting and to get a smooth estimate of the manifold, we can use a quadratic regularisation term, as exposed in Sect. 3.1.2. Ideally, we would directly try to identify the continuous manifold M that generated the dataset and constrain its smoothness through the Laplace-Beltrami operator ∆ M . In practice, M is not known and because of the nite nature of the considered datasets at hand, we need to resort to an approximation. Instead, we model it as a graph structure G and the Laplacian of the graph can be seen as a discrete approximation of the Laplace-Beltrami operator de ned on the manifold [Chung, 1999;Belkin & Niyogi, 2001]. Noting µ := (µ 1 , . . . , µ K ) T ∈ R K×D , we therefore constrain the graph smoothness through

µ 2 G = K i=1 K j=1 a ij µ i -µ j 2 2
= 2 Tr{µ T Lµ}, (4.6)

which converges to M ∇µ 2 , a measure of the manifold smoothness, in the in nite data limit [Hein et al., 2005;[START_REF] Singer | [END_REF]. This quantity can also be seen as the total length of the graph since

µ 2 G = K i,j=1 w ij . Minimising µ 2
G together with a data-driven term would hence intuitively lead to a shorter and smoother graph, similarly to what is done in Kegl et al.

[2000] but without strictly xing the length during the optimisation.

Regularisation terms

Early formulations of principal curves, based on the minimisation of the mean-squared projected distance, already suggested the need of regularisation [Hastie & Stuetzle, 1989]. In particular, Duchamp & Stuetzle [1996] show that principal curves in the plane are saddle points of the mean-squared objective function, making regularisation an indispensable way to constrain the form of the solution (see Sect. 3.1.2) when this kind of cost function is used1 . In the present probabilistic formulation, the "curve" is introduced as a graph prior structure embedded through the regularisation term. It imposes a constraint on the behaviour of the average position of Gaussian clusters µ through the graph smoothness de ned by Eq. (4.6) to be added to the log-likelihood. This prior allows the inclusion of the geometric assumption has about the observed distribution of being generated by an underlying one-dimensional structure. In a Bayesian setup, this term can also be seen as a prior distribution over the parameter space, and more precisely as a Gaussian prior on the graph weights through the interactions between parameters of the model µ k ,

log p(µ) = - λ µ 2 µ 2 G , (4.7)
with λ µ the precision parameter of the Gaussian prior distribution. We later refer to this parameter either as the precision of the prior on µ or as a regularisation term on the loglikelihood, the two being equivalent. As we saw previously, this term corresponds to the length of the graph and will be added to the log-likelihood that we aim at maximising. It is hence equivalent to a soft minimisation of the total length of the graph structure. In the elastic map formulation of Gorban & Zinovyev [2005], a physics analogy is proposed in which Eq. (4.7) can also be seen as a stretching energy. In this analogy, graph nodes are considered as linked together by elastic bonds with elasticity coe cients λ. Such a formulation of the principal graph problem alleviates some of the drawbacks of the principal curve in which selfintersecting curves are not allowed [Hastie & Stuetzle, 1989;Kegl et al., 2000]. By restricting the graph to a chain topology with elements of the Laplacian matrix given by

l ij = 2δ K i,i - δ K i,j+1 -δ K i,j-1 , with δ K i,j
the Kronecker delta function, we can also derive the same type of curve constraints as in Yuille [1990] and Tibshirani [1992], making the graph formulation more general in terms of handled topologies.

The direct application of the EM procedure to maximise the regularised log-likelihood de ned by the sum of the log-likelihood and Eq. (4.7) leads to an estimate of Θ. In this setup, the variances learnt by the procedure would depend only on the local distribution of datapoints. However, natural manifestations of heteroscedastic patterns, such as trees or blood vessels, exhibit a linear evolution of the sampling size without sudden variations. For instance, the width of a tree is continuously expanding from small branches to the trunk and the size of the vessel network is smoothly enlarging from capillaries to wide arteries. For these physically motivated reasons, the local size of the underlying continuous structure as measured by the variances of the model can be considered to evolve smoothly along the graph. To incorporate this idea in the formalism, we use again the non-Euclidean proximity measures on the graph structure for variances update. To this end, an additional prior distribution on variances is used, based on the local neighbourhood of a node. To obtain a closed-form expression of the new update equation, we use the conjugate prior for variances of a Gaussian likelihood.Formally, we rely on the inverse-Gamma distribution distribution with shape parameter 1 + λ σ and scale parameter λ σ σ 2 N k de ned such that the mode of the distribution is located at the mean variance of the neighbouring nodes, namely σ 2

N k = 1/|N k | i∈N k σ 2 i with N k = {i | a ik = 1} and |N k | = d kk the degree of node k. Mathematically, we write log p(σ 2 k ) = -λ σ log σ 2 k + σ 2 N k /σ 2 k + cst, (4.8)
the prior distribution for the σ 2 k parameter and use the same λ σ ≥ 0 to constrain all Gaussian components.

Finally, a prior distribution is added for mixing coe cients to avoid singular solutions to happen when a node of the graph is paving an underdense region, like galaxies standing in voids for instance, making its amplitude going to 0. This can be achieved by assuming a Gaussian prior centred on uniform coe cients on the set of datapoints not represented by the background noise, namely (1 -α) /K. Hence, we have

log p(π k ) = - λ π 2 1 -α K -π k 2 + cst, (4.9)
where λ π ≥ 0 controls the force of this prior.

The regularised mixture model

The full prior distribution on the parameter set Θ can be written as the summation of all individual terms,

log p(Θ) = log p(µ) + K k=1 log p(σ 2 k ) + K k=1 log p(π k ). (4.10)
As we saw in Sect. 3.3, the optimal values of parameters can be estimated using the EM algorithm allowing the maximisation of log-likelihood in an alternating procedure. In the context of the proposed graph regularised mixture model (GRMM), we aim at maximising the log-posterior (that we equivalently call regularised log-likelihood) log p(Θ | x) ∝ log p(x | Θ) + log p(Θ). For this purpose of maximum a posteriori estimation, EM can still be used with only minor changes to the introduced equations (3.12) and (3.13). Note that the E-step remains unchanged since the added term only depends on Θ and that this rst step is a maximisation of the lower bound over the latent variables only. In the M-step, we now seek to solve

Θ (t+1) = argmax Θ Q(Θ, Θ (t) ) + log p(Θ), (4.11)
where Q(Θ, Θ (t) ) is de ned through Eq. (3.12) as the expectation of the completed loglikelihood over the latent variable distribution p(Z | X, Θ). We recognise here the general form for optimisation problems teased in Eq. ( 3.3) of Sect. 3.1.2 with the quadratic association cost hidden in the log N (x i , θ k ) term of the log-likelihood and where log p(Θ) constrains the form of the solution with strengths {λ µ , λ σ , λ π }.

For the log-likelihood as de ned by the particular mixture model of Eq. (4.5) and the prior de ned by Eq. (4.10), responsibilities of Gaussian and uniform background components, respectively noted p ik and p bkg i , can be computed during the E-step as p(Z | X, Θ (t) ) using Bayes' theorem and the current parameter values

Θ (t)                p bkg i = αρ(x i ) K j=1 π j N (x i | θ j ) + αρ(x i ) , p ik = π k N (x i | θ k ) K j=1 π j N (x i | θ j ) + αρ(x i )
.

(4.12)

Update equations for each parameter are then derived during the M-step of Eq. (4.11) as

                                         α (t+1) = 1 N N i=1 p bkg i , π (t+1) k
= 1/N N i=1 p ik + λ π 1 -α (t+1) /K 1 + λ π , µ (t+1) k = N i=1 x i p ik /σ 2 k + 2λ µ K j=1 a kj µ (t+1) j N i=1 p ik /σ 2 k + 2λ µ K j=1 a kj , σ (t+1) k = N i=1 p ik x i -µ k 2 2 + 4λ σ σ 2 N k D N i=1 p ik + 4λ σ 1/2
.

(4.13)

When not speci ed, parameters not indexed by time correspond to time t. This is especially important in the E-step in which all parameters are expressed at time t and in the computation of µ (t+1) k

. It is indeed interesting to note the contribution of all µ (t+1) j in the update of µ (t+1) k in Eq. (4.13). It is hence more convenient to write this update equation matricially for µ ∈ R K×D .

µ (t+1) = ΓS -1 + 2λ µ L -1 S -1 R T X, (4.14)
where S is a diagonal K ×K matrix with s kk = σ 2 k , Γ a diagonal K ×K matrix of average datapoints explained by the k th component, i.e. γ kk = N i=1 p ik and R ∈ R N ×K the responsibility matrix for Gaussian components such that r ik = p ik .

Algorithm and illustrative results

Algorithm (1) sums up all the steps for the learning of the proposed principal graph from a given set of measurements X and a graph construction prior G. It extends the GMM by embedding a topological prior through G constraining the position of centres, propose an handling of outliers and the learning of the local width of structures. In this model, the topology of Algorithm 1 Graph regularised mixture model (GRMM) Input: Data: X ∈ R N ×D , hyper-parameters: Y = {λ µ , λ σ , λ π }, initialisation: Θ (0) , G (0) . Output: Θ (t) and G (t) . while R t ≥ do Compute the adjacency matrix A of G (t) E-step: Compute responsibilities p ik and p bkg i using equations (4.12) M-step: Compute new parameters Θ t based on responsibilities using equations (4.13) and (4. Bottom left is the one from the SimplePPT algorithm [Mao et al., 2015[Mao et al., , 2017] ] with σ = σ 0 and identical initialisation nodes. In both cases, the shaded areas correspond to the 1-σ k circles centred on µ k . The bottom right panel is the result from the ElPiGraph procedure [Albergante et al., 2020] with 100 nodes initialized with 70 taken randomly over X and for di erent values of the trimming radius R 0 with the same elasticity parameters λ = 0.01, the length constraint and µ = 0.01 the bending constraint. the graph can be either xed or updated during the optimization. This is particularly important since the prior approximation on the noisy pattern with outliers may not be suitable to describe the nal smooth underlying structure. For instance, a bifurcation in the prior graph may not be relevant in the nal one. Consequently, it may be useful to update the topological prior based on the current estimate of {µ} K k=1 . When updating the topology at each iteration, A cannot be understood as a prior anymore but has to be included as a parameter in the model making the chosen graph construction itself acting like a prior. In the case of tting an MST topology, we can embed the integer programming optimization problem de nition of the MST from Eq. (4.4) in the regularized log-likelihood hence leading to estimate both Θ and A in the M-step which simply involves computing the new graph G t at iteration t given a graph construction process and the positions {µ} The bottom left panel of Fig. 4.4 illustrates the principal graph learned by the SimplePPT algorithm on the same dataset and using the same parameters and initialisation as our construction. Many branches are falling in the background noise and, because of its xed-variance scheme, it fails at capturing the ridges of both the large and small variance branches. Choosing a large value for σ 2 0 biases the graph in the small branch, while a low one irremediably creates a lot of spurious branches in the large-variance part of the pattern, even in the absence of outliers. With its contracting iterative scheme, the Subspace Constrained Mean Shift [SCMS, Ozertem & Erdogmus, 2011;Genovese et al., 2014;Chen et al., 2014] algorithm would su er less from the latter exposed problem. Yet, it does not provide a properly-speaking curve, but a set of independant projected points that are to be linked a posteriori and the SCMS is also usually used together with a pre-processing step to lter datapoints standing in low-dense areas such as proposed in Chen et al. [2015] hence involving an additional non-trivial parameter impacting the quality of the result. The sensitivity of the graph to outliers is handled by the ElPiGraph framework [Albergante et al., 2020] through a trimming radius R 0 discarding distant datapoints from the update of a node position. In the bottom right panel of Fig. 4.4 can be found three realisations of this algorithm with di erent values of R 0 , showing the robustness of ElPiGraph to uniform background noise with a ne-tuned value. The choice of R 0 however remains scale-dependent and can alter the recovered structures in case of heteroscedastic patterns. It is also not obvious to tune in real applications and comes in addition to other hyper-parameters already complex to choose. Moreover, the resulting graph do not embed a description of the local width as opposed to the GRMM which ts all parts of the pattern in its center. Note nonetheless that the ElPiGraph algorithm handles better the dense area at the center of the dataset when R 0 = 0.3 with a single bifurcation thanks to an added terms to the cost function minimising the overall graph complexity. The algorithm we develop here [Bonnaire et al., 2021b], by unifying all these aspects of robustness and local size description in a single formulation, is able to provide a principal graph that visually shows a smoothly evolving adaptive scale along the graph structure. Even though the graph is initialized with only K = 100 nodes taken randomly over X with variances ∀k, σ (0) k = 0.08, there are no nal nodes standing outside of the pattern and the resulting estimate of the background level is 25%. Note also that, on patterns showing no outliers and with branches of the same sizes, all algorithms provide similar results with a smooth graph passing in the middle of the structure.

-neighbourhood k-nn, k = 6 MST

About graph priors 4.4.1 Basic graph constructions

The choice of the prior graph construction depends on the guess of the underlying manifold topology on which the data presumably live. Manifolds learning techniques for dimensionality reduction based on graphs usually rely either on a complete graph with weights obtained from a heat-kernel w ij = exp{ µ i -µ j 2 2 /s} with the choice of s being data-dependant Belkin & Niyogi [2003], either on a k-nearest neighbours graph. Such constructions are convenient for the precise purpose of dimensionality reduction in which the mapping between the high and low dimensional spaces is supposed to preserve the local neighbourhood of the data, assuming the manifold to be locally Euclidean [Belkin & Niyogi, 2003]. In our context of ridge estimation, we nd that such local constructions, in addition to depend on a free parameter, are also creating a lot of redundant and spurious edges hence degrading the pattern extracted from Algorithm (1), as shown in the middle panel of Fig. 4.5 where the "S" shape of the pattern is not correctly extracted. Note however that, with a proper post-processing of the graph prior, pre-processing of the input dataset, or ne-tuning of the parameter for the graph prescription ( or k), we could end up with a clean estimate of the ridge. The MST prior, on the other hand, is tting well the expected underlying shape and has the advantage of being parameter-free, scale-free and can handle general patterns by assuming a tree-like topology.

The average graph prior

Although the minimum spanning tree exhibits some nice features discussed above, it has a limited representative power for general datasets due to this latter-mentioned tree topology that cannot represent cycles, as shown in the top left panel of Fig. 4.6. By using a non-cycling topology, the regularised graph becomes even more inaccurate if the data embeds holes because of the optimisation tending to shorten graph extremities hence emphasising the absence The two circles in the top right and bottom left panels illustrate that the hole not captured by the MST varies with the sub-samplings. This instability is exploited in the average graph prior to obtain a cycling topology.

MST (X) MST (X b ) X b MST (X b ) X b AGP (X)
of cycles. Several graph constructions allow such a feature, as for instance the k-nearest neighbours graph. This solution, in addition to create long-range edges for isolated nodes that would turn in branches reaching outliers or spurious cycles, is not based on an optimisation of the total length of the graph, which is convenient for convergence properties of the algorithm when updating the prior (see Sect. 4.5).

Because the MST results from a global minimization of the total Euclidean distance as can be seen in Eq. (4.4), the obtained preferred path is very sensitive to random removals of datapoints (see top right and bottom left panels of Fig. 4.6). We exploit this particular aspect by merging the idea of a graph with the minimum total length and the handling of one-dimensional holes in the dataset to propose an empirical construction based on the computation of MSTs obtained from a set of B realisations of random sub-samplings of the set of nodes. Formally, the average adjacency matrix is Ā = B b=1 A b /B, where A b is the adjacency matrix of the MST computed the b th random sub-sampling of µ using a fraction K b /K of points. This approach, proposed in the context of pattern extraction to provide an uncertainty measurement [Chen et al., 2014;Bonnaire et al., 2020;Albergante et al., 2020], is exploited in the present case to combine all the realizations to obtain a single graph construction. This operation is also done a priori on non-regularized MSTs hence reducing the computational cost by avoiding the need of running Algorithm (1) multiple times.

The average adjacency matrix Ā represents the frequency of appearance of an edge during B realisations of MST and hence the probability, for each pair of centres, to be linked. To illustrate the method, we build a discrete Voronoi dataset consisting of intersecting straight lines surrounding low-density areas visible on the top panel of Fig. 4.7. Such a dataset aims at visually reproducing the cosmic web with elongated intersecting 1D structures and was used to assess the quality of the detected structures in Aragón-Calvo et al. [2010] or Chen et al.

[2015] but also to train a U-net architecture in Aragon-Calvo [2019]. The bottom panel shows the distribution of edge probabilities for several ratios K b /K. For a large range of values of this ratio, we clearly distinguish two populations of edges: those with high probabilities and those with almost-zero probabilities. In the high probability population, we retrieve, in the three cases, all K -1 edges of the original MST computed over µ plus 27 new ones which correspond to the exact number of closed cycles in the arti cial dataset. These additional edges are shown in bold blue in the top panel of Fig. 4.7. When the ratio K b /K becomes smaller, the high probability mode tends to be centred at lower and lower probability until the pattern is so much altered by the sub-sampling that it is not retrieved and the distribution becomes centred at low probability. The proposed method hence allows to take advantage of the inherent instability of the MST to recover additional edges inducing cycles independently of their scale. It solely and indirectly depends on the edge length required to close the cycle without imposing neither a formal de nition for the cycle nor a hard threshold in edge length.

Even though observed in the example of Fig. 4.7, the retrieval of all the K -1 edges of the MST is not theoretically guaranteed. To ensure it, the resulting set of edges is obtained by the union of all MST edges and those in the high probability mode leading to the non-singular symetric adjacency matrix

(A) ij = max (A MST ) ij , Ā>m ij , (4.15)
with Ā>m the thresholded average adjacency matrix Ā at level m such that it isolates the high probability mode only. By doing so, the proposed graph construction is an extension of the MST containing all its edges with additional ones sharing similar probabilities of appearance during all the random computations.

For the sake of illustrative comparison, we add, as the turquoise blue line of the top panel of Fig. 4.7, the result of the cycling topology as de ned by the framework of persistent homology Edelsbrunner et al. [2002]. In particular, we use the 1D-homologically persistent skeleton (HoPeS) of Kurlin [2015] built as the MST completed with edges that are creating persistent 1D homologies in the dataset. After the extraction of those critical edges, it is not trivial to obtain a persistence threshold that captures the desired cycles. Here, we rely on the bootstrap procedure that, together with the stability of the persistence diagram allow a statistically wellde ned computation of a threshold [Chazal et al., 2018]. We see that some additional edges are creating high-scale cycles due to the noisy datapoints leading to inaccuracies in the representation of the underlying pattern. Even though these undesired cycles could be handled using the sub-level sets analysis of the distance-to-measure function of Chazal et al. [2011Chazal et al. [ , 2018]], it does not allow an easy extraction of a graph on the inputted datapoints as required by our application. A pre-processing of the data by removing points in low-density areas or a careful ne-tuning of the persistence threshold are also possible but this is at odds with the proposed formalism supposed to handle such datapoints. One hint that these spurious branches come from the noisy measurements is that, when varying the randomness of the sampling, we end up with a di erent pattern (plotted as the orange line on the top panel of Fig. 4.7) while the one traced by the proposed average graph remains the same. It is also worth emphasising that, in both realisations of the 1D persistent skeleton, the two small-scale cycles at the center of the dataset are not captured because of the too high persistence threshold imposing a lower-bound on the scale of the detected cycles while they are observed in our average graph prior.

Convergence and time complexity 4.5.1 Convergence analysis

Convergence towards a local maximum and monotonic increase of the log-posterior through iterations are guaranteed by the EM procedure when xing ∀t, G t = G 0 [see McLachlan & Krishnan, 1997, for a detailed analysis of EM convergence properties]. When updating the topology at each iteration and using a prior based on the MST, convergence as well as monotonic increase of the regularised likelihood are still guaranteed [Mao et al., 2017], but it does not remain true for any general graph construction if not based on optimisation procedures such as de ned by Eq. (4.4). The left panel of Fig. 4.8 shows the convergence for the learning of the principal graph from Fig. 4.4. We see that the regularised log-likelihood given by log p(X | Θ) + log p(Θ)2 is getting maximised during the procedure, as theoretically predicted by EM. Since this quantity can be computationally costly to evaluate, we can also rely on the increment in parameter values as a stopping criterion. As an example, the evolution of µ (t) -µ (t-1) 2 is shown in dashed red on the left panel of Fig. 4.8, indicating that nodes of the graph are not signi cantly moving after roughly 50 iterations, which is also where the regularised log-likelihood gets stable.

Time complexity

The E-step equation (4.12) requires the computation of the responsibilities taking O(N DK) operations to complete while the M-step equations ( 4

Runtimes

A Python implementation3 of the proposed Algorithm (1) provides the principal graph from Fig. 4.7 in less than three minutes on a modern laptop with our single-core implementation. Figure 4.8 quanti es the evolution of the runtime for di erent number of input points, with xed K = 1000 nodes in black and the evolution with K for xed input N = 10000 in red. Using the MST prior computed using the Kruskal algorithm applied on the Delaunay Tessellation yields a fast algorithm, able to deal with large-scale datasets, as it can be the case in cosmological applications. The implementation additionally exploits adapted data structures to encode the sparse matrices encountered in the algorithm, like the adjacency or Laplacian matrices A and L making it also e cient from a memory point of view. Besides, it makes use of binary search tree structures [Bentley, 1975] to optimise the computation of responsibilities requiring the evaluation of squared distances x i -µ k 2 2 under our assumption of spherical covariance matrices in the Gaussian mixture model.

Hyper-parameters and initialisation

The impact of parameters

Hyper-parameters of the full model are K and Y = (λ µ , λ σ , λ π ). K denotes the number of Gaussian components used in the mixture model while λ µ , λ σ , λ π are all related with shapes of prior distributions on the parameter indicated as subscripts. 1) for an arti cial dataset made of three branches with linearly evolving standard deviations converging to a spherical Gaussian cluster. In all cases, K = 350 nodes are used with the same random initialisation over X. Black dots are datapoints, coloured lines are the set of edges learned by Algorithm (1) and gray shaded areas correspond to the 1-σ k circles centred on µ k . (top row) From left to right, quadrants corresponds to several values of λ µ , λ σ and λ π . (bottom row) Probability distributions of the impacted parameters. From left to right: edge weights w ij = µ i -µ j 2 , variances of components σ 2 k , and mixing coe cients π k .

E ect of λ µ . λ µ controls the force of smoothness constraint and corresponds to the precision of the prior Gaussian distribution put on edge weights. From Eq. (4.7), and emphasised by the upper left quadrant of Fig. 4.9 showing regularized graphs obtained with increasing values of λ µ from 0 to 200/σ 2 0 , we see that the higher λ µ , the more {µ k } K k=1 are pulling themselves along the graph structure leading to constrain the graph length ij w ij by shortening its extremities. The bottom left panel of Fig. 4.9 shows the nal distribution of edge weights, in other words the distances between linked nodes in the graph, w ij . We clearly observe the mode of the distribution being translated to lower values when λ µ increases, indicating lower and lower distances between connected nodes. E ect of K. At a xed value of λ µ , a large increase of the number of nodes K can induce over-tting. Sticking with a similar extracted pattern would hence require the increase of λ µ . This is for instance illustrated when comparing Fig. 4.4 obtained with K = 100 and Fig. 4.9 with K = 350 and required a larger value of λ µ to provide a similar pattern. One way to reduce the interlink between these two parameters and the dependency of the pattern on K is the growing grammar proposed in Gorban & Zinovyev [2009] and followed-up by Albergante et al. [2020] in which the graph is initialized with a small value of K to then add some nodes iteratively using prede ned rules. It also comes with a way to penalise "complex" graphs by adding a term to the log-likelihood and has the advantage to diminish the computational cost at early iterations. E ect of λ σ . λ σ acts on the shape of the inverse-Gamma prior distribution of Gaussian variances σ 2 k . Equation (4.13) teaches us that, as for other hyper-parameters, when λ σ → 0, the prior is cancelled and variances are updated through the usual EM equation. When λ σ increases, the shape of the inverse-Gamma law is more and more constrained leading σ 2 k to be updated mainly through σ 2

N k and, ∀k, σ 2 k σ 2 0 with σ 2 0 the value used to initialise the algorithm. Eventually, a high enough value for this parameter thus leads to a xed scale version of the algorithm, similar as those presented in Mao et al. [2017] and Bonnaire et al. [2020], as shown in the middle bottom panel of Fig. 4.9 where the distribution of log 10 σ 2 k is more and more centred around log 10 σ 2 0 . As illustrated in the upper middle quadrant of Fig. 4.9, when λ σ increases, the structure shows a smoothly evolving variance along the graph structure with multiple scales described at the same time. This is also highlighted by the lower panel where the variance distributions obtained from regularized graphs with small values of λ σ are spreading over several order of magnitudes and show three distinct modes corresponding to the characteristic scales of the three branches in the dataset. These estimates tend to be more and more biased toward σ 0 when λ σ increases.

E ect of λ π . Finally, λ π is acting on the amplitude π k , corresponding to the proportion of points being represented by the Gaussian component k and is controlling the force of the prior of uniformly distributed Gaussian mixtures. This parameter has a very low impact on the overall result but can avoid singular solutions to happen in very low-density regions paved by graph nodes. The upper right panel of Fig. 4.9 shows the low impact of the parameter despite a large range of tested values, from 0 to 10. Although the obtained distributions of mixing weights π k are clearly di erent, as seen in the lower right panel tending to be centred at (1 -α) /K when λ π increases, the obtained graph structure remain unchanged, so as the local extent measured by the variances.

Even though the impact of these hyper-parameters is interpretable and that there is a wide range of values providing similar results, the "quality" of the obtained graph still depends on the settings, and, to the best of our knowledge, there is no well-de ned method to choose regularisation parameters independently from user tests or external information. In that regard, the theoretical work of Gerber & Whitaker [2013] in which is proposed a modi cation of the L 2 -form of the objective function of the principal curve formulation. By minimising this new cost, from which admissible principal curves are minima, they are freed from any regularisation parameter and can hence be of interest to alleviate the complexity of the model selection process.

Initialisation Initialisation rules

There are four sets of parameters to initialise, namely {µ k } K k=1 , {σ k } K k=1 , {π k } K k=1 and α. A simple and direct strategy to initialise positions of Gaussian components is to choose µ (0) = X. By doing so, we ensure that the observed point cloud distribution is well paved by centres. Note however that, from Sect. 4.5, when K N , the complexity scales with N 3 . For large datasets, it may be interesting to rst reduce the complexity by initialising the model with K N using for instance sub-samplings, noise reduction techniques or simple clustering methods like the K-Means or ducial GMM algorithms as the case K = N will generally produce more clusters than needed to pave the dataset. When no prior knowledge on the local size of structures, variances can be initialized as ∀k ∈ {1, . . . , K}, σ

(0) k = σ (0) 0 . In this context, σ (0)
0 can be chosen as a prior guess on the average size of structures or through rules borrowed from density estimation methods [Heidenreich et al., 2013]. In practice, we can choose σ (0) 0 using a modi ed version of the Silverman's rule [Silverman, 1986] as

σ (0) 0 = A 0 [N (D + 2)] -1 D+4 σ min , (4.16)
where A 0 is a constant, N is the number of datapoints, D is the dimension of the data and σ min is the minimum standard deviation over all directions. Taking A 0 = 1 leads to the Silverman's rule and is the optimal estimate for an underlying Gaussian distribution. As argued by Chen et al. [2015], when the data are not Gaussian anymore, A 0 should be optimised as a free parameter.

The α parameter is the evaluation of the level of outliers in the dataset that should not be paved by the Gaussian components. Its value depends on the application and data at hand. In our experiments, we x α (0) = 0.10 and then let it adjust itself during the learning. As an example, results from Fig. 4.9 were obtained by starting with this guess and then quickly converges towards a value numerically indistinguishable from 0. Finally, mixing coe cients are initially assumed to be uniformly distributed and ∀k ∈ {1, . . . , K}, π

(0) k = 1 -α (0) /K.

Impact of the initialisation on the learning

Because the proposed algorithm relies on the EM procedure to maximise the log-posterior, it shares the same drawbacks, namely the convergence towards a local maximum, as exposed in Sect. 3.3. This makes, in principle, the full optimisation scheme dependant on the initialisation of Θ. One way to obtain a graph independant from the initialisation is to use the prescriptions from statistical physics established in Sect. 3.4. As already stated in Chapter 3, such simulated annealing approaches were introduced to track the global maximum of the loglikelihood [Kirkpatrick et al., 1983;Ueda & Nakano, 1998]. This solution however comes with a sizeable increase of the computational cost and, in all our experiments, did not signi cantly improve the quality of the extracted pattern.

Illustrative application: Road network

In Chapter 5, we shall see the application of the proposed method for the detection of the lamentary pattern of the cosmic web based on halos or galaxies. In this section, the context of road network detection from noisy GPS measurement is used as a showcase. The detection of roads is usually carried on images acquired by satellites, a task for which many algorithms have been proposed [see for instance Merlet & Zerubia, 1996;Stoica et al., 2004;Wang et al., 2016]. However, with the expansion of GPS technologies and their integration in smartphones, the update of maps obtained by satellite imaging techniques became possible at small cost. One of the objective of map reconstruction methods is to produce street maps from a set of sampled positions collected by travelling vehicles (like taxis or volunteers). Reaching this goal requires sophisticated algorithms to be deployed to handle the complex patterns observed, the high level of noise and outliers present in these datasets but also the uneven sampling of several trajectories. Main roads are indeed traced by hundreds of trajectories, while some low-level ones are traced only by a few, as illustrated in the top left panel of Fig. 4.10.

When applying our Algorithm (1) with the average graph prior on the 67 193 datapoints Berlin dataset, we obtain the graph from the top right panel of Fig. 4.10. Note that no preprocessing of the dataset nor post-processing of the resulting graph structure have been carried out. We see that the principal graph could use some topological re nements to make The graph was obtained using the average graph topology with K = 7300 and σ 0 = 0.003, Y = (10/σ 2 0 , 5, 1). Red circles highlight features that could not be caught by a non-cycling topology. (bottom left) Same as for the top panel, but overplotted on the ground truth map extracted from OpenStreetMap. (bottom right) Probability distributions of variances for nodes in the corresponding circled coloured regions of the top panel. it look smoother, removing waving branches and closing some of the remaining dangling branches. For more details about particular processing of inputs and outputs in the context of graph structure learning for road networks, see [START_REF] Huang | [END_REF]. The obtained graph passes through the entire dataset, correctly paving regions with both high and low densities. We can also observe that the proposed algorithm recovers a large variety of cycle sizes, from very small ones, as for instance in the red shaded region with intersections or roundabouts, but also larger ones, as in the top-left corner. When comparing it with the ground truth map obtained from OpenStreetMap4 in the bottom left panel, we clearly distinguish that some roads are not well-traced with several datapoints completely out with respect to the road map due to the poor quality of sampled data, as it is the case in the top right corner. However, the algorithm succeeds in proposing a robust version that do not take into account all such datapoints.

The proposed algorithm o ers the possibility to estimate the local variance around the inferred principal graph. In the present dataset for instance, it is possible to investigate further small portions of roads where the estimated σ (t) k is high or has a broad distribution to spot roads that are either wider or noisier than others. The bottom right panel of Fig. 4.10 reports the distribution of variances for nodes standing in rectangular regions of the top panel. When using variances of Gaussian clusters as a proxy of the road width, we conclude that main roads with multiple lanes like those in the green or purple rectangles have larger sampling standard deviation, by up to a factor of two, than the low-level road in the blue rectangle. These estimates of road widths however, depend on the quality of the sampling and some roads can appear wider than they actually are because of spurious points arti cially increasing the estimate locally or at the extremities of some branches of the graph. This can be seen when inspecting the variance of the σ (t) k distributions where the purple one is much larger than the others because of the local noise in the sampling. Finally, the proposed cycling graph captures correctly some topological features of the road network, such as the most prominent roundabouts or road intersections highlighted in shaded reddish areas on the two top panels of Fig. 4.10. The representation of such features would not be permitted by tree-based topologies.

Summary and prospects

In the present chapter, we merged concepts from mixture models and graph theory to build a new framework for principal curve estimation that alleviates some drawbacks of the existing de nitions.

In summary, the proposed formulation:

1. Provides a natural connection of the points through a graph structure, as opposed to several ridge nders, that only results in a set of projected points stands linked a posteriori such as in the SCMS algorithm.

2. Embeds a built-in robustness to outliers of the pattern. Most of the algorithms require a pre-processing step [Stanford & Raftery, 2000;Ozertem & Erdogmus, 2011] to remove outliers exhibited by real-world datasets. Here, this di culty is circumvented by directly handling outliers in the mixture model through an added uniform component.

3. Is able to learn the local size of the one-dimensional structure through the variance of the Gaussian clusters making it a heteroscedastic version of the SimplePPT algorithm proposed in Mao et al. [2015Mao et al. [ , 2017]]. In particular, we showed that the same update equations for nodes can be derived as a particular case of the proposed formalism.

4. Can work with any prior topology given by a graph construction. We additionally propose a graph that includes cycles of various scales and compare it with the mathematically well-de ned framework of persistent homology [Edelsbrunner et al., 2002;Kurlin, 2015].

5. Relies on simple, fast and well-established procedures such as the Kruskal and EM algorithms making it suitable for handling a large number of datapoints.

By combining mixture models and graph theory, we have been able to take advantage of the probabilistic formulation to get a description of the local size of the sampling around the approximated one-dimensional manifold. One of the interesting perspectives of this work will be to extend the formalism for the detection of d-dimensional structures in R >d datasets with for instance applications to the detection of walls in the cosmic web that are 2D structures embedded in a 3D space. Leads of extensions in that direction may be found in the use of more general graph topologies, like those based on k-nearest-neighbours, and by relaxing the spherical assumption for Gaussian clusters used in the present version of the algorithm. These extensions may also trigger the need to adapt the cost function to constrain the full surface/volume of the data rather than solely the nodes of the graph.

The overall optimisation scheme of the proposed algorithm may be another avenue of future work. We have shown that it exhibits a stable behaviour for a large range of model's hyper-parameters and initialisation giving freedom to the user to retrieve the underlying patterns without requiring a cautious ne-tuning. The current optimisation, based on the tuning of free regularisation parameters, makes the extraction of a satisfactory pattern dependent on multiple trials and heuristics. Recent literature [Fischetti & Stringher, 2019] propose ways of alleviating these limitations by resorting to simulated annealing (also discussed in 4.3.1).

Finally, the proposed algorithm was devised for the speci c task of detecting the lamentary pattern of the cosmic web. Even though these data show a high level of complexity with di erent local sampling densities, loops of various sizes, features of di erent scales, the proposed algorithm requires no pre-processing to remove spurious measurements and provides a global description of the pattern including properties such as local length or width. In Chapter 5, we will see how this algorithm was compared to other web-nder methods and used to build a graph representation of the cosmic web through the distribution of galaxies in simulations.

Part III

Analysis of the Cosmic Web pattern

Chapter 5. In this chapter, we present how the algorithm developed in Chapter 4 can be used for the identi cation of the principal graph of the cosmic web. It partly exposes results from Bonnaire et al. [2020] and some from Gouin et al. [2021]. We show how the graph can be post-processed to identify the lamentary pattern as a whole together with an estimate of the positional uncertainty of the spine, but also to extract individual laments and their properties for carrying statistical analyses. We apply and compare the laments obtained from the galaxies of several large-scale hydrodynamical simulations, namely EAGLE, IllustrisTNG and Magneticum and then focus on the role of laments in shaping galaxy clusters in simulations.

The principal graph of the Cosmic Web

Context and motivations

We have stressed in Sect. 2.3 the primordial role of the cosmic web lamentary structure and Sect. 2.4 focused on the inherent challenges of carrying out an accurate detection of cosmic web elements. The sparse sampling of galaxies in observation is also an additional di culty that comes with its own systematic e ects and limitations (like the completeness and spatial coverage of the survey). It is however essential to be able to extract cosmic web environments in both data and simulations to get a better understanding of how the large-scale structures formed and evolved through time. Because they are easier to detect, the nodes were the rst to draw researchers' attention which led to a rather clear picture of their average properties like galaxy content, density pro les and gas content based on several catalogues [e.g. Abell et al., 1989;Ryko et al., 2014] but also on their information content in terms of cosmological parameters [Yoshida et al., 2000;Peter et al., 2013]. Filaments on the other hand received an attention only more lately, permitted by the evolution of wide spectroscopic surveys [such as York et al., 2000;Colless et al., 2001] and by the elaboration of sophisticated tools to identify components other than nodes in the cosmic web. Filaments, acting like cosmic highways linking together large overdense nodes, play a key role in the dynamics of the Universe but also in shaping the evolution of galaxies that has been studied in both data and simulations. While the local density-mass relation between a tracer of the matter distribution and its environment is now well-established, making galaxies (or halos in simulations) more massive in dense environments like clusters [Dressler, 1980;Kau mann et al., 2004;Baldry et al., 2006], the relationship between other properties of both tracers and large-scale structures is still under investigation. Many recent studies for instance point out that galaxies are less e cient in forming stars in the core of laments than in their periphery [Alpaslan et al., 2016;Malavasi et al., 2017;Kraljic et al., 2018;Bonjean et al., 2020] suggesting a particular process occurring when a galaxy enters a lament. This result also matches a local density in uence as laments have been shown denser in their inner regions [Bonjean et al., 2020;Galárraga-Espinosa et al., 2020]. Beyond density, the local orientation of laments has also an impact on the spin of galaxies reported by many works in both data and simulations [Hahn et al., 2007;Codis et al., 2012Codis et al., , 2015;;Ganeshaiah Veena et al., 2018, 2019;Kraljic et al., 2020].

All these observations and conclusions allowing to push forward our understanding of the complex statistical interplay between matter tracers, like galaxies, and the large-scale structures predominantly made of nodes and laments have only been possible thanks to the detection of such objects in the galaxy distribution [e.g. Abell et al., 1989;Ryko et al., 2014;Tempel et al., 2014;Chen et al., 2016;Rost et al., 2020;Malavasi et al., 2020b;Duque et al., 2021]. It is however crucial, given the di erent results provided by many methods to identify laments [Libeskind et al., 2017], to corroborate the correlations observed in various datasets and by several methodologies in order to avoid any bias induced by the de nition of laments and by the choice of the algorithm.

In this chapter, we rst present T-ReX, a lament detection method based on the mathematical formulation exposed in Chapter 4. In the cosmological context of the detection of the lamentary pattern drawn from subhalos or galaxies, we begin by extracting the overall spine of the lamentary structure together with its spatial uncertainty. We assess the robustness of the method to sparse sampling, in addition to provide comparisons with other web-nders applied on same datasets. We then explain how individual laments can be de ned from the learnt principal graph to build catalogues gathering their positions with some geometrical properties like their lengths, curvatures, local and averaged widths. The proposed algorithm is also able to associate to each input datapoint (galaxy or halo) a probability to reside in a given lament. This association is successfully compared to the one provided by a physical rather than geometrical web nder (see Sect. 2.4.2) applied on the full dark matter distribution. T-ReX is able to retrieve 80% of the galaxies as belonging similarly to laments, even though with a much smaller number density of tracers as input. We then carry out a comparison of the properties of laments identi ed individually from di erent samples of galaxies in several simulations that are EAGLE, IllustrisTNG and Magneticum. Finally, T-ReX is applied to the large-scale simulation of galaxies IllustrisTNG and is used to study the interplay between the nodes of the cosmic web and their surrounding lamentary pattern. In particular, we focus on the correlations between the connectivity of galaxy clusters and their physical and dynamical properties.

Filamentary pattern detection

In this rst section, we focus on the detection of the lamentary pattern as a whole and explain how the algorithm presented in Chapter 4 can be used through multiple realisations based on random sub-samplings of the input dataset to obtain a picture of the lamentary structure of a dataset together with a spatial uncertainty.

T-ReX: Tree-based Ridge eXtractor

The minimum spanning tree (MST) is a graph with a tree topology that is linking all the input points together with the minimum total distance (see Sect. 4.2 for a more formal introduction). It has a long history in cosmology and has been the rst method to exhibit the lamentary structure of the cosmic web in early galaxy distributions mapped in the sky by Barrow et al. [1985]. Among its most appealing features are its unicity, the absence of free-parameter, its scale-invariance but also its easiness to compute since it relies on fast and well-established methods like the Kruskal algorithm. All these features make the MST a well-suited tool for the study of the large-scale distribution of matter in the Universe as proposed by numerous papers on the topic [Pearson & Coles, 1995;Bhavsar & Splinter, 1996;Colberg, 2007;Park & Lee, 2009;Alpaslan et al., 2014bAlpaslan et al., ,a, 2016;;Naidoo et al., 2020;Bonnaire et al., 2020;Pereyra et al., 2020b,a]. However, one of the main drawbacks of the MST formulation is that, by linking all datapoints, it results in a locally spiky geometry, as can be seen in the right panel of Fig. 5.1, that several works recently propose to alleviate. Pereyra et al. [2020b] for instance used a post-processing of the MST in which an ad hoc selection of tree branches are smoothed by means of a B-spline interpolation while Bonnaire et al. [2020Bonnaire et al. [ , 2021b] ] and related Chapter 4 present a way of incorporating the MST in an optimisation scheme embedding the graph as a prior. In the cosmological context, the algorithm established in Chapter 4 is entitled T-ReX standing for "Tree-based Ridge eXtractor". Let us recall the main components of the model, the optimisation problem solved by the algorithm and how these can be interpreted in the context of cosmological datasets.

In the mixture model framework, the probability that a matter tracer stands at a position x i is given by

p(x i | Θ) = K k=1 π k N (x i , θ k ) + αρ(x i ), (5.1)
where Θ is the set of model's parameters, N (x i , θ k ) is the Gaussian probability density function centred on x i -µ k with variance σ 2 k , ρ(x i ) is the uniform distribution over the convex hull of the point-cloud distribution. The rst term of this equation models the graph nodes standing on the lamentary pattern, paving the distribution while the second one handles datapoints that are not part of it, standing outside the pattern, in voids or walls embodied by a uniform distribution. We have seen in Sect. 4.3 that a double maximisation procedure can be used to estimate the smooth graph solving in particular To summarise, the T-ReX algorithm keeps the fundamental idea of the MST-based approach by representing the topology of the cosmic web as an interconnected network based on discrete representation of tracers but provides additional features not allowed by previous de nitions that are: (i) a smooth version of the MST incorporated in the core of the formalism; (ii) a handling of outliers to represent datapoints outside the lamentary pattern; (iii) a description 1 Additional terms include the priors on uniform weights with strength λ π , smooth evolution of variances along the graph with strength λ σ , uniform background noise and weights π k . 2 In the sense of the local maximum a posteriori.

Θ (t+1) = argmax Θ - N i=1 K k=1 p ik x i -µ k 2 2 σ 2 k data delity term -λ µ K i=1 K j=1 a ij µ i -µ j 2
of the local width of laments through variances σ 2 k learnt during the optimisation; and (iv) the probabilistic association of tracers to individual laments (see Sect. 5.3.3 for a more detailed presentation of this feature).

To initialise graph nodes in the algorithm, we use in the entire section a pruned version of the initial MST using all the input datapoints. It consists in a simple denoising operation cutting all the nodes standing in branches of the tree at a level l. Strictly speaking, we iteratively remove all nodes of degree one in the graph structure, similarly to Barrow et al. [1985] or Colberg et al. [2005]. By doing so, we remove the most spurious parts of the structure corresponding to nodes that are more likely to be found in physically irrelevant regions for the underlying pattern (i.e. underdense regions). This approach is iterative, meaning that a pruning at level l removes iteratively nodes of degree 1 a total of l times. To give a representative image of this procedure, it acts like the iterative peeling of an onion, attributing to each node a depth in terms of layers to peel before we reach it and starting from extremities [Hébert-Dufresne et al., 2016]. Note that, even though we prune the tree to initialise the number K and positions µ (0) of graph nodes, they are still used in the full optimisation and contribute to the shaping of the nal smooth version of the graph.

Previous MST-based methods usually perform, in addition to this pruning, a removal of all edges above a given physical length. In our case, this operation is not applied to avoid the introduction of a new parameter that is not easy to tune, but also because we argue that all connections, even 'long' ones, can provide information about the underlying structure. Of course, as a result, if two unconnected parts of a network are given as an input to the presented method, they will end up connected.

In summary, the T-ReX method estimates a smooth MST and relies on the set of parameters that are: l the pruning level to initialise the graph which implicitly de nes K the number of nodes and their initial position, Y = (λ µ , λ σ , λ π ) the prior strengths and σ (0) k , the standard deviations of Gaussian graph nodes3 .

Filamentary pattern extraction from Illustris subhalos

In the rest of the section, we make use of the Illustris simulation outputs 4 [Vogelsberger et al., 2014]. It consists in a set of large-scale hydrodynamical simulations with di erent resolutions in which an initial set of particles distributed over a 75 Mpc/h box is evolved forward in time from high redshift to z = 0. From the resulting distribution at z = 0, halos of dark matter are identi ed using a Friend-of-Friend algorithm [FoF, Davis et al., 1985]. To mimic a galaxy survey, we consider structures inside halos, the subhalos, identi ed with the Sub nd algorithm [Springel et al., 2008] and provided within the Illustris package. In the left panel of Fig. 5.1 is shown a thin 5 Mpc/h slice of the subhalo distribution obtained from the Illustris-3 simulation.

As previously mentioned, the MST highlights one particular path linking datapoints together but does not provide any idea of its uncertainty or reliability. It is also restricted by its tree topology that has no loops and cannot represent holes but only connected components in the cosmic web. Both of these issues can be overcome by introducing a robust representation that takes into account the variations in the input distribution. To do so, we build B di erent samples {X b } B b=1 from the initial one X and compute the regularised MST for each of them in a similar fashion as in bootstrap approaches, also evoked in Sect. 4.4.2.

From the B realisations of regularised MSTs, one can construct a map I characterising the probability, in a frequentist meaning, of a position x g on a grid to be crossed by a realisation of a tree:

i(x) = 1 B B b=1 1 H b (xg)=1 , (5.3)
where 1 A is the indicator function and H b is the binary histogram obtained from the graph nodes µ b . The random nature of I thus comes from the random sub-sampling of X and not from Algorithm 1 that is a deterministic optimisation step.

The left panel of Fig. 5.2 shows a probability map obtained from a 5 Mpc/h depth slice in which the intensity of each pixel corresponds to the frequency that an edge of the MST crossed it. This way, we quantify the reliability of the various paths in the input domain. In practice, to build I(x g ), we use both the graph nodes µ b and the set of edges linking vertices encoded that contains information on the paths used and consequently should be taken into account in the nal distribution. Edges are thus sampled and counted in the computation of H b for Eq. ( 5.

3).

In what follows, we may refer to a quantity called the superlevel set of those maps de ned as Γ p (I) = {x g | i(x g ) ≥ p}. Those sets are used to threshold the probability maps and keep only regions with a probability higher than p. We can see, in the right panel of Fig. 5.2, that the highly probable regions of the map are tting what one would expect for the underlying distribution while the overlap of the superlevel set Γ 0.25 (I) with the DM distribution allows us to see that high probability paths (above 0.25 in this case) are tracing the most prominent part of the network. It is worth noting that the agreement is particularly interesting given that the input of the algorithm are subhalos and not DM particles. The zoomed-in region clearly shows that small scales are also recovered where high probability paths follow the ridge in the DM distribution.

Performance evaluation

Robustness to sparse samplings

In order to assess the robustness of the method against the datapoint sampling density used for the detection of ridges, we reduce the number of subhalos in the initial dataset by keeping only those with a mass M ≥ M cut . In practice, we investigate how the original lamentary map is spatially close to the recovered ones when M cut varies. Figure 5.3 shows probability maps obtained for increasing values of M cut leading to sparser and sparser input, namely 100%, 83%, 60%, 31% and 10% of the initial subhalos in the slice respectively corresponding to M cut = {0, 0.85, 1.35, 3.22, 11} × 10 10 M /h. Visually, probability maps show a nice stability, even when the sparsity is high: patterns are pretty much the same when we keep at least 60% of the most massive objects hence recovering the essential part of the structure. In Fig. 5.4 is exhibited the spatial proximity between the di erent maps by representing, for each I J , where J denotes the fraction of galaxies we kept to compute the map, the cumulative distribution of {d J

x } x∈Γ 0.25 (I 100 ) de ned, for a position x in the set Γ 0.25 (I 100 ), as

d J x = min
x ∈Γ 0.25 (I J )

xx 2 .

(5.4) Hence d J x corresponds to the closest distance from a position x in the original skeleton obtained by keeping all subhalos, namely Γ 0.25 (I 100 ), to a given thresholded map Γ 0.25 (I J ). This way, the distribution of d J

x measures how far the original pattern is from the one obtained with J% of the datapoints.

In more than 95% of the cases, the original pattern nds a closest point in the 83% and the 60% maps at less than 1.8 Mpc/h, showing that structures found in the three maps are spatially close and about the thickness of typical laments [Cautun et al., 2014]. When M cut increases, the lamentary pattern traces the most prominent parts of the structure with a loss of some small scales and hence highlights coarser and coarser structures. Even though the pattern is rough with only 31% of the datapoints used, we still observe a correlation with previous maps highlighting coherent structures with 90% of the original pattern being retrieved at less than 3 Mpc/h. As expected, an unrealistic scenario where we use only 10% of the datapoints associated with the most massive subhalos degrades the reconstruction of the lamentary pattern. Yet, the recovered structures show a coarse but coherent connectivity between regions. This illustrates the ability of T-ReX to recover the underlying structure with high stability with respect to missing information in the input distribution of datapoints. x } between positions of the binary maps Γ 0.25 (I 100 ) obtained with increasing mass threshold M cut to the one with J% of the datapoints. M cut = {0.85, 1.35, 3.22, 11} × 10 10 M /h leading respectively to 83%, 60%, 31% and 10% of the total number of subhalos in the slice.

Comparison with other algorithms

In this section, we apply T-ReX on the 3D distribution of halos obtained from a 200 Mpc/h Gadget-2 simulation5 and compare our results with some other existing procedures. These have also been run on the same dataset in a review by Libeskind et al. [2017] to propose a comparison of the main existing procedures to classify elements of the cosmic web using either dark matter particles or halos as inputs. Although the review considers a dozen of di erent methods, we focus the comparison on three procedures, namely Nexus+, DisPerSE and Bisous, so that we have a broad set of di erent approaches using respectively scale-space representation, topological considerations, or stochastic approach to recover the lamentary pattern. Nexus+ [Cautun et al., 2013] is a classi cation algorithm inspired by image processing and based on ltering techniques leading to state-of-the-art environment classi cation able to identify clusters, laments and walls. The main idea is to assume that the local morphology of the density eld fully encodes the environmental information. Eigenvalues of the Hessian of the density eld are thus used to compute an environmental signature in each voxel of the smoothed eld. The key idea is to compute this signature for a set of smoothed elds with a log-Gaussian lter over a range of di erent scales to highlight structures of di erent sizes. Physically motivated criteria are then used to threshold signature values and attribute a classi cation to each volume element. Bisous [Stoica et al., 2007] is a publicly available 6 stochastic method based on halo positions that identi es the lamentary structure using a set of random parametric cylinders. Filaments are modelled as aligned and contiguous small cylinders of a given size in the galaxy distribution. The Bisous model generates two maps allowing to extract laments spine; one characterising the probability to nd a lament at a given position called the visit map and an other one corresponding to the lament orientation eld. This way, spines are de ned as dense regions and are aligned with the axis of the di erent cylinders.

Note that not only these methods have very di erent mathematical de nitions for what they all call clusters, laments, and walls, but they also have been run with di erent input, using either DM particles or halos. We applied T-ReX to the full halo distribution of the 3D simulated box (281 465 halos in total) and built a 100 × 100 × 100 grid map like other methods. For T-ReX, this means that the nal probability map is computed over a 100 3 grid in which all visited voxels are considered part of the lamentary structure. As T-ReX is using 1D objects (segments of the smooth graph) sampled over the input space, it is preferable, for illustration and comparison, to give its lamentary pattern a 'thickness' by smoothing the obtained probability map. Whenever a voxel is classi ed as part of the lamentary structure, a smoothing is thus performed over its 26 direct neighbours. In what follows, we call this version T-ReX s while the original result is referred to as T-ReX us .

For illustration, following Libeskind et al.

[2017], we show in Fig. 5.5 the results of the classi cation provided by each method for a 2 Mpc/h depth slice from which FoF halos were extracted (top left panel of Fig. 5.5). We note that all methods have been run over the full 3D cube and this is a projected slice of the detection. It is also worth noting that T-ReX identi es the lamentary pattern as a whole and does not classify the environment into clusters, laments and walls as Nexus+ and DisPerSE do. To perform the comparison, we must look at the full pattern provided by each method and compare it with our extracted skeleton. We observe that T-ReX provides a satisfactory connectivity of the halos through the slice. In its smoothed version, it leads to thicker laments compared to the results of Nexus+ and Bisous but thinner ones than Disperse, and retrieves most of the structures ( laments, walls, and clusters) obtained by the Nexus+ algorithm.

Even though these methods have been developed with di erent approaches, it is interesting to see whether they agree or not in the detection of the lamentary pattern. To do so in a quantitative way, we could use the proximity measurement of Eq. (5.4) but as the resulting patterns are presented on a 2 Mpc/h grid, the distance between them would not be accurate. Hence, we introduce a similarity measurement as follows: considering the answers provided by two detection methods, H 1 and H 2 , such that H • (x) = 1 if the cell centred at x g is part of the lamentary structure and 0 otherwise, the similarity measurement is de ned as: Of course, such a simple metric does not provide the full information on the similarity between the considered patterns. This measure must then be used in tandem with others, or with visual inspection, as we have done here. Table 5.1 shows the similarity indices between all considered methods for the entire 3D cube. We observe that 85% 5.1) but not reciprocally (last column). This is due to the sparse and unconnected detection provided by the Bisous method. The thick skeleton of DisPerSE also tends to contain a large fraction of other skeletons (fourth column of Table 5.1) but it lls so much volume that it is not contained in the latter (fourth line).

S(H 1 , H 2 ) = |H 1 ∩ H 2 | |H 1 | , ( 5

Identi cation of individual laments

In the previous section, we have shown how the detection of the overall lamentary structure can be done with the graph-based algorithm and that the identi cation agrees with some other de nitions. The gridded output representing the probability of a cell belonging to the pattern can be of interest for using it as a mask in large scale cross-correlation analyses. However, it is not trivial to de ne individual objects that are laments from such an output. One way to achieve this is to use and post-process the principal graph learnt from the application of the T-ReX algorithm on the full dataset made of the N tracers at positions X = {x i } N i=1 .

A graph-based de nition for laments

In a graph, one can associate to each node i a degree deg(i) corresponding to the number of direct neighbours (see Sect. 4.2). Naturally, three types of nodes can be distinguished: i) Extremity nodes with deg(i) = 1; ii) Junction nodes with deg(i) = 2; and iii) Bifurcation nodes with deg(i) > 2. One simple way to de ne laments based on the MST is hence to use branches of the graph, a branch being the set of connected edges linking an extremity node to a bifurcation or a bifurcation to a bifurcation as illustrated in the left panel of Fig 5.6.

Characteristics of individual laments

At this stage, we consider an individual lament as a set of M nodes which are actually Gaussian components and a subset of the K centres µ. Centres composing a lament are found at positions Euclidean distance w ij = f i -f j 2 . Let us also de ne R the ridge of the lament corresponding to the one dimensional piecewise linear line connecting nodes. Filaments characteristics that can be extracted from the graph are listed and de ned below.

F = {f m } m=1...M with F ∈ R M ×D , a
Geodesic length. The geodesic length L of the lament is de ned as the sum of all edge weights forming the ridge,

L = M i,j=1
w ij .

(5.6)

Inspecting the left panel of Fig. 5.6, the length of a lament is for instance given by summing all edges of same colour.

Curvature. The simple proxy we use to describe the shape of the lament ridge is how much the geodesic length di er from the Euclidean length. The curvature is thus de ned as

γ = 1 - f 1 -f M 2 L , (5.7)
with f 1 and f M the two extremities of the branch, i.e. a node of degree 1 or higher than 2, as illustrated in left panel of Fig. 5.6. Consequently, γ ∈ [0; 1] and the closest it is to one, the more the lament deviates from a straight line.

Local width. The local width of a lament is de ned by the set of variances of the Gaussian component composing the graph nodes paving its ridge, {σ 2 k } M k=1 . A one-point summary of this distribution can be the average of standard deviations leading to an estimate of the average radius of the lament, noted

r = M m=1 3σ m /M, (5.8)
such that locally, the radius is chosen to be represented by three times the standard deviation of the Gaussian cluster paving the ridge. Note that this de nition of radius for lament takes into account the spatial variations in the extension of the lament through the local variances of Gaussian graph nodes and is not a xed-length de nition.

Radial distance. It is possible to de ne for each galaxy in the catalog a projected distance to the ridge R of the lament. This distance is what we call the radial distance and corresponds, for a datapoint at position x i , to

∆ i = d(x i , R) = min x ∈R x i -x 2 .
(5.9)

This measurement can be used to provide either the full distribution of datapoint distances around the lament or as a proxy to the radial extension of the lament using the mean of the distribution, that we note ∆, alternative to the previous de nition of r.

Positional uncertainty. In Sect. 5.2.2, we used multiple realisations of the algorithm to obtain an estimate of the positional uncertainty of the ridge. We can nonetheless still obtain an idea of the uncertainty at the graph nodes level. Assuming B regularised graph realisations, we can associate to each node k of the full graph an uncertainty in the form of a con dence sphere B(f k , h k ) centred at position f k and with radius h k , similarly to what was done in Chen et al. [2015]. Radii of the uncertainty spheres are provided by the mean projected distance of a given node k to the ridge for each bootstrap realisation that we can write

h 2 k = 1 B B b=1 d 2 (f k , R b ).
(5.10)

It is noteworthy to specify that, in practice, this step is realised before any lament extraction and is independant from it, operating on the full lamentary pattern. Thus, in Eq. (5.10), the k index refers to any node in the graph and R b to the full 1d ridge of the lamentary pattern of the realisation b. Hence, the positional uncertainty of graph nodes can be de ned as the union of all uncertainty spheres

U (p) = K k=1 B(f k , ph k ), (5.11)
where p indicates the level of the uncertainty band, similarly to a p-sigma uncertainty.

Association of galaxies

T-ReX provides a probabilistic version of the regularised graph in which each node of the lament ridge is actually a Gaussian cluster with mean µ k and variance σ 2 k . It is also equipped with a robustness to outliers handled by an added uniform background distribution. In the EM procedure, we compute what is called the responsibility p ik (see more details in Sect. and Sect. 4.3.1) characterising the probability that a given tracer x i is drawn from a particular component of the mixture model, hence made of K Gaussian clusters and one uniform background component. Mathematically, we recall that we can write

p bkg i = p(z i = K + 1 | x i , Θ) = αρ(x i ) K j=1 π j N (x i | θ j ) + αρ(x i ) , p ik = p(z i = k | x i , Θ) = π k N (x i | θ k ) K j=1 π j N (x i | θ j ) + αρ(x i )
, where θ j = µ j , σ 2 j encodes the parameter of the Gaussian indexed j and z i is the latent variable encoding the cluster attribution of datapoint x i . What hence interests us is to nd the most probable value of this assignation for each datapoint. First, a datapoint can be attributed either to the overall lamentary pattern or to the background, namely if K k=1 p ik > p bkg i (resp. <), the tracer i has a higher probability of being generated by the Gaussian components (i.e. the laments) than the background noise. We have seen that a lament can be de ned as the set of graph nodes linking two extremities or bifurcations. A datapoint belonging to the lamentary structure can then be uniquely associated to the lament hosting the graph node maximising the probability p ik over all nodes, namely ẑi = argmax p ik . For a given lament, we can thus derive a set of galaxies associated to graph nodes that are forming the ridge R. Such points are shown as coloured datapoints in the right panel of Fig. 5.6 while those associated with the background are in grey.

Until now, the procedure was focused on associating a datapoint either to a lament or to the background. However, the graph can also be used to investigate a possible de nition for nodes. These latter are de ned in the cosmic web as dense regions linked together by laments. In the graph, these may be represented as dense bifurcations, meeting points of several branches. To investigate further this hypothesis and to assess the overall environment classication scheme, we perform a comparison with a physical web nder, Nexus+ [Cautun et al., 2013], applied on the EAGLE hydrodynamical simulation [Schaye et al., 2015]. The Nexus+ algorithm was run (Marius Cautun, private communication) on the dark matter density eld and the classi cation was then propagated at the level of galaxies depending on the cell environment (node, lament, wall or void) they belong to. On the other side, T-ReX was run using as an input the set of galaxies with stellar mass M > 107 M which yields a total of N gal = 142 392 galaxies in the L = 100 Gpc length box of EAGLE. We then extracted the laments and bifurcations in the regularised graph computed with parameters l = 25, λ µ = 100, λ σ = 5 and λ π = 1. Note that the high value of λ µ is required given the low stellar mass threshold used to trace the pattern with which we expect to nd a large number of galaxies in walls and voids. We hence use a high pulling prior to avoid the detection of spurious laments, even though it may prevent the algorithm from nding the most tenuous ones.

The bifurcations in the graph are de ned as nodes with degree ≥ 3 and we additionally compute a local density for each of them de ned as the number of galaxies residing in the 3σ k region around their centre, with σ k the learnt variance of the bifurcation divided by the enclosed volume of 4π (3σ k ) 3 /3. This results in a total of 285 bifurcations with various densities n bif . We can expect that only dense bifurcations trace nodes of the cosmic web. De ning the mean number density of galaxies ng = N gal /L 3 0.143, we explore three density thresholds factors of ng , one keeping almost all bifurcations, even low dense ones, one intermediate and one very restrictive keeping only 10% of them, respectively given by 5n g , 30n g and 100n g . The obtained classi cations compared to the Nexus+ ones are shown in Table 5.2. We see that for a low value of the number density threshold (i.e. keeping almost all bifurcations), we retrieve most of the galaxies classi ed as belonging to nodes by Nexus+ as well (99.4%) but with a 26.5% contamination of galaxies standing in laments according to the Nexus+ algorithm. These values tend to both decrease when the density threshold increases. These results illustrate the ability of T-ReX to identify nodes and galaxies residing in them as bifurcations in the graph structure. The lament classi cation in T-ReX is the result of galaxies satisfying the criterion K k=1 p ik > p bkg i , meaning that they have a higher probability of being part of the lamentary pattern than being in the uniform background, and that are not already classi ed as being in nodes, with n bif > 30n g . We see that with this de nition, we retrieve 70.2% of the galaxies classi ed as in laments by Nexus+ and 34.3% of galaxies in walls. The missed galaxies in laments can be explained by the high value of λ µ hence leading to a graph where the tenuous laments are missing because they are not well-traced by galaxies, but also by the density threshold for nodes galaxies in which 13% of them are classi ed in nodes. Exploring further the walls contamination shows that 74% (resp. 95%) of the galaxies identi ed by T-ReX in laments are actually standing at less than 0.5 Mpc (resp. 2 Mpc) from a galaxy tagged as lament by Nexus+. It means that the "misclassi cation" 7 is only locally induced by the borders of walls close to laments which are not precisely detected by T-ReX when using the galaxies.

We conclude that both methods agree well in classifying the galaxies in the di erent environments, even though T-ReX do not, by construction, distinguish between voids and walls and simply attribute the datapoint to the "background" component. Focusing on galaxies in laments and nodes, Fig. 5.7 visually support the good agreement between the T-ReX and Nexus+ classi cations in the 3D disposition of galaxies in the simulation (galaxies in laments and nodes are respectively in green and red). The Nexus+ distribution of nodes is quite sparse, and this is the reason we did not follow the density threshold giving the smallest contamination to assign galaxies in laments. It is also remarkable to recall that the T-ReX results are obtained using solely the galaxies to trace the pattern and do not resort to the dark matter . Galaxies in green are those standing in laments while red ones are found in nodes. Note also that we only show the galaxies classi ed in these environments, showing only around 65% of the galaxies in both cases.

particles which number density is nearly 24 000 times larger.

Filaments characteristics in simulations

In this section, we propose to study the statistical characteristics of laments as de ned in Sect. 5.3.2 obtained from the set of galaxies in three hydrodynamical simulations. In addition of di erent recipes for baryonic processes, these simulations have also di erent mass and volume resolutions consequently drawing di erent pictures of the cosmic web. The assessment of laments characteristics is hence of importance to calibrate as well as possible the simulations with respect to observations and improve the quality of the physics governing the formation and evolution of galaxies.

Simulations and principal graphs

We use multiple datasets to compute several principal graphs, extract the corresponding laments and their characteristics. These simulations are brie y described below, together with the parameters used for the running of the T-ReX algorithm to obtain the analysed laments.

EAGLE. As already presented, EAGLE [Schaye et al., 2015] is a (100Gpc) 3 box hydrodynamical simulation. The cosmological parameters used to evolve the simulation are consistent with Planck15 results [Planck Collaboration XIII et al., 2016] with Ω Λ = 0.693, Ω m = 0.307, Ω b = 0.04825, σ 8 = 0.8288, n s = 0.9611 and h = 0.6777. We use the identi ed galaxies with stellar mass M * > 10 7 M yielding a total of 142 392 galaxies and consequently, a mean number density of galaxies of ng 0.142. The catalogue of laments is obtained by running T-ReX in initialised with a pruned MST with l = 25, and σ (0) k = 1 Mpc. Values of the hyperparameters are λ µ = 100, λ σ = 5 and λ π = 1. We discard from the analysis all the small branches with L < 2σ (0) k Mpc which leads to a total of 572 laments in the catalogue.

IllustrisTNG. The suite of hydrodynamical cosmological simulations IllustrisTNG [Nelson et al., 2019] follows the evolution of dark matter, gas, stars, and black holes on a moving mesh from redshift z = 127 to z = 0. The input cosmology of the simulation is consistent with the Planck15 one as well with Ω Λ = 0.6911, Ω m = 0.3089, Ω b = 0.0486, σ 8 = 0.8159, n s = 0.9667 and h = 0.6774. We use here the largest simulation box IllustrisTNG300-1 with a size length of 302.6 Mpc. Galaxies are identi ed by means of the SubFind algorithm [Springel et al., 2001] and we restrict the analysis using galaxies with M * ≥ 10 8 M , to highlight a di erent setup and number density than the EAGLE case. In total, we get 603 630 galaxies leading to a mean number density ng 0.0218 which is hence almost 5 times smaller than the EAGLE sample. The T-ReX algorithm is consequently run using l = 15, σ (0) = 1 Mpc and with λ µ = 30, λ σ = 5 and λ π = 1. Note that these values are lower than in the EAGLE case due to the lower number density of galaxies and higher mass threshold which hence require less smoothing and denoising of the initial structure. Using the same threshold of L > 2σ (0) , we end up with a total of 5 492 laments identi ed.

Magneticum. As a last sample of galaxies, we use the z = 0.066 snapshot of the Magneticum simulation Box2/hr [START_REF] Hirschmann | [END_REF] with a size of 500 Mpc. The simulation uses the cosmological parameters consistent with WMAP7 data [Komatsu et al., 2011] with Ω Λ = 0.728, Ω m = 0.272, Ω b = 0.0456, σ 8 = 0.809, n s = 0.963 and h = 0.704. Similarly to the case of IllustrisTNG, we restrict the analysis to the set of galaxies with M * ≥ 10 8 M leading to a total of 1 363 468 galaxies and a mean number density ng = 0.011, a value twice as small as that of IllustrisTNG. The used parameters are exactly the same as in the IllustrisTNG case leading to 11 675 laments.

Comparison of laments characteristics

From each obtained principal graph, we extract individual laments together with their characteristics as de ned in Sect. 5.3.2. The probability distribution functions of some of these properties are displayed in Fig. 5.8. The top left panel is focusing on the length L of laments, which is showing an exponential tail also reported by many previous ndings [such as Bond et al., 2010;Galárraga-Espinosa et al., 2020;Malavasi et al., 2020b;Rost et al., 2020]. At small lengths, all the samples provide similar distributions with a peak of the distribution around 5 Mpc, as also reported in Galárraga-Espinosa et al. [2020]. At larger lengths, the di erences between the simulations can be attributed to their di erent volumes which is indeed larger for Magneticum than for TNG which in turn is also larger than the EAGLE simulation, hence allowing for longer laments. The curvature distribution are closely similar with a large number of low-curvature laments in all considered catalogues and only few highly curved ones, as shown in the top right panel of Fig. 5.8. In addition to the previous quantities for laments, we also compute the mean galaxy density of each lament de ned as ρ f = n f / (Lπ4r 2 ), where n f is the number of galaxies associated individually to the lament. In the bottom left panel, we show the distribution of lament overdensities de ned as ρ f /n g where ng is the average number density of galaxies in the overall input volume depending on the simulation box. The density of laments spans a broad range in all cases, from tenuous ones with galaxy overdensities close to 0 and up to ∼ 100 for dense bridges of matter also drawn in dark matter analyses of the cosmic web like Cautun et al. [2014]. In the bottom right panel, we can also see that the radii as de ned by Eq. (5.8) are distributed over few Mpc around the lament spine in all the simulations with a peak slightly below 2.5 Mpc for TNG and EAGLE and around 2.8 Mpc for Magneticum. Even though depending on the de nition used to de ne the width, these results are also consistent with previous measurements of lament extensions in simulations exhibiting peak of the radii distribution between 2 and 3 Mpc [Colberg et al., 2005;Bond et al., 2010;Cautun et al., 2014] using the dark matter. The di erences between the three distributions are mainly due to the number density of objects that is higher for EAGLE than for TNG and Magneticum, this latter having the smallest. Note also that the radius of laments has been shown highly sensitive to the time-evolution of structures with thinner laments at low redshift as a result of the gravitational collapse and that Magneticum has a slightly higher redshift than the two other ones. These results on the spatial radial extent of laments are also consistent with the observational ndings of Bonjean et al. [2019] with a characteristic radius of 7.5 Mpc.

When investigating further the relations between individual properties of laments, we nd strong dependencies on the curvature and densities with the spatial extent of the lament. In the left panel of Fig. 5.9 is shown the evolution of the curvature γ with the length L while the right panel displays the evolution of the laments overdensities ρ f /ρ b . All the laments in the studied simulations show the same trends with shorter laments appearing, on average, straighter and denser than their longer counterparts. This result is in line with those of Galárraga-Espinosa et al. [2020] who identify two populations of laments with short ones mainly standing in overdense regions and longer ones connecting lower density regions in the Universe. We hence additionally show that small bridges connecting dense clusters are also respectively straighter than the long laments. Note also that, even though the trends are similar for the di erent simulations, the absolute values of the quantities vary. This may be due to the di erent baryonic physics models that are impacting the distribution of matter around lament, as shown by Galárraga-Espinosa et al. [2020]. This work constitutes a very rst step in studying the statistical properties of laments drawn from di erent simulations. However, the end goal is to apply the identi cation tool on actual galaxy surveys. To do so, several observational e ects must be taken into account that we leave for future investigations.

The impact of the cosmic web on cluster properties in simulations

In Sect. 5.3.3, we introduced a de nition of nodes in the cosmic web as bifurcations in the graph structure based on the assumption that nodes can be found at the densest intersections 0 25 50 75 100 between laments. The local topology and geometry of the density eld near these massive halos can be probed by the connectivity, a quantity encoding the number of laments a node is connected to. The connectivity of halos must decrease with the cosmic time due to the Universe accelerated expansion which disconnects cosmic nodes from the network of laments [Pichon et al., 2010]. Therefore, the statistics of halo connectivity is expected to depend on the growth factor and hence constitute a topological constraint on dark energy [Codis et al., 2018]. On the the other hand, the lamentary pattern surrounding the nodes (quanti ed by the connectivity) can heavily impacts their characteristics such as their internal properties and density pro les [see e.g. Contigiani et al., 2021]. In this section, we summarise our investigation of the impact of the connectivity on the physical properties of galaxy clusters (morphology, dynamical state, and mass assembly history), detailed in Gouin et al. [2021].

L [Mpc]

Data, lamentary pattern and connectivity

For this analysis, we use the set of halos detected from the largest resolution of the IllustrisTNG simulations [Nelson et al., 2019] with a box size of 302.6 Mpc (see Sect. 5.4.1). Groups and halos are detected in the simulation by means of the FoF algorithm [Davis et al., 1985] with a linking length of 0.2 Mpc/h. To trace the lamentary pattern, we additionally make use of the set of subhalos identi ed with the Sub nd algorithm to detect substructures inside host halos [Springel et al., 2001]. From the set of FoF sample, we select 2522 halos representing galaxy groups and clusters with masses M 200 ≥ 1 × 10 13 M /h at z = 0. We de ne R 200 as the radius enclosing a mass of M 200 characterising a mean overdensity of 200 times the critical background density. Notice also that we discard 79 halos that are less distant than 3R 200 from the edges of the simulation box to focus our analysis on the large-scale environments around halos.

The anisotropy in the large-scale environment of groups and clusters is quanti ed by the local number of laments that are connected to them, the so-called connectivity κ. This proxy is a powerful tool to understand the geometry of the underlying density eld, as discussed in Codis et al. [2018]. To detect the laments in the simulation, we use the T-ReX algorithm applied on the set of subhalos with M * ≥ 10 9 M , following Galárraga-Espinosa et al. [2020, 2021]. From the output graph, we associate to each group and cluster the closest node of the graph, and the local connectivity κ is the number of intersecting laments in a sphere of 1.5R 200 radius around the graph node (a similar de nition adopted in previous works like Darragh Ford et al. [2019]; Sarron et al. [2019]). In addition to the 79 previously removed halos, we remove 24 more that are low-mass groups considered as standing too far away from the graph in terms of projected distance (> 1 Mpc/h). As a result, our nal sample of groups and clusters consists of 2419 halos, for which we estimate the connectivity. The cluster connectivity is illustrated in Fig. 5.10, and one can see that the T-ReX lamentary structure (computed from the galaxy distribution) traces well the DM distribution around halos. In addition, Fig. 5.11 shows the probability distribution function of the connectivity for four mass bins. The distribution of massive clusters (M 200 > 10 14 M /h) peaks around κ ∼ 3, 4 meaning that they are mostly connected to around three or four laments, and is spread over large κ values (up to κ = 6). In contrast, the connectivity statistics of the lowest mass groups (10 13 M /h < M 200 < 2 × 10 13 M /h) strongly peaks at κ = 2, suggesting that low-mass groups are preferentially located inside laments, whereas massive clusters are more likely located at the nodes of cosmic web, connecting more than two laments. This thus con rms that massive structures have a higher connectivity than low-mass ones, as also found by previous works such as Aragon-Calvo et al. 10 13.5 ] M /h is low with nearly 80% of the objects standing in a lament or at its extremity with κ < 2 while the most massive ones with M 200 > 10 14.2 M /h are more connected with κ > 3.

Impact of connectivity on the growth and shapes of clusters

It is already well known that the mass of halos are strongly correlated with their shape [Aragon-Calvo et al., 2010;Codis et al., 2018;Darragh Ford et al., 2019;Sarron et al., 2019;Malavasi et al., 2020b]. Beyond the driving mass e ect, we investigate the in uence of the largescale cosmic web environment on the shape of the mass distribution in clusters. Their morphology, as de ned by the ellipticity , measures the local anisotropy traced by the dark matter particles. Similarly to Jing & Suto [2002], we compute the ellipticity as the ratio (λ 1 -λ 3 ) / (λ 1 + λ 2 + λ 3 ) where λ i is the ith largest eigenvalue of the local mass tensor. Starting from a sphere, the procedure is made iterative, shrinking the axes, such that the ellipsoid encloses a total mass of M 200 . An illustration of the resulting ellipsoids is shown in Fig. 5.10 for four clusters. The rst row of Table 5.3 quanti es the correlation between κ and through the Spearman rank correlation coe cients ρ sp for two sets of clusters. The correlation coe cient increases when the mass increases and reaches ρ sp ∼ 0.24 when M 200 ≥ 10 14 M /h, hence stressing that the connectivity is also closely linked to the ellipticity, beyond the e ect of the mass dependence. This is also shown in the left panel of Fig. 5.12 in which, in a given mass bin, groups with a high connectivity are more elliptical on average than low-connectivity groups and clusters.

The observed impact of lamentary structure on the cluster shape must be the result of di erent accretion phases and it is natural to think that the connectivity is also closely linked to the way clusters accrete matter. To con rm this assumption, we compare the connectivity to a proxy of the accretion of a halo at z ∼ 0. We therefore introduce the instantaneous mass accretion rate de ned as

dM 200 dt z∼0 = M 200 (t + dt) -M 200 (t) dt .
(5.12)

To compute it, we use the ten last snapshots of the simulation and perform a linear regression to obtain the value of dM 200 /dt. We show in the right panel of Fig. 5.12 the evolution of the the instantaneous mass accretion rate at z = 0 as a function of mass for three bins of connectivity (low connectivity with κ < 3, mildly connected with κ = 3 and highly connected with κ > 3). We see that, at xed mass, highly connected clusters tend to grow faster than low-connectivity groups and clusters, also emphasised by the Spearman correlation coe cients from the second row of Table 5.3. Intuitively, one can assume that more mass is feeding a cluster when this latter is connected to a larger number of laments. These results hence highlight that the halo environment in the cosmic web is impacting its shape, independently of its mass: highly connected clusters accrete more matter and this fast accretion must disturb their mass distribution consequently increasing their ellipticity.

Impact of cluster dynamical states on the connectivity

Beyond the shape of a cluster as traced by its ellipticity, we can focus on the correlation between the connectivity of a node in the cosmic web and its dynamical state. We quantify M 200 > 1 × 10 13 M /h M 200 > 5 × 10 13 M /h ρ sp p-value ρ sp p-value and κ 0.11 2.15 × 10 -7 0.17 5 × 10 -4 κ and dM/dt z∼0 0.33 3.8 × 10 -60 0.35 4 × 10 -13 κ and χ DS -0.13 4.6 × 10 -11 -0.17 5 × 10 -4

Table 5.3. Spearman rank correlation coe cients ρ sp and corresponding p-values between halo properties and the connectivity for all the groups and clusters in the sample (M 200 > 1 × 10 13 M /h), and for the 408 most massive groups and clusters (M 200 > 5 × 10 13 M /h). the level of dynamical relaxation, called the relaxedness, de ned by [Haggar et al., 2020] as

χ DS = 3 ∆r 0.07 2 + f sub 0.1 2 + η-1 0.15 2 , (5.13)
where ∆ r is the centre of mass o set between the density peak and the barycentre of the mass distribution in the halo, divided by the virial radius, f sub the subhalo mass fraction de ned as M sub /M tot where M sub is the sum of the subhalo masses except the most massive one, and η is the virial ratio characterising the level of virialisation of the halo de ned as η = 2T /|W |, with T and W the kinetic and gravitational potential energy respectively. Intuitively, a cluster that has a low value of f sub < 0.1 (few substructures), a low o set of the centre of mass ∆ r < 0.07 and that is close to virial equilibrium with |η -1| < 0.15 can be considered as relaxed, leading to χ DS ≥ 1 [Kuchner et al., 2020]. Groups and clusters that do not ful l this criterion are considered as unrelaxed ones.

It is admitted that the mass is also driving the relaxedness of clusters with more massive ones being less relaxed on average [see e.g. Power et al., 2012;Kuchner et al., 2020] since they formed later and are still in their formation phase. Beyond this mass dependence, we show in the left panel of Fig. 5.13 the evolution of the connectivity as a function of the mass for the two categories of relaxed and unrelaxed clusters. At a xed mass bin, we see that unrelaxed clusters are more connected to the cosmic web than relaxed ones, a result which appears in the last row of Table 5.3 as a weakly-negative correlation between the two quantities whose absolute value grows with the mass threshold. This shows that independently of the mass, the connectivity is impacting the dynamical state of groups and clusters.

The in uence of mass growth history

In addition to the higher accretion rate of high-connectivity clusters shown in Sect. 5.5.2, we exhibited in Sect. 5.5.3 that these latter are also less relaxed on average. All these results suggest that the connectivity is actually tracing di erent mass assembly histories (MAH) of clusters. Previous works already point out a relation between dynamical state and MAH [Power et al., 2012;Mostoghiu et al., 2019]. Therefore, we wish to investigate how a cluster large-scale environment in the cosmic web in uences its MAH. In addition to the relaxedness, we focus here on two quantities linked with the MAH of clusters given by the formation redshift z f at which the halo reached half of its M 200 mass at z = 0 [Cole & Lacey, 1996] and the continuous mass accretion rate Γ 200 = ∆ log M 200 /∆ log a, with a the scale factor. In the right panel of Fig. 5.13 we show the evolution of the connectivity for three subsamples of clusters with di erent MAH: (i) the early-formed, relaxed and slowly-accreting clusters (in orange); (ii) the early-formed, unrelaxed and fast-accreting clusters (in yellow); and (iii) the lately-formed, unrelaxed and fast-accreting clusters (in blue). We observe that older and more relaxed clusters are weakly-connected, while the unrelaxed ones are more connected to the cosmic web at xed mass. Finally, the young and unrelaxed population of clusters in blue are signi cantly more connected to the cosmic web. These objects are strongly a ected by the infalling matter and can be thought as the result of recent merger events disturbing the matter distribution and increasing the connectivity [Klypin et al., 2016;Darragh Ford et al., 2019;Vallés-Pérez et al., 2020].

All these observations advocate for a strong impact between the mass assembly history of clusters and the way they are embedded in the cosmic web. Old relaxed clusters are more spherical and slowly accreting matter with a small value of the connectivity while young and still in formation clusters are unrelaxed, more elliptical and connected in the cosmic web by numerous laments feeding them abundantly.

Summary and perspectives

Given the primordial role of laments in the cosmic web, we investigated in this chapter a possible de nition based on the principal graph formulation established in Chapter 4 not only for the laments but also for the nodes of the cosmic web. We particularly showcased two possible applications of the proposed method, entitled T-ReX, aiming either at detecting the lamentary structure as a whole, together with a positional uncertainty of the ridge, or at de ning individual laments as branches of the principal graph. We showed that this de nition based on the probabilistic framework of T-ReX allowed the derivations of many interesting characteristics for the statistical analyses of laments that we carried out from multiple samples of galaxies in simulations. Finally, we used the smooth graph learnt by the algorithm to perform a thorough analysis of the properties of galaxy clusters depending on their spatial embedding in the cosmic web as measured by the connectivity.

In this chapter, we covered several topics related to the lamentary structure of the cosmic web and, more precisely:

1. We showed that the T-ReX algorithm performs as good as other web nders for the detection of lament ridges but adding the information of the uncertainty on the ridge and using only a sparse distribution of the input traced by halos in simulations.

2. We exposed how the principal graph can be used to identify the cosmic web elements that contain the largest mass fraction, namely nodes and laments, as dense bifurcations and branches in the graph structure respectively. By deriving from the obtained graph di erent characteristics for laments, we were able to exhibit an exponential tail in the distribution of their length and their curvature. The radii of laments resulting from local measurements in the T-ReX algorithm are spread over few Mpc with a peak between 2 and 3 Mpc. All these results were shown to be in agreement with previous ndings of laments statistics.

3. The galaxies used to determine the graph can be associated consistently with each individual lament, to nodes, or to the background in a probabilistic setup. This association was compared to a physical web nder extracting multi-scale laments and we nd a very good agreement with the advantage of having at hand a method relying on a much sparser sampling of the matter distribution.

4. By studying the connectivity of clusters in the cosmic web, we con rmed the mass-connectivity relation exhibited by previous works. We also showed that the cosmic environment around clusters signi cantly impact their shapes and the way they accrete matter such that, at xed mass, halos in nodes (with high connectivities) have a larger accretion rate than those standing in laments (weaklyconnected). These results led us to study multiple scenarios of mass assembly his-tory for galaxy clusters and how they connect to the cosmic web. In particular, old relaxed groups and clusters have a small connectivity while young, fast-accreting and unrelaxed galaxy clusters have more surrounding laments on average.

We sketched an analysis of the laments properties traced by the distribution of galaxies in hydrodynamical simulations. The natural next step will be to study such characteristics as extracted from actual galaxy surveys like the Sloan Digital Sky Surveys [York et al., 2000] or the Dark Energy Survey [Abbott et al., 2016], in a similar manner to [Malavasi et al., 2020b;Rost et al., 2020]. Samples of reliably detected laments with their characteristics (size, radius, curvature, etc.) have already proven to be crucial for the study of the distribution of baryons in laments [e.g. Tanimura et al., 2020a] or study their properties in other observables like Sunayev Zel'dovich e ect or X-rays [e.g. Tanimura et al., 2020c,b]. However, constructing samples of reliable laments necessitates to overcome some observational complications. The two main di culties being that (i) most of the large-scale galaxy surveys are photometric observations with relatively large errors on the estimated redshift; and (ii) observations are carried out in redshift-space inducing some distortions of the spatial distribution of galaxies (see Sect. 2.2.2). Both of these issues create elongations of the distribution along the line-ofsight (LoS) that could be handled in the T-ReX formalism by allowing the spherically Gaussian components composing the graph nodes to be elongated in the LoS direction. This proposed modi cation could result in a built-in approach to reduce the e ect redshift-space distortions and the blurring induced by photometric redshift measurements produced on the extracted lamentary pattern. Note however that this would come with an increase of the computational cost since the responsibilities from the Expectation-Maximisation procedure will require the full computation of Mahalanobis distances (x i -µ k ) T Σ -1 k (x i -µ k ) instead of isotropic L 2 norm which can easily be optimised.

In addition to the previously-mentioned interests, the detection of reliable laments and their characteristics is key to improve our understanding of the structure formation. As we have seen, the connectivity in simulations is strongly correlated with the physical properties of clusters and has also been shown to depend on the underlying cosmology and on the redshift of the data distribution in Codis et al. [2018]. A further analysis of how such an information perform and how it can be combined with existing probes to constrain cosmological models is also an interesting perspective that matches the scienti c question of the next Chapter 6.

Finally, in the T-ReX formalism, the lamentary structure is modelled as a graph, and we exploit only partially the graph information to constrain the overall smoothness of the resulting structure. However, the graph encodes much more information at the nodes and edges level. Assigning for instance local physical properties to graph nodes (like densities, masses, etc.) may o er the opportunity to link the topology of the cosmic web to the physics of tracers, similarly to Coutinho et al. [2016]. Additionally, the statistics of edges in the MST was found sensitive to cosmology in previous works [Hong & Dey, 2015;Hong et al., 2016;Naidoo et al., 2020], showing the ability of graph-derived statistics to encode a signi cant amount of information about the underlying cosmological model. Chapter 6.

Constraining cosmological parameters with cosmic environments

"It should be quite straightforward. "

J. K Chapter 5 investigated a possible de nition of the cosmic environments that are nodes and laments from a distribution of matter tracers (galaxies or halos) and studies their physical properties through a graph framework. In this chapter, we make use of cosmic environments de ned physically through the eigenvalues of the Hessian matrix of the gravitational potential and traced in simulations by the full dark matter distribution. Focusing on the two-point statistics derived from the environments and carried out in both real and redshift spaces, we present the rst quantitative measurement of the informative power of the di erent cosmic environments about the underlying cosmological parameters as compared to the analysis of the matter power spectrum.

Context and introduction

6.1.1 The matter power spectrum as a cosmological probe

The most basic statistics that one can build from centred cosmological elds such as the overdensity eld δ or convergence maps are based on the two-point correlation function or its Fourier-equivalent, the power spectrum. This latter summary statistics quantitatively expresses the covariance between Fourier modes of the eld (see Sect. 1.3.2). Beyond its simplicity, the two-point statistics fully encodes the information of Gaussian random elds, particularly well describing some cosmological elds, like the early distribution of matter. Theoretical predictions, allowed for instance by linear perturbation theory, depicts the late-time matter power spectrum as depending on the initial one with great accuracy at large scales (k < 0.15 h/Mpc, see Fig. 2.2 and related discussion). In particular, the linear theory permits to describe the impact of each cosmological parameter of the ΛCDM model on the late-time matter power spectrum based on the one from the initial density eld at the linear scales. As such, the interpretability of the two-point statistics is well-known and well-understood at linear scales [Heath, 1977;Peebles, 1980], even for non-Gaussian overdensity elds, making the matter power-spectrum a keystone for the statistical analysis of large-scale structures since their rst observations in the early 80s.

The cosmic environments as an alternative probe

When the eld is non-Gaussian, such as the late-time matter distribution in the Universe, the two-point statistics is not carrying all the information about the underlying eld. Even though still informative, the matter power spectrum is subject to degeneracies among parameters of the ΛCDM model which prevent it from fully constraining di erent values. For instance, changes in the matter density Ω m , in the normalisation of the power spectrum σ 8 and in the summed neutrino mass M ν are known to produce similar e ects on the matter power spectrum over a wide range of scales.

In Sect. 2.3.1, we discussed several statistics derived from topological de nitions of the matter distribution or based on machine-learning data compression methods to incorporate the information that leaked into higher-than-two order moments. Alternatively, cosmic environments also exhibit particular dependencies with cosmological parameters. The hierarchical formation of structures makes nodes a wealthy source of information [White & Frenk, 1991] depending on the matter and dark energy contents of the Universe but also on the amplitude of the initial density uctuations [for a review, see Allen et al., 2011]. Their number counts, shapes, mass pro les and evolution with redshift have been shown particularly ecient in partially breaking the degeneracy between Ω m and σ 8 occurring in a matter power spectrum analysis [Bahcall et al., 1997;Bahcall & Fan, 1998;[START_REF] Holder | [END_REF]. The always larger and more complete availability of cluster samples obtained from di erent observables like optical, X-rays and millimetre wavelengths enabled to considerably improve the cosmological constraints derived from these statistics [see e.g. Mantz et al., 2015;Salvati et al., 2018;DES Collaboration et al., 2020;Corasaniti et al., 2021]. In Costanzi et al. [2013] is also shown that varying M ν induces di erent abundances of massive clusters at xed primordial conditions and Villaescusa-Navarro et al. [2014] exhibit a scale-dependence of the bias for those massive tracers in Universe with massive neutrinos. Voids, on their side, are objects of lowdensity making them the component of the cosmic web the least a ected by the non-linear collapse of matter. This property is particularly interesting to probe the accelerated expansion of the Universe and study the dark energy [Lee & Park, 2009;Lavaux & Wandelt, 2012;Pisani et al., 2015]. The measure of their sizes, shapes, counts and corresponding evolution with redshift are key quantities related to the underlying cosmology [van de Weygaert & Platen, 2011;Hamaus et al., 2014Hamaus et al., , 2015]]. Numerous works also point out the interest of studying the e ect M ν in voids since the large thermal velocities of neutrinos, coupled with the volumedominating property of voids, make neutrinos contribute to a large extent to the overall mass voids enclose. In particular, voids have been measured smaller and denser when M ν increases in massive neutrino simulations [Villaescusa-Navarro et al., 2013;Massara et al., 2015;Kreisch et al., 2019]. The constraints brought by these two extreme environments that are voids and nodes are combined by Bayer et al. [2021] who show that the combination of information provided by the halo mass function and the void size function leads to considerable improvement over the matter power spectrum constraints in real space. Kreisch et al. [2021] recently shown that using the same statistics derived from halos in the simulations also yields sizeable gains.

In an attempt to incorporate local information about the underlying pattern into the twopoint analysis, several works rely on a weighted version of the matter clustering in which a mark is assigned to each source, based for instance on the local luminosity [Beisbart & Kerscher, 2000;Sheth et al., 2005] or density [White, 2016]. This latter version, the markedby-density power spectrum up-weights the low-density parts of the eld and is of particular interest for discriminating between several cosmologies [Valogiannis & Bean, 2018;Armijo et al., 2018] and constraining cosmological parameters in real-space [Massara et al., 2021].

In this picture, the di erent components of the cosmic web, from its densest to its emptiest parts are a promising way to provide complementary information about the underlying cosmological model. They show their own sensitivities to some cosmological parameters whose e ects are limiting the constraints obtained from the matter power spectrum. We hence seek to quantify the information brought by the combination of low, intermediary and high density environments represented respectively by voids, walls, laments and nodes that should be able to break some degeneracies among parameters allowing to improve the constraints over a direct analysis of the matter clustering.

In this chapter, we undertake, using large N -body simulations, a quantitative analysis of the cosmological information content of all cosmic environments at linear and nonlinear scales (up to k ∼ 0.5 h/Mpc) and in both real and redshift spaces. After introducing the N -body simulations from the Quijote suite and how the environments are theoretically and practically de ned through the local tidal anisotropies, we explain in what manner the Fisher formalism can be used to assess the constraining power of a statistical representation of an observable. Equipped with these methodological aspects, we show that the information carried by the two-point statistics of the cosmic environments is superior to a matter power spectrum analysis both in real and redshift spaces leading to a sizeable gain in the constraints put on cosmological parameters of the underlying model. In particular, the combination of the di erent environmental sensitivities breaks some key degeneracies in several planes and mostly between matter-related parameters like M ν -σ 8 or M ν -Ω m . After drawing these conclusions, we expose some caveats of the proposed study and discuss their potential impact on the presented results together with how they may limit the perspective of applying such an analysis to observational data. et al., 2020] is a publicly available1 large suite of N -body simulations. With 44 100 simulations spanning more than a thousand cosmological models, each with multiple realisations, it is the ideal dataset to perform statistical cosmological analyses as it allows to build accurate covariance matrices and compute derivatives for any cosmological representation. Each simulation consists of a set of 512 3 particles (and 512 3 neutrinos in massive neutrinos cases) that are evolved forward in time from z = 127 to z = 0 using a tree-PM Gadget-3 code [Springel, 2005] in a L = 1 Gpc/h size box. The ducial cosmology is a at ΛCDM cosmology with parameters consistent with Planck Collaboration VI et al. [2020]: Ω m = 0.3175, Ω b = 0.049, h = 0.6711, n s = 0.9624 and σ 8 = 0.834. With these parameters, and assuming a zero mass for neutrinos (M ν = 0), 15, 000 random realisations are computed. The Quijote suite then provides 500 realisations by varying individually each parameter, xing the others at their ducial values. The stepsizes are: dΩ m = 0.010, dΩ b = 0.002, dh = 0.020, dn s = 0.020, and dσ 8 = 0.015. Additionally, 500 realisations using several sum of neutrinos mass are also computed, with M ν = m ν = {0.1, 0.2, 0.4} eV, that we will refer to as M + ν , M ++ ν , and M +++ ν cosmologies respectively. All these information about the Quijote suite of simulations are summarised in Table 6.1.

Data & Methodology

In the present analysis, we aim at studying quantitatively the cosmological constraints obtained from the two-point summary statistics derived in the di erent cosmic web environments in both real and redshift spaces. To do so, we mimick RSDs (see Sect. 2.2.2) in every simulations by displacing all particles (dark matter particles and neutrinos if any) through Eq. ( 2.1) along the third Cartesian axis of the box hence assuming the plane parallel approximation.

Cosmic web segmentation

In this theoretical work, we make use of the T-web algorithm introduced in Hahn et al. [2007], and later extended by Forero-Romero et al. [2009] to identify cosmic environments through the tidal tensor T . Using prescriptions originating from the linear growth of perturbations in the Zel'dovich approximation [Zel'dovich, 1970], this web nder de nes environments in a physical way based on the local level of tidal anisotropy. From the discrete set of particle positions, we rst rely on a B-spline interpolation scheme [START_REF] Hockney | [END_REF]Sefusatti et al., 2016] to estimate the density eld ρ(x) on an N 3 g regular grid. For our purpose, we adopt an interpolation at the order four, namely the Piecewise-Cubic Spline (PCS) scheme, in which the mass of a particle is spread over the 4 3 = 64 closest cells. By noting d = N g xx p 2 /L with x the center of a grid cell, x p the particle position and L the size of the box length (assuming a cubic box), PCS weights are given by (6.1) This choice of interpolation order represents a good trade-o between the accuracy of the reconstructed eld and its computational time.

     (4 -6d 2 + 3d 3 )/6 if d ∈ [0, 1[ , (2 -d) 3 /6 if d ∈ [1, 2[ , 0 otherwise.
From ρ, one can derive the gravitational potential Φ by solving the Poisson equation

∆Φ(x) = 4πGρ(x), (6.2)
where ∆ is the Laplacian operator and G the gravitational constant. It is convenient to write this equation in terms of the reduced potential Φ r (x) = Φ(x)/4πGρ so that the Eq. (6.2) satis es ∆Φ r (x) = δ(x), with δ(x) = ρ(x)/ρ -1 the overdensity. Solving this reduced version of the Poisson equation in Fourier space using a discrete approximation of the Laplacian operator (in our case, a 7-point approximation) holds an estimate of Φ(x) on the grid. From the gravitational potential, the tidal tensor can be obtained in each grid cell x as

T i,j (x) = ∂ 2 Φ(x) ∂x i ∂x j , (6.3)
leading to the eld of eigenvalues λ 1 (x) ≤ λ 2 (x) ≤ λ 3 (x). The cosmic environment associated with a grid cell x is obtained depending on the number of eigenvalues below a given threshold λ th as de ned in Table 6.2.

The T-web algorithm hence segments the density eld at the cell level. To build individual overdensity elds for each environment, we simply propagate the classes at the particle level by assigning the same environment signature to all hosted particles in a given cell. From these four disjoint sets of particles, we build four corresponding overdensity elds {δ V , δ W , δ F , δ N } estimated with the PCS interpolation scheme. The full density eld δ is hence decomposed into the four environmental elds and respects the linear combination (6.4) where f α denotes the mass fraction of the environment α, namely N α /N , with N the total number of particles2 . In Fig. 6.1 is shown this decomposition with the contribution of each In our implementation of the T-web formalism, the potential is being smoothed at a scale of σ N Mpc/h before the classi cation, resulting in three parameters for the full segmentation procedure that are: N g , the total number of grid cells, λ th the threshold for the eigenvalues of the tidal tensor and σ N . After assessment of the e ect of each parameter on the classi cation in a physically reasonable range (from 1 to 3 Mpc/h), we concluded that the smoothing scale and the number of cells were not of great impact on both the mass and volume fractions of the resulting environments. We consequently adopt for this work σ N = 2 Mpc/h and N g = 360 leading to a grid size of 2.77 Mpc/h. The λ th parameter, however, has a signi cant e ect on the classi cation, both in terms of volume and mass fractions in the di erent environments [Forero-Romero et al., 2009]. The impact of this parameter is illustrated in Fig. 6.2 in which are displayed the averaged mass fractions f α for each cosmic environment as drawn by the T-web algorithm with three values of λ th that are {λ - th , λ fid th , λ + th } = {0.2, 0.3, 0.4}. The ducial value of 0.3 is the one corresponding roughly to the threshold at which voids percolate for our cosmological volume of 1 (Gpc/h) 3 [Forero-Romero et al., 2009]. It is also the value adopted in many other works for classi cation by means of the T-web prescriptions such as Martizzi et al. [2019]. The left (resp. right) panel of Fig. 6.2 focuses on the real-space (resp. redshift-space) case where an increasing value of λ th from 0.2 to 0.4 is attributing more particles in voids and less in laments and nodes. Comparing the two panels also illustrates that more mass is being associated with laments and less in nodes in the redshift space. This is mainly due to the Finger-of-God distortions squashing clustered regions and breaking their isotropic nature making them appearing elongated in the line-of-sight direction (see Sect. 2.2.2). Quantitatively, varying the threshold in eigenvalues from 0.2 to 0.4 yields a factor of two between the obtained mass fractions in voids in both real and redshift spaces. Consequently, in order to derive constraints from environments that are robust to uncertainties in the identi cation of the environments and also indirectly to changes in the de nitions o ered by the variety of methods [see e.g. Libeskind et al., 2017], we embed λ th as a nuisance parameter in the formalism such that all presented constraints are marginalised over it. et al., 2014;Libeskind et al., 2017] re ects their di erent gravitational histories (see also Sect. 2.4.1). Voids are for instance environments of low-density spanning most of the volume while nodes are highly non-linear and dense objects enclosing a similar value of the mass as voids but in only few percents of the volume. Filaments span a broad range of densities from long and tenuous arteries to small and dense bridges linking clusters together. Figure 6.3 shows the ratio between the average mass fractions in each environment when a cosmological parameter varies and the average obtained with the ducial cosmology. The error bars (represented as the bars around points and crosses and as the grey shaded area for ducial simulations) are the 3σ con dence intervals. Many parameters are causing sizeable changes in these proportions and, unsurprisingly, parameters related to matter density, like σ 8 and Ω m , have the largest impact. The most important variations are induced by σ 8 , for which an increase (resp. decrease) is leading to a larger (resp. smaller) mass fraction in dense environments. This is in agreement with the de nition of σ 8 that is measuring how matter clusters at a scale of 8 Mpc/h. The impact of neutrino mass, even though smaller in comparison, is still signi cant and fraction ratios lie outside the 3σ con dence regions. When M ν increases, it basically leads to make dense environments more massive, similarly to σ 8 . All these di erent e ects observed in the mass fractions already suggest that each cosmological parameter has a di erent impact on the environments that will be even more re ned when inspecting the clustering statistics in each environment through the power spectra.

δ = f V δ V + f W δ W + f F δ F + f N δ N ,
f α / f fid α Ω + m Ω + b h + n + s σ + 8 Ω -m Ω -b h - n - s σ - 8 V W F N

Power spectra in cosmic environments

The auto power spectrum P αα (k) is de ned as the covariance of Fourier modes (see Sect. 1.3.2) of the density eld δ α , with α ∈ {V, W, F, N} denoting a given environment. More generally, for two overdensity elds δ α and δ β , the cross power spectrum is given by

P αβ (k)δ D (k 1 + k 2 ) = 1 (2π) 3 δα (k 1 ) δβ (k 2 ) , (6.5)
with k = k 1 2 , and δ referring to the Fourier transform of δ. In redshift space, the peculiar velocities of sources is causing a dependence of the power-spectrum with the line-of-sight inducing a breaking of the density eld isotropy property which consequently alters the spatial pattern (see Sect. 2.2.2). This e ect can be taken into account using a multipole expansion of the power spectrum expressed as

P s,αβ (k) = 2 + 1 2 1 -1 P αβ (k, µ)L (µ)dµ, (6.6)
with µ = k • n/k, the angle with the line of sight, P αβ (k, µ) the 2D power spectrum obtained by binning both in k and µ, and L the Legendre polynomials. In this work, we rely on the three rst non-zero multipoles of the power spectrum in redshift-space, P s,αβ =0 (k), P s,αβ =2 (k), P s,αβ =4 (k), respectively called monopole, quadrupole and hexadecapole. These ≤ 4 orders are the only non-vanishing moments in the linear approximation of the distortions and encode the full 2D information at linear scales [Kaiser, 1987]. The corresponding Legendre polynomials are

L (µ) =      1 if = 0, (3µ 2 -1) /2 if = 2, (35µ 4 -30µ 2 + 3) /8 if = 4.
(6.7) Note that Eq. (6.6) generalises Eq. (6.5) and, under isotropy conditions for the overdensity elds, all poles P (k) with > 0 are exactly 0. To avoid confusion, we refer to the real-space monopole as simply P (k) and leave the subscript for redshift-space spectra, in addition to the superscript s. In the present study, in expressions (6.5) and (6.6) is in fact not directly appearing the overdensity eld δ but its deconvolved evaluation. Indeed, since δ is estimated by the PCS smoothing interpolation scheme, it deforms the shape of the estimated power spectrum [Jing, 2005]. To correct for this e ect, we rst deconvolve the elds δ α by applying the window function in Fourier space

W (k) = i 1 - 4 3 s i + 2 5 s 2 i - 4 315 s 3 i -1 , (6.8)
with s i = sin πk i /2k Nyq and k Nyq = πN g /L the Nyquist frequency. Additionally, to avoid any bias induced by aliasing e ects, we restrict the analysis to the modes of P (k) below half of the Nyquist frequency, k Nyq /2 = 0.57 h/Mpc. In our case, k max = 0.5 h/Mpc, allowing us to take into account both large-scales and non-linear ones in a robust manner. Finally, because of the discrete nature of the input, namely the dark matter particles, we also subtract the shot noise from power spectra estimated (6.5) and (6.6). Even though the number of particles is very large and we expect the shot noise contribution to be small at the scales of interest, autospectra P αα (including also P mm ) are subtracted by the quantity 1/n α where nα = N α /V .

6.4 Constraining power of cosmic environments

Fisher formalism for information content quanti cation

Considering a set of model parameters θ ∈ R d (in our case, cosmological parameters), we assume that the vector s(X) ∈ R n is a statistic built from an observable (here, the binned power Fig. 6.4. The top panel shows the normalised power spectra f 2 α P αα (k) obtained in real space for each environment α ∈ {V, W, F, N} compared to the one from all dark-matter particles in black. Plain lines are the averages over the N fid = 7, 000 ducial simulations and the shaded areas are the 1-σ con dence intervals. The grey area depicts the range of k > k max excluded from the analysis. In the bottom panel are found the ratios between the averaged normalised spectra in the environments and the matter one (black from the top panel). spectra drawn from the overdensity elds) following a Gaussian distribution s ∼ N (s, Σ). Its log-likelihood can hence be written

log p(s | θ) = - 1 2 (s -s) T Σ -1 (s -s) - 1 2 log |Σ| + cst, (6.9)
where the constant comes from the normalisation of the distribution. A general way to quantify the information carried by s on θ is to use the Fisher information matrix I(θ). From the Fréchet-Darmois-Cramér-Rao inequality, its inverse I(θ) -1 corresponds to a lower-bound on the variance of any unbiased estimator drawn from s hence assessing the e ciency of the representation. Elements of the Fisher matrix are de ned as the variance of the derivative of the log-likelihood, namely 6.10) which can also be written in terms of the 2 nd derivative of the log-likelihood under some smoothness constraints (which are ful lled in the Gaussian case), 6.11) where E θ is the expectation taken over the distribution p(s | θ). This latter equation is intuitively explaining how the amount of information is measured. A sharp log-likelihood around θ implies a huge increase with small changes of the parameters, making the representation s very sensitive to variations dθ. On the other hand, a weakly-curved log-likelihood with a locally at behaviour advocates for a poor description since its sensitivity with changes in the parameters is small. Under the Gaussian assumption described above and by further considering a covariance matrix Σ independant from cosmological parameters θ, mainly because this contribution is expected to be small and source of underestimation of errors [Carron, 2013;Kodwani et al., 2019], it yields

I(θ) i,j = E θ ∂ log p(s | θ) ∂θ i T ∂ log p(s | θ) ∂θ j , (
I(θ) i,j = -E θ ∂ 2 log p(s | θ) ∂θ i ∂θ j , (
I(θ) i,j = ∂ s ∂θ i T Σ -1 ∂ s ∂θ j .
(6.12)

The non-linear operation of the inversion to compute the precision matrix Σ -1 is actually leading to a biased estimate, even though the covariance may be computed using the classical unbiased estimation. Still under the previously-established Gaussian assumptions, the unbiased estimate of the precision matrix is given by [Kaufman, 1964;Hartlap et al., 2007]

Σ -1 = N fid -n -2 N fid -1 Σ-1 , (6.13)
where N fid is the number simulations at the ducial cosmology, n is the length of the summary statistics vector s and Σ = (s -s) (s -s) T / (N fid -1) is the unbiased estimate of the covariance matrix.

The partial derivatives of the summary statistics with respect to parameters of the model can be computed using the variations of cosmologies provided by the Quijote suite of simulations. Considering the set of studied cosmological parameters {Ω m , Ω b , h, n s , σ 8 }, we can estimate numerically

∂ s ∂θ i s(θ i + dθ i ) -s(θ i -dθ i ) 2dθ i . (6.14)
In the case of massive neutrino simulations, M ν ≥ 0 with a ducial value at 0.0 eV. For this parameter, we thus cannot rely on Eq. (6.14) and instead estimate the derivative using the four-point forward approximation

∂ s ∂M ν s(4M + ν ) -12s(2M + ν ) + 32s(M + ν ) -21s(M ν = 0.0) 12M + ν .
(6.15)

Because massive neutrino simulations in Quijote are initialised using the Zel'dovich approximation and ducial ones using the 2 nd order Lagrangian perturbation theory, the quantity s(M ν = 0.0) is computed using the ducial simulations initialised with the Zel'dovich approximation also. In all the presented results, if not mentioned otherwise, the numerical estimation of derivatives and covariances have been respectively made with N deriv = 500 and N fid = 7000 realisations. In Sect. 6.4.4, we discuss the impact of these numbers and assess the numerical stability of the results.

Real-space auto-spectra

We rst study the constraining power of summary statistics derived from the set of power spectra in distinct environments P VV , P WW , P FF , P NN and their combination, P comb , for cosmological models in real space. The two key ingredients of the Fisher-based quanti cation of information appear in Eq. (6.12) as:

(i) The partial derivatives of the statistics with respect to the cosmological parameters. Intuitively, a summary presenting large variations with parameters of the model carries more information than if it barely varies;

(ii) The covariance matrix Σ measuring the relations between elements of the summary statistics. Naturally, having elements varying the same way limits the overall informative power of the summary.

Figure 6.5 shows the rst constituent, the derivatives ∂P αα (k)/∂θ i while Fig. 6.6 plots a proxy of the second ingredient through the normalised version of the covariance matrix, namely the correlation matrix C, whose elements are de ned as

C ij = Σ ij Σ ii Σ jj . (6.16)
The rst striking observation when inspecting the correlation matrix for P mm is that it quickly becomes highly non-diagonal with correlation coe cients C ij = 0.5 at scales of ∼ 0.3 h/Mpc. Such high couplings between scales are expected at low redshifts, Fourier modes being more and more correlated with time, as a result of the non-linear evolution of the matter distribution [Blot et al., 2015]. These non-diagonal terms are intrinsically reducing the representative power of the matter power spectrum to constrain cosmological parameters, independently of how it varies with these latter. This insu ciency of P mm (k) is also marked by the shape of the derivatives, quite monotonous with similar structures for most parameters, except Ω m , Ω b and M ν where the wiggles are signatures of an impact on the baryonic acoustic oscillations. On the opposite side, spectra drawn from cosmic environments are showing di erent patterns in the derivatives. Taking the example of the Ω m parameter in the top left panel of Fig. 6.5, the change of sign occurs at di erent scales, which seem to follow the order of average density, namely from void to node. This pattern is also observed for other parameters like Ω b , h, n s or M ν . The correlation coe cients of the environmental statistics visible in the bottom panel of Fig. 6.6 also display less o -diagonal cross-correlation coe cients of high values, at the exception of the node-node case. Indeed, P NN Fourier modes are highly correlated with values C k 1 k 2 ∼ 0.5 for k 1 , k 2 ∼ 0.2 h/Mpc, which is a signature of the highly non-linear environment it represents. These overall observations suggest that the environments bring di erent information on the set of cosmological parameters, and that, when combined together, they may break degeneracies and allow to put tighter constraints on the underlying cosmological model. One way to quantify this gain is the Fisher formalism in which the marginalised 1σ con dence ellipses can be obtained from the information matrices (6.12). These latter are shown in Fig. 6.7 when using the statistics of the matter power spectrum or the one from each environment either individually or combined all together. Table 6.3 delivers the complementary information of the marginalised σ θ i constraints obtained in the di erent cases and de ned as 6.17) where I ii is the ith diagonal element of the Fisher information matrix. As already hinted by the analysis of the partial derivatives and the correlation matrix, the marginalised errors obtained from the matter-matter power spectrum P mm are high. The corner plot especially shows us the degeneracies among parameters in almost all panels, observed for instance in the M ν -σ 8 panel for which the matter power spectrum behaves similarly when varying either M ν or σ 8 , making it di cult for the P mm statistics to disentangle the two e ects. When inspecting the ellipses obtained from the individual spectra from the environments, we clearly distinguish di erent orientations for several parameters, such as the void and lament environments in the M ν -Ω m , M ν -σ 8 or M ν -Ω b projected spaces. This observation is emphasised by Fig. 6.8 in which we show a zoom over the M ν -Ω m and M ν -σ 8 planes. This suggest a di erent type of information delivered by the two environments, which, when combined all together, tightens up the constraints as quantitatively shown in Table 6.3 with improvement factors from 4.5 to 17.1 for all the ve cosmological parameters considered and 15 for the sum of neutrino mass. Unsurprisingly, the constraints put by individual environments are, to a lesser extent, also better than a matter power spectrum analysis in real space. As discussed previously, we however see the broad ellipses and hence poor constraints obtained from the node environment due to the high level of correlations between the P NN coe cients, which, together with the parameter degeneracies, lead to high errors on cosmological parameters. Focusing on M ν , we also see that the best constraints are obtained by the void environment, as theoretically expected and stated in previous works [Pisani et al., 2015;Massara et al., 2015;Kreisch et al., 2019].

σ θ i = 1 [I(θ) -1 ] ii , (
In White [2016] is exposed a way to build an estimate of the matter power spectrum weighted by the local density as

m(x, R, p, ρ ) = 1 + ρ ρ + ρ R (x) p , (6.18)
where ρ R is the density eld smoothed by a top-hat lter of radius R, p the factor of enhancement of low (resp. high) density areas if p > 0 (resp. p < 0) and ρ a density parameter to tune. Confronting results from Table 6.3 to those obtained with the marked power spectrum analysis of Massara et al. [2021] in real-space shows that the combination of autospectra in cosmic environments yields similar constraints for all the studied parameters of the simulation. Similarly, Bayer et al. [2021] constrains the same set of cosmological parameters using the real-space matter power spectrum from Quijote together with the void size and halo mass functions (respectively VSF and HMF). They report 1σ marginalised errors of {0.0063, 0.037, 0.23, 0.10, 0.0069, 0.096} on the set of parameters {Ω m , Ω b , h, n s , σ 8 , M ν } meaning that the combination of auto-spectra in environments performs better in most cases with improvement factors of {0.5, 4.1, 3.0, 3.4, 3.5, 1.6} compared to the VSF+HMF statistics in real space. It is also noteworthy that λ th is set free in the analysis and Fig. 6.7 shows that this parameter is well-constrained by most environments. Quantitatively, voids, walls, laments, nodes and their combination are respectively leading to marginalised error over the threshold of σ λ th = {0.0046, 0.0090, 0.0071, 0.0100, 0.0006}, indicating that the results are robust to changes in this threshold. Even though λ th may in uence the identi ed cosmic structures, it does not much a ect the constraints. This is also encouraging in the sense that it leaves room to other de nitions of cosmic environments to be applied and still ending up with similar results. These obtained constraints are nonetheless dependant on the maximum scale involved for the power spectra coe cients k max . Figure 6.9 illustrates this dependency by showing the evolution of σ θ i for all the parameters and derived statistics with the value of k max . The rst conclusion that we can draw is that the information extracted from P mm is saturating when k max increases. This saturation when going to smaller scales, pointed out by previous analyses [Takahashi et al., 2010;Blot et al., 2015;Chan & Blot, 2017], is mostly induced by the degeneracies among parameters for this precise summary statistics which do not lead to any further improvement on the constraints at mildly non-linear scales when k max > 0. scales. In particular, when restricting the analysis to k max < 0.2 h/Mpc, environments are not individually constraining better the set of cosmological parameters, even though their combination still lead to an improvement over the matter-matter analysis, for all considered values of k max . It is however interesting to note the better performance of the mass-dominating environment, the laments, that are, at large scales k max < 0.2 h/Mpc, bringing the tightest constraints for almost all parameters. When going to smaller scales, laments start to saturate and the volume-dominating environments, namely voids and walls are providing similar, if not tighter, constraints for most of the studied parameters. Another quantity of interest to measure the power of a representation is given by the signal-to-noise ratio (hereafter SNR) that describes the reachable accuracy of the statistics measurement given the covariance matrix. In general, the SNR of a summary statistic s ∈ R n is de ned as

SNR(s) = √ s T Σ -1 s, (6.19)
with Σ -1 the corresponding precision matrix de ned by Eq. (6.13). Figure 6.10 shows the evolution of the SNR for the matter and environmental power spectra as a function of the maximum scale k max . We again observe the attening when k max approaches non-linear scales at 0.25 h/Mpc for SNR(P mm ). Although environment-dependent spectra also suggest such a plateau, it happens at much lower scales and higher values of the SNR, except for nodes again which saturate at the lowest value among all of the studied statistics. The shapes of the SNR evolution with k max from environments combination also suggest that there is room left for a further increase when going to even smaller scales, where the matter analysis will not be able to improve. Quantitatively, the SNR obtained from the combination of environments is eight times higher than the one from the matter auto-spectrum at k max = 0.5 h/Mpc and explains also partly the better constraints put on cosmological parameters from Fig. 6.9 and Table 6.3.

Redshift-space auto-spectra

The real-space results demonstrate the power of the cosmic web in breaking degeneracies among cosmological parameters to bring, at all scales, signi cant improvements over the matter power spectrum constraints. This latter shows quick saturation at mildly non-linear scales which limits the e ciency of this statistics in constraining parameters of the model, even when pushing the analysis to small scales. Observations however usually rely on sources mapped in the sky and for which the redshift is used as a measure of distance. The so-called redshift space is well-known to have a signi cant impact on the matter clustering statistics. The two main di erences between the real monopole P (k) and its redshift-space counterpart P s,mm =0 (k) are (see Fig. 2.3 for an illustration): (i) a power boost at large scales due to the coherent motion of matter escaping from voids and moving towards dense regions [Kaiser, 1987]; (ii) a decrease of power at small scales where Finger-of-God e ect is spreading particles around initially residing in a spherical overdensity [Jackson, 1972]. These two e ects are clearly observed when comparing the redshift and real space matter power spectra shown in Fig. 6.11. Spectra de- rived from cosmic environments in redshift space, obtained by applying the same procedure as in real space with identical parameters, are shifted at large scales. This shift appears as a power boost for laments and walls and as a decrease of power for voids and nodes with respect to their real-space counterparts. These two latter environments are the most subject to the decrease of power at all scales. Unexpectedly, the node environment is the most a ected by RSDs at small scales, with a considerable power decrease due to the Finger-of-God e ect, while the damping is much less marked and occurs at smaller scales for laments and walls. It is also particularly emphasised for nodes since they may also be more di cult to detect by the classi cation procedure since they now appear as elongated structures in the line-of-sight (see for instance Fig. 2.4 and corresponding discussion). The individual impact of each cosmological parameter on the matter power spectrum is also di erent in redshift space. As an example, due to the distortions in the line-of-sight direction, a higher value of σ 8 is not implying a simple shift of the spectrum as in real space. Instead, it suppresses additional power at small-scales (large k) due to the Finger-of-God distortions inside non-linear clustered structures. It has been shown that the e ect of massive neutrinos can be mimicked by a decrease of σ 8 on the redshift-space monopole [see e.g. Villaescusa-Navarro et al., 2018;Hahn et al., 2020], which in turn has a similar impact as a shift of Ω m . This is illustrated in the left panel of Fig. 6.12 showing the residuals P s,mm =0 (k) θ i /P s,mm =0 (k) fid -1 with θ i being either σ - 8 , Ω - m or M + ν . The shape dependencies of the matter power spectrum for variations of σ 8 and M ν have been shown similar at scales k > 0.1 h/Mpc in Villaescusa-Navarro et al. [2018] and it is possible to nd a value of σ 8 tting at some percent levels the spectrum obtained for M + ν . Monopoles obtained in the cosmic environments however show various dependencies for changes in σ 8 and M ν , as delineated in the right panel of Fig. 6.12. Consequently, a change in σ 8 cannot reproduce the e ect of massive neutrinos in all environments, inducing a breaking of degen- eracy between the two parameters. The di erences between real and redshift monopoles can be appreciated when comparing the derivatives respectively in Fig. 6.5 and in the top panel of Fig. 6.13. Notable changes are observed in the σ 8 and M ν panels in which the matter spectrum P s,mm =0 (k) is showing a broader range of amplitudes suggesting more changes with these parameters in redshift than in real space and hence di erent constraints obtained for it. The monopoles of the environments, on their side, are still showing di erent shapes from each other, highlighting di erent variations with cosmological parameters. These various e ects illustrate the ability of statistics built from the combination of the spectra from di erent environments to break degeneracies between cosmological parameters and subsequently improve the constraints over the sole information from the matter power spectrum. The second ingredient of the Fisher analysis, the correlation coe cients between wavenumbers of the matter and environmental monopoles C k 1 ,k 2 de ned in Eq. (6.16), are respectively shown in the bottom-left block-submatrices of the left and right panels of Fig. 6.14. Compared to the realspace matrices obtained in Fig. 6.6, the redshift-space case show much less high-amplitudes o -diagonal terms. Taking the same example as in real-space, C k 1 ,k 2 ∼ 0.20 for k 1 , k 2 ∼ 0.30 h/Mpc. This phenomenon is also visible for the environment-dependent spectra, and more particularly in the case of nodes where the correlation between modes is also reduced.

All the discussed e ects on the matter power spectrum P s,mm =0 in redshift space are leading to tighter constraints on cosmological parameters shown in the rst row of Table 6.4. Almost all parameters are getting considerable gain over the monopole of the real-space analysis, except σ σ 8 which remains roughly at the same value. As we have been arguing in the previous Fig. 6.12. A closer look at the impact of σ 8 , Ω m and M ν on the matter power spectrum and those derived from cosmic environments. Are shown the residuals of matter (left panel) and spectra in cosmic environments (right panel) P s,αα =0 (k) θ i /P s,αα =0 (k) fid -1 with θ i = σ - 8 (plain line), M + ν (dashed line) or Ω - m (dotted dashed line). paragraph, the various sensitivity of cosmic environments to some cosmological parameters are still leading to a sizeable decrease of the marginalised errors σ θ i when all of them are combined together. Even though they perform individually worse than the matter two-point analysis, their combination is allowing, by breaking some degeneracies, to bring improvement factors of 1.4 to 3.8 on cosmological parameters and 5.5 on M ν . Adding the next non-zero multipole order in the matter analysis, namely the quadrupole P s,mm =2 , is not changing signi cantly the constraints on Ω b , h and n s but improves drastically those on Ω m , σ 8 and M ν respectively with factors 1.6, 6.2 and 4.6, as shown in the bottom-half part of Table 6.4. These tighter constraints obtained by the matter power spectrum in redshift space are explained by the breaking of degeneracies in the M ν -σ 8 plane allowed by the quadrupole. Note however that the absolute value of these constraints are optimistic since our analysis rely on dark matter particles. Using tracers like halos or galaxies would considerably reduce the constraining power of the two-point statistics [see for instance Table 2 of Hahn et al., 2020]. This improvement is highlighted by much more features observed in the derivatives of the quadrupole with parameters like σ 8 and M ν of the bottom part of Fig. 6.13. In this case, the 320 Fourier coe cients resulting from the combination of monopoles and quadrupoles in cosmic environments lead to improvement factors of 1.0 to 2.3 for cosmological parameters and 2.4 for the summed neutrino mass. In particular, we denote no further improvement of the error on σ 8 , already well-constrained by the P s,mm =0 + P s,mm =2 statistics. This is also illustrated in the 1σ θ i marginalised con dence ellipses presented in Fig. 6.15 where the ellipses from environmental combination are superimposed on the matter ones for almost all paired parameters with σ 8 . We still observe, however, some di erent orientations for ellipses in several planes like the M ν -σ 8 or Ω m -n s which are at the origin of the non-negligible improvements for the constraints on M ν and n s over the matter monopole and quadrupole analysis. Exactly as for the real-space case, these gains are observed for all considered k max ∈ [0.11, 0.5] h/Mpc scales, as shown in Fig. 6.16 and especially marked by a larger improvement factor when considering only the largest scales k max < 0.20 h/Mpc. It is also interesting to note the saturation of the void environment at roughly 0.20 h/Mpc which then sees its constraining power improving again at scales of 0.35 h/Mpc. Overall, while the matter power spectrum shows a plateau for all parameters, the combination of environments still suggest further improvement when going to smaller scales for some parameters like Ω m , Ω b , or h. Finally, in Fig. 6.17 are shown the evolution of the SNRs with the maximum scale k max for the monopole-only analysis in dotted line and the one adding the quadrupole in solid line. The rst remarkable feature is the saturation of information for P s,mm =0 which slowly increases from 0.20 to 0.50 h/Mpc while increases much faster when adding the quadrupole. This is coherent with the observed constraints by the Fisher analysis. On the other hand, the environmental SNR are not changing much when adding the quadrupole, except for nodes which show a regain at small scales. Quantitatively, the SNR of the combined environmental clustering statistics is twice the one of the matter at all scales when using both monopoles and quadrupoles. Finally, adding the information of the next non-zero multipole = 4 in the analysis does not improve further the constraints at the considered scale of 0.5 h/Mpc, both for the matter power spectrum and the ones obtained in the cosmic environments.

Stability and convergence analysis

In this Fisher forecast, we resort to numerical computations of the precision matrices de ned in Eq. (6.13) but also of the derivatives from Eq. (6.14) and (6.15). To avoid biased results induced by a non-convergence of these quantities, it is essential to check the stability of the derived constraints under reduction of both N fid , the number of simulations used to compute the covariances and N deriv , the number of simulations for the derivatives. We focus here on the convergence of the constraints σ θ i derived from the combination of power spectra in all environments yielding the maximum length among all the studied statistical summaries with n = 160 in real space and n = 320 when combining the monopole and quadrupole in redshift space. Note that the convergence of the matter power spectrum is already studied in Villaescusa-Navarro et al. [2020]. Figures 6.18 and 6.19 show how the marginalised constraints σ θ i individually behave respectively in real and redshift space when varying N fid and N deriv . .17. Evolution of the SNR with the maximum scale for the power spectra used in the Fisher analysis in redshift space, namely k max , for the monopoles P s =0 only (dashed lines) and monopoles plus quadrupoles P s =0 + P s =2 (plain lines).

We see that, for all parameters, convergence at a ±1% level is obtained when N fid ∼ 5000 and N deriv ∼ 300 in real space while these values are respectively shifted to 5250 and 350 in redshift space. These results show the good convergence properties of the analysed statistics, excluding any bias induced by numerical instabilities in the computation of Fisher constraints.

Conclusion and perspectives

In this work, we have carried out the rst quantitative analysis of the cosmological information content of all cosmic web environments based on the two-point clustering statistics in Fourier space. The auto-spectra derived from environments were computed from individual density elds identi ed with the T-web formalism which relies on the local curvature of the gravitational potential to depict nodes, laments, walls and voids. Using the large suite of Quijote simulations, we were able to carry out a Fisher analysis by computing the partial derivatives and the covariance matrix of the extracted statistics in a statistically robust manner in the non-linear regime with k max ≈ 0.5 h/Mpc. We compared the constraints put by the environments upon the matter power spectrum analysis on the set of ve cosmological parameters {Ω m , Ω b , h, n s , σ 8 } and the summed neutrino mass M ν . In particular, we nd that • The power spectra computed in the cosmic web environments show di erent shape dependencies when varying some cosmological parameters such as M ν , Ω m and σ 8 in real and redshift spaces that are not simple translations of the matter power spectrum. These variations originate from the intrinsic di erences in densities and hence evolution histories of each environment, where the observed structures at z = 0 are imprinted di erently depending on the cosmology.

• From a quantitative point of view, the combination of power spectra in the environments at linear and non-linear scales (k max = 0.5 h/Mpc) leads to the possibility of breaking some key degeneracies between parameters of the model which consequently allows to tighten the constraints with improvement factors of {8.0, 4.5, 6.7, 17.1, 6.5, 15} respectively on parameters {Ω m , Ω b , h, n s , σ 8 , M ν } over the power spectrum in real space. Redshift-space constraints, although already narrowed down when using the joint analysis of the matter monopoles and quadrupoles, are getting even tighter when combining the environmental auto-spectra allowing factors of improvements up to 2.3 on n s and 2.4 on the sum of neutrino mass.

• The constraints obtained from the combination of two-point statistics in Fourier space in the cosmic environments are superior for the whole range of maximum scales analysed k max ∈ [0.11, 0.5] h/Mpc and this statistics yields a 8 (resp. 2) times higher signal-to-noise ratio than the matter power spectrum in real space (resp. redshift space).

The matter component used in this analysis, the cold dark matter, is however not a direct observable. Observational analyses rely on statistics derived from biased and visible tracers of the matter, such as galaxies. The rst issue of handling such biased tracers is that they may trace di erently the cosmic environments. For instance, using the halos of the Quijote simulations with the same mass threshold as used in Hahn et al. [2020], namely M > 3.2 × 10 13 M /h, would considerably alter the detection of low-density environments like voids or walls that we do not expect to host halos of such mass [Cautun et al., 2014]. It is then necessary to infer the underlying dark matter clustering taking this bias e ect into account (see Sect. 2.2.2). Bias usually comes in the analysis as an additional nuisance parameter that needs to be inferred from the data itself, thus contributing to increase of the size of the parameter space and may also lead to additional degeneracies among parameters reducing consequently the constraining power of the statistics. In our case, assuming a linear biasing scheme, we would need to take an individual bias term into account for each environment {b V , b W , b F , b N }.

It is also well-known that adding constraints on the cosmological parameters from the independant cosmic microwave background measurements as a prior in such analyses of the matter power spectrum can considerably reduce the obtained errors. In this work, we however did not consider additional priors given that the aim was to carry a theoretical quanti cation of the information content of cosmic environments only. Adding such priors could however be of interest when targeting real observations, together with matter tracers, which will be the purpose of future works.

The estimates of the matter power spectrum weighted by the local density proposed by White [2016] yields a non-linear transform of the overdensity eld that we exposed in Eq. (6.18). Such a weighted version of the power spectrum has been shown to contain more information than the standard matter power spectrum in real-space by Massara et al. [2021]. It is however well-established that cosmic environments are complex objects and cannot be uniquely described in terms of density thresholds [see e.g. Cautun et al., 2014;Libeskind et al., 2017]. In this context, it would be interesting to link the presently exposed analysis of the environments with a mark depending on both the density and the level of local tidal anisotropy. This would have the vast advantage of relieving the analysis from detecting the environments with a pre-xed de nition.

In conclusion, we have shown that there is signi cantly more information contained in the density eld when analysing independently the cosmic environments and then using their combination rather than directly relying on the matter density eld summarised by solely its power spectrum, as it is traditionally done. The sizeable improvements in the constraints on all cosmological parameters brought by our environment-dependent analysis, even in the ideal case addressed in the present study, opens up the possibility to take advantage of the spectra and cross spectra in environments for the optimal exploitation of future large galaxy surveys such as DESI [Levi et al., 2013] or Euclid [Laureijs et al., 2011].

Conclusion

"On n'aura pas de poste permanent, mais on aura bien rigolé. "

V. B

The initial matter distribution, a nearly-Gaussian density eld, evolved through billions of years under the e ect of gravity eventually drawing the complex picture of the cosmic web that astronomers, astrophysicists and cosmologists are together trying to observe, describe and model. Throughout this thesis, we explored multiple facets of the cosmic web with, as underlying guideline, the extraction of relevant physical or cosmological information from cosmological datasets of the large-scale structures.

Naturally, such a question is tightly coupling multiple research elds among which are data science and statistics. This is hence by rst taking a step back from the cosmological questions that we proposed to build representations of generic point-cloud datasets. By resorting to a statistical physics reformulation of the clustering, we were able to improve our understanding of how the structure of a dataset is driving the partitioning of datapoints into multiple groups. The identi cation and follow-up of successive phase transitions occurring during a simulated annealing allowed us to extract information about the number, sizes, and hierarchical embeddings of the multiple sub-structures in the data. We showed, for di erent regularised models, that the quantity at the origin of the transitions can explicitly be derived and exactly computed for the rst transition and approximated for the following ones.

Building upon this clustering procedure, we have established a graph regularised model to represent continuously-structured datasets. By assuming that the data are living on a latent one-dimensional manifold, and modelling this latter as a graph, we built a new framework for the estimation of principal graphs. Taking advantage of probabilistic foundations, the algorithm we proposed was shown particularly e cient to unveil the pattern in complex datasets with multiple scales and high levels of noise and outliers. Beyond its mathematically appealing properties, such as a guaranteed convergence towards a local maximum during the optimisation, it is also fast, making it suitable for the description of large datasets.

Such multi-dimensional datasets in which a representation must be learnt to allow further scienti c analyses can arise in various elds, ranging from biology to cosmology. In our case, we used the previously-exposed principal graph method, T-ReX, to carry out an in-depth analysis of the lamentary structure of the cosmic web as depicted by galaxies in hydrodynamical simulations. We exposed how the graph representation of the interconnected network of galaxies can be used to de ne individual laments. By focusing on a set of three simulations, we established the main properties of the laments statistics such as their length and curvature distributions with their characteristic exponential tails showing only rare long and highly-curved laments. In this intricated cosmic network, a special focus was put on laments themselves but also in the way they interact with nodes of the cosmic web. We showed that these anchors can be readily expressed in the T-ReX formalism as dense bifurcations, leading to the identi cation of the environments of galaxies in agreement with the physical de nition of locally isotropic regions in the matter density eld. The connectivity of nodes, quantifying their local embedding in the cosmic web, was also shown to be particularly sensitive to the assembly history of clusters. In particular, we provided evidences for a scenario in which early-formed and unrelaxed clusters are, on average, more connected to the lamentary pattern than old and relaxed clusters.

All these results emphasise the importance of identifying reliably the di erent environments to make the best use of cosmological observables in order to understand the formation of structures at all scales, from galaxies to clusters and how the cosmic web is shaping these elements of the Universe. By undertaking a thorough quantitative analysis of the two-point information in the cosmic web environments, we showed that the combined use of nodes, laments, walls and voids e ciently breaks degeneracies between cosmological parameters to which these environments have di erent sensitivities. We thus tightened the constraints on cosmological parameters, over the use of the full matter power spectrum, by up to one order magnitude. We thus pictured the cosmic environments as a gold mine of information about the Universe itself, able to bring tremendous knowledge about the underlying cosmological model.

This thesis was conducted in the perspective of optimising the modelling, the characterisation and the utilisation of large-scale structure datasets for cosmological analyses. In this context, more and more importance is particularly given to topological and geometrical representations of the cosmic web enabling new insights in the study of the structure formation and evolution at multiple scales. This growing interest is obviously fuelled by the considerable advances from the observational, theoretical and statistical communities. On the observational side, the next generation of large-scale surveys will map the large-scale matter distribution at all wavelengths from the optical to the radio with Euclid, the Vera Rubin Observatory, DESI and SKA. These new data, their cross-correlations and combinations, will not only allow us to constrain the cosmological parameters of the standard cosmological model with accuracies orders of magnitudes better than what is currently achieved but they will also allow the testing of alternative theoretical models proposed to explain the Universe formation and evolution. It is paramount that the scienti c exploitation of these surveys is accompanied with an extensive e ort aiming at devising optimised methods to extract meaningful information and hence address the cosmological questions.

The work presented in the thesis mainly targeted the topological representation of the cosmic web as a graph structure, focusing of the global characteristics of the matter distribution and at present time. Graphs built from the spatial proximity of galaxies are a promising way to build new summary statistics based on quantities derived from graph theory such as the degree (the number of edges connecting a given galaxy), the assortativity (the correlation between linked galaxies) or the shortest paths (set of shortest geodesic paths on the graph linking two galaxies) that have already been shown to encode cosmological information [Hong & Dey, 2015;Hong et al., 2016;Naidoo et al., 2020]. Studying the correlations between the physical properties of graph nodes (galaxies) and graph properties (degree, assortativity, length, etc.) also o er a unique opportunity to link the topology of the cosmic web to the physics of tracers.

Another avenue of research interesting to follow is the study of both the evolution and dynamics of the large-scale lamentary pattern as the temporal morphism of the extracted graph structures. Indeed, whilst the proposed work, as well as the current literature gives importance to the pattern at present time, or studies it independently at several time snapshots, only few works address the time evolution as a set of successive events. This could in particular allow the interpretation of the topological modi cations of the graph structure, i.e. the local variations in the number of branches and the related size of cycles in the topology, as thermo-dynamic events of the cosmic pattern. By building a model bringing key information of the time evolution of the lamentary pattern as viewed by the successive merging of primordial laments, it would provide a perfect playground for the testing of hypotheses, such as the cosmic web detachment and other environmental quenching e ects argued by several works [see e.g. Aragon Calvo et al., 2019].

The complexity of datasets encountered in cosmology, involving a wide range of scales, linear and non-linear physics, makes it an adapted eld to apply and develop machine learning methods, which were shown particularly e cient in solving problems intractable so far based on such large and complex data in reasonable times. The outstanding results obtained by the rst applications of deep learning methods in astrophysics are however often limited by the di culties in interpreting the obtained models. It is only with the joint e ort of various communities, notably from computer science and statistical physics that these problems could be readily tackled. In particular, the structure of the data and how this latter is embedded in space (symmetries, lower-dimensional manifolds, etc.) is expected to play a major role in the learning dynamics [see e.g. Goldt et al., 2020]. The work presented in this thesis explored this aspect in the simple context of clustering. An exciting perspective would be to develop more fundamental relations between the data structure and the features that are learnt in the context of simple neural networks model to better understand what makes them particularly successful for certain tasks.

On the application side of the machine learning, many recent works investigate the possibility to recover the early and/or late time large-scale dark matter distribution from the observation of sparse tracers like halos, galaxies or clusters at redshift z = 0 [see for instance [START_REF] Jasche | [END_REF]Leclercq et al., 2015;Schmittfull et al., 2017]. This mapping problem is made complex by the non-linear relation between the non-Gaussian density eld and its sparse discrete representation by the distribution of galaxies. For the purpose of inferring non-linear relations between two or three dimensional elds, machine learning have already been shown particularly powerful. The recent availability of large amount of simulations targetting the speci c needs to perform statistical analyses and machine learning applications like Quijote [Villaescusa-Navarro et al., 2020] makes it now possible to explore deep learning methods for such cosmological investigations. Coupled to the previous point on their interpretability, such models, if well-understood, could help improving our knowledge on the interplay between dark and baryonic matter but more so on the evolution of the Universe.

[Wilding et al. 2020] W , Georg, N , Keimpe, W , Rien Van D., et Résumé : La distribution de matière dans l'Univers se présente sous une structure complexe que l'on appelle la toile cosmique. Dans cette disposition spatiale, des régions denses, les noeuds de la toile cosmique, sont reliés par des ponts de matière, les filaments, qui se trouvent à l'intersection de structures planaires moyennement denses appelées murs qui définissent eux-mêmes les bords de vastes régions vides. Cette distribution, façonnée par la gravité depuis des milliards d'années, contient de précieuses informations sur le modèle cosmologique sous-jacent mais également sur les conditions initiales de l'Univers et son évolution. La détection et l'étude des éléments de la toile cosmique, qui jouent également un rôle fondamental dans la formation et l'évolution des galaxies, constituent de véritables défis nécessitant la conception d'outils sophistiqués pour traiter la complexité des structures multi-échelles qui la compose.

Avec pour ambition d'identifier et de caractériser les différents environnements, cette thèse propose plusieurs approches pour analyser des jeux de données spatialement organisés au moyen de méthodes d'apprentissage non supervisé fondées sur les modèles de mélanges. En particulier, des principes dérivés de la physique statistique sont utilisés pour mieux appréhender et comprendre la dynamique d'apprentissage d'un algorithme de classification non supervisé. Nous exposons comment utiliser ce parallèle avec la physique statistique afin d'explorer le jeu de données et obtenir des informations sur sa structure. Afin d'identifier la structure filamentaire de la toile cosmique, nous construisons en-suite une version régularisée de la procédure de classification pour apprendre itérativement une représentation du jeu de données, que l'on suppose généré par une structure uni-dimensionnelle sous-jacente. La méthode modélise cette structure latente par un graphe qui est intégré comme un a priori dans la formulation Bayésienne du problème menant à l'estimation d'un graphe principal passant au centre de la distribution de matière tracée par les galaxies. Nous montrons que cette formulation est particulièrement adaptée à la description des filaments cosmiques puisqu'elle permet la description de leurs propriétés géométriques (longueurs, épaisseurs, etc.) ainsi que l'association, pour les traceurs (galaxies, halos), d'une probabilité d'appartenir à un filament donné. L'algorithme proposé dans la thèse est appliqué avec succès à des simulations numériques. Ces applications ont notamment permis l'étude des relations entre la connectivité des amas de galaxies dans la toile cosmique et leurs propriétés dynamiques et morphologiques. Enfin, nous réalisons, à partir d'un ensemble de simulations à N -corps, une étude approfondie de l'information cosmologique contenue dans les environnements de la toile cosmique (noeuds, filaments, murs et vides). Il est notamment montré que l'analyse des environnements permet de lever les dégénérescences entre certains des paramètres du modèle faisant de la toile cosmique une sonde alternative permettant d'améliorer significativement les contraintes sur les paramètres cosmologiques vis-à-vis des analyses conventionnelles.

Abstract:

The late-time matter distribution depicts a complex pattern commonly called the cosmic web. In this picture, the spatial arrangement of matter is that of dense anchors, the nodes, linked together by elongated bridges of matter, the filaments, found at the intersection of thin mildly-dense walls, themselves surrounding large empty voids. This distribution, shaped by gravitational forces since billions of years, carries crucial information on the underlying cosmological model and on the evolution of the large-scale structures. Detecting and studying elements of cosmic web, playing also a key role in the formation and evolution of galaxies, are challenging tasks requiring the elaboration of optimised methods to handle the intrinsic complexity of the pattern made of multi-scale structures of various shapes and densities.

With the aim of identifying and characterising the cosmic web environments, we propose several approaches to analyse spatially structured point-cloud datasets, not restricted to cosmological ones, by means of unsupervised machine learning methods based on mixture models. In particular, we use principles emanating from statistical physics to get a better understanding of the learning dynamics of a clustering algorithm and expose how statistical physics can be used to explore the data distribution and obtain key insights on its structure. In order to identify the filamentary part of the pattern, its most prominent feature, we propose a regularisation of the clustering procedure to iteratively learn a non-linear representation of structured datasets, assuming it was generated by an underlying one-dimensional manifold. The method models this latent structure as a graph embedded as a prior in the Bayesian formulation of the problem to estimate a principal graph passing in the ridges of the matter distribution as traced by galaxies or halos. We show that this formulation is especially wellsuited for the description of the filaments since it allows the description of their geometrical properties (lengths, widths, etc.) and associates to each tracer a probability of residing in a given filament. The resulting algorithm is successfully used to detect filaments in state-of-theart numerical simulations. It also allows us to study the relation between the connectivity of galaxy clusters to the cosmic web and their dynamical and morphological properties. Finally, based on a large suite of N -body simulations, we perform a comprehensive analysis of the cosmological information content based on the two-point statistics derived in the cosmic web environments (nodes, filaments, walls and voids). We show that they can break some degeneracies among key parameters of the model making them a suitable alternative probe to significantly improve the constraints on cosmological parameters obtained by standard analyses.

Université Paris-Saclay Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France

IntroductionI

  Emergence of large-scale structures 5.2 (left) Probability map I obtained from subhalos displayed in the left panel of Fig. 5.1 with B = 100 and N b = 0.75. The resolution of the probability map is 250Kpc/h. (right) Superlevel set Γ 0.25 (I) (red squares) overplotted on the DM distribution together with subhalos (black dots). . . . . . . . . . . . . . . . . . 5.3 Probability maps with increasing mass threshold M cut . . . . . . . . . . . . . . 5.4 Robustness to sparse samplings by the cumulative distribution of distances {d J x } 5.5 Identi cation results provided by four detection methods on a randomly chosen 2 Mpc/h depth slice of the full 3D detection for each method. . . . . . . . . . . 5.6 Illustration of the de nition of individual laments . . . . . . . . . . . . . . . 5.7 Galaxy distribution of the EAGLE simulation coloured by their environments by Nexus+ and T-ReX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Probability distribution functions of several quantities extracted from the three catalogues of laments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Evolution of the average curvature and density of laments as a function of their length L for the three catalogues . . . . . . . . . . . . . . . . . . . . . . . 5.10 Illustration of the ellipsoidal shapes and large-scale environments of four simulated galaxy clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Probability distribution function of groups and clusters connectivity, κ, for four mass bins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Ellipticity and mean instantaneous mass growth dM/dt as a function of mass for three connectivity bins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 Evolution of the connectivity as a function of mass for relaxed and unrelaxed clusters and with di erent mass assembly histories. . . . . . . . . . . . . . . . 6.1 Overdensity eld computed in the several cosmic environments shown in a slice of a Quijote simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Averaged mass fractions f α in the T-web environments for distinct values of λ th in real and redshift spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Ratio of environmental mass fractions between di erent cosmologies and the ducial ones for an eigenvalue threshold of λ fid th . . . . . . . . . . . . . . . . .

  INTRODUCTION (T. Bonnaire, A. Decelle & N. Aghanim, Cascade of phase transitions for multiscale clustering, Phys. Rev. E 103 012105 (2020) [arXiv:2010.07955]). Chapter 4 introduces a new formulation of the principal graph learning problem presented in T. Bonnaire, A. Decelle & N. Aghanim (Regularisation of Mixture Models for Robust Principal Graph Learning, under review in IEEE

1. 1

 1 The homogeneous universe . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.1 Distances in an expanding universe . . . . . . . . . . . . . . . . . . . 11 1.1.2 The dynamics of the homogeneous Universe . . . . . . . . . . . . . . 12 1.2 The birth of large-scale structures . . . . . . . . . . . . . . . . . . . . . 13 1.2.1 Linear perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.2 Zel'dovich formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 Statistical descriptions of the matter distribution . . . . . . . . . . . . 16 1.3.1 Discrete random elds . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.2 Correlation functions and poly-spectra . . . . . . . . . . . . . . . . . 16 1.4 The ΛCDM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.1 Presentation of the model . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.2 Cosmological parameters and matter power spectrum . . . . . . . . 19

Fig. 1

 1 Fig. 1.1. Galaxy distribution from the Center for Astrophysics Redshift Survey [CfA, de Lapparent et al., 1987]. A galaxy cluster at the center is connected by elongated laments.

Fig. 1 . 2 .

 12 Fig. 1.2. The 3D linear matter power spectrum at z = 0 drawn from di erent observables. The solid black line is the theoretical expectation given the best-t Planck Collaboration VI et al. [2020] cosmology while the dotted line is the non-linear power spectrum. Figure reproduced from Chabanier et al. [2019].

Fig. 2 .

 2 Fig. 2.1. The sophistication of simulations from the seventies to nowadays. (left) 2D distribution of points moved according rst non-linear approximations in cosmological simulations. Image from [Doroshkevich & Shandarin, 1978]. (right) A 2D slice of the dark matter density eld from the Illustris simulation [Vogelsberger et al., 2014].

Fig. 2 . 3 .

 23 Fig. 2.3. Schematic illustration of the e ect of redshift space distortions for spherical overdensities (represented as the black circles) at large and small scales. RSDs only alter the position of objects that have a non-zero velocity component along the line of sight (coloured points only).

Fig. 2 .

 2 Fig. 2.4. E ect of redshift-space distortions in N -body simulations. (left) Representation of log 10 (2 + δ) in real space on a 2D slice of the Quijote simulation. (right) Same in redshift space, assuming Z as the line of sight and displacing particles according Eq. (2.1) with a = 1 and H = H 0 .

3 Fig. 2

 32 Fig. 2.5. The limitation of power spectrum analyses. (left) Overdensity eld log 10 (2 + δ) of a 2.77 Mpc/h depth slice from a N -body simulation. (right) Random phase reshu ing of the left panel. Both elds have the exact same evaluation of the power spectrum since P (k) is blind from phase information.

Fig. 2 .

 2 Fig. 2.6. Illustration of the complexity of cosmic web pattern. (left) The lamentary structure detected by NEXUS+ in the EAGLE [Schaye et al., 2015] simulation. Image from Fig. 1 of Cautun et al. [2014]. (right) Density PDF in each environment of the cosmic web as detected by NEXUS+. Plot from Fig. 13 of Cautun et al. [2014].

Fig. 2 .

 2 Fig. 2.7. (top) A slice from the Illustris simulation showing the estimated density eld δ drawn from dark matter particles. (bottom) The galaxy counterpart. The sparse sampling of the pattern by the visible matter makes the extraction of the pattern more complicated in observations than in simulations where the full information is available.

Figure 3 Fig. 3 . 2 .

 332 Fig. 3.2. Illustration of the interest of mixture models. Black dots are datapoints generated by three Gaussian clusters. Green, orange and blue ellipses are 1-σ con dence ellipses assuming a Gaussian mixture model with K = 3 while the black ellipse is obtained by maximising a Gaussian likelihood from the data. Discussed in Sect. 3.2.

Fig. 3 .

 3 Fig. 3.4. (left) Four con gurations of the system for di erent values of the temperature during the annealing procedure. Black points are datapoints, coloured crosses are positions of cluster centres. From top left to bottom right, σ 2 = {33.59, 13.76, 0.76, 0.10}. Grey shaded areas correspond to 1-σ circles. (right) Evolution of the average energy E(Z) of Eq. (3.18) of the system as a function of the temperature σ 2 . The black dashed dotted line corresponds to the critical temperature T hard c = 26.87 while the grey ones illustrate the temperatures of the successive further transitions.

Fig. 3 .

 3 Fig. 3.5. (left) Displacement of K = 25 centres during the annealing procedure for a dataset with ve spherical Gaussian clusters. Colours indicate in which nal cluster the center ends. (right) Evolution of the ratio Γ k /σ 2 as a function of σ 2 . Black stars correspond to the scales of successive transitions, the black vertical line to T hard c and coloured ones indicate the size of the clusters as de ned by the maximum eigenvalue of the empirical covariance. The black dashed curve shows to the evolution of Q as de ned in Eq. (3.25) that we identify as an order parameter. This quantity is not represented for σ ≤ 1 since the number of physical clusters K r begins to be higher than the number of generated clusters q.

Fig. 3

 3 Fig. 3.7. (left) Displacement of K = 25 centres during the annealing procedure for a dataset made of ten 5D spherical Gaussian clusters visualised in the plane of the two rst principal components. Colours depend on the macro-cluster the component stands in at the last iteration. (right) Evolution of the ratio Γ k /σ 2 as a function of σ 2 , the hard annealing parameter. Coloured vertical lines indicate the actual size of the corresponding Gaussian cluster or macro-cluster (grey dashed lines, rst from the right in each panel) as de ned by the maximum eigenvalue of the empirical covariance.

Fig. 3 .

 3 Fig. 3.8. (left) Arrows indicate the displacement of K = 25 centres during the soft annealing procedure for a dataset made of six spherical Gaussian clusters (black points). Colours relate to the cluster in which the component ends. Red crosses and grey dashed circles respectively indicate the positions and variances xed a posteriori when the center undergoes its last split before remaining above the Γ k /σ 2 k = 1 line. (right) Evolution of the ratio Γ k /σ 2 k as a function of σ 2 . The vertical black line corresponds to T soft c . The inset gure shows the evolution of the ratios max Q/Q th and σ2 /σ 2 true , when varying the contrast between the two nested clusters.

Fig. 3 .

 3 Fig. 3.9. Evolution of T graph c

Fig. 3 .

 3 Fig. 3.10. (top left) Displacement of K = 100 components during the hard annealing of a tree branches dataset with di erent sampling standard deviations. Black dashed line corresponds to the rst principal direction. Colours refer to branches in which centres end. (top right) Learned structure when stopping the annealing for components reaching the temperature σ 2 γ k . Red lines are edges of the graph and grey shaded areas are 1-σ k circles. (bottom) Evolution of the ratio γ k /σ 2 as a function of σ 2 . Vertical lines indicate the used variance for the generation of branches. The black vertical line corresponds to the value of T graph c predicted by the linear stability analysis of the model in Eq. (3.40).

Fig. 4 . 2 .

 42 Fig. 4.2. Several graphs built from the set of datapoints in black showing a noisy "S" shape with N = 220 points among which 20 are outliers. From left to right: the -neighbourhood with visually shown, the k-nearest-neighbours (k = 6) and the minimum spanning tree.

Fig. 4 . 3 .

 43 Fig. 4.3. Schematic view of the principal graph modelling proposed in this work. Datapoints are in black and represent a noisy heteroscedastic sampling of an underlying sinewave shown in dashed red. The model we propose is based on a coupling between a mixture model and a graph G linking Gaussian clusters together with an MST topology prescription. Blue crosses are averages of Gaussian {µ k } and grey shaded areas are the associated spherical covariance matrices {Σ k }. Edges of the MST linking Gaussian clusters are shown in dark blue.

Fig. 4

 4 Fig. 4.4. Illustrative comparison of three procedures to learn a principal graph on an arti cial dataset made of N = 2666 datapoints with three branches of linearly evolving standard deviations converging to a spherical Gaussian cluster shown in the top left panel. Top right is the principal graph learned by the proposed GRMM algorithm (1) with K = 100 initialized randomly over X, Y = {5/σ 2 0 , 10, 1} and σ 2 0 = 0.01.Bottom left is the one from the SimplePPT algorithm[Mao et al., 2015[Mao et al., , 2017] ] with σ = σ 0 and identical initialisation nodes. In both cases, the shaded areas correspond to the 1-σ k circles centred on µ k . The bottom right panel is the result from the ElPiGraph procedure[Albergante et al., 2020] with 100 nodes initialized with 70 taken randomly over X and for di erent values of the trimming radius R 0 with the same elasticity parameters λ = 0.01, the length constraint and µ = 0.01 the bending constraint.

  K k=1 . When re ning the graph, update equation for µ k of the SimplePPT algorithm proposed in Mao et al. [2015] can be derived from Eq. (4.14) by considering xed variances, no outlier distribution and an MST prior.

Figure 4 .

 4 Figure 4.4 provides an illustrative comparison of several algorithms to identify a principal graph in point-cloud datasets. The top right panel of Fig. 4.4 shows the graph learned by the GRMM Algorithm (1) for a dataset made of three branches with linearly evolving standard deviations σ ∈ [0.015, 0.15] and 25% background noise uniformly added in the square.The bottom left panel of Fig.4.4 illustrates the principal graph learned by the SimplePPT algorithm on the same dataset and using the same parameters and initialisation as our construction. Many branches are falling in the background noise and, because of its xed-variance scheme, it fails at capturing the ridges of both the large and small variance branches. Choosing a large value for σ 2 0 biases the graph in the small branch, while a low one irremediably creates a lot of spurious branches in the large-variance part of the pattern, even in the absence of outliers. With its contracting iterative scheme, the Subspace Constrained Mean Shift[SCMS, Ozertem & Erdogmus, 2011; Genovese et al., 2014; Chen et al., 2014] algorithm would su er less from the latter exposed problem. Yet, it does not provide a properly-speaking curve, but a set of independant projected points that are to be linked a posteriori and the SCMS is also usually used together with a pre-processing step to lter datapoints standing in low-dense areas such as proposed in Chen et al.[2015] hence involving an additional non-trivial parameter impacting the quality of the result. The sensitivity of the graph to outliers is handled by the ElPiGraph framework[Albergante et al., 2020] through a trimming radius R 0 discarding distant datapoints from the update of a node position. In the bottom right panel of Fig.4.4 can be found three realisations of this algorithm with di erent values of R 0 , showing the robustness of ElPiGraph to uniform background noise with a ne-tuned value. The choice of R 0 however remains scale-dependent and can alter the recovered structures in case of heteroscedastic patterns. It is also not obvious to tune in real applications and comes in addition to other hyper-parameters already complex to choose. Moreover, the resulting graph do not embed a description of the local width as opposed to the GRMM which ts all parts of the pattern in its center. Note nonetheless that the ElPiGraph algorithm handles better the dense area at the center of the dataset when R 0 = 0.3 with a single bifurcation thanks to an added terms to the cost function minimising the overall graph complexity. The algorithm we develop here[Bonnaire et al., 2021b], by unifying all these aspects of robustness and local size description in a single formulation, is able to provide a principal graph that visually shows a smoothly evolving adaptive scale along the graph structure. Even though the graph is initialized with only K = 100 nodes taken randomly over X with variances ∀k, σ

Fig. 4

 4 Fig. 4.5. The regularised versions of the graph priors shown in Fig. 4.2 obtained from Algorithm (1) with K = N = 220, Y = {0.5/σ 2 0 , 0, 0} and σ 2 0 = 0.1. From left to right: the -neighbourhood with visually shown, the k-nearest-neighbours (k = 6) and the minimum spanning tree.

Fig. 4

 4 Fig. 4.6. Illustration of the interest of combining MSTs obtained from random sub-samplings of the data. (top left) The regularised MST computed over the all the input datapoints. (top right) The regularised MST computed over a random sub-sampling of the input with N/N b = 0.75 (black crosses). Grey datapoints are those not used. (bottom left) Same as top right for another sub-sampling realisation. (bottom right) The graph obtained using the average graph prior.The two circles in the top right and bottom left panels illustrate that the hole not captured by the MST varies with the sub-samplings. This instability is exploited in the average graph prior to obtain a cycling topology.

FigFig

  Fig. 4.7. Illustration and comparison of the average graph prior. (top) Black points and grey crosses are two sampling realisations of an arti cial 2D dataset obtained from a Voronoi pattern. Red edges are those from the MST and bold dark blue ones are those added by the high probability mode with B = 500, K b /K = 0.75 and thresholded at m = 0.35. The turquoise blue and orange lines are the set of edges from the regularised graph obtained with a prior given by the 1D-homologically persistent skeleton [Kurlin, 2015] with the two realisations of the dataset. When not explicitly visible, it means that the three lines overlap perfectly. (bottom) Probability distributions of edge probabilities Ā ij for several ratios K b /K.

Fig

  Fig.4.9. Illustration of the impact of hyper-parameters Y = {λ µ , λ σ , λ π } in Algorithm (1) for an arti cial dataset made of three branches with linearly evolving standard deviations converging to a spherical Gaussian cluster. In all cases, K = 350 nodes are used with the same random initialisation over X. Black dots are datapoints, coloured lines are the set of edges learned by Algorithm (1) and gray shaded areas correspond to the 1-σ k circles centred on µ k . (top row) From left to right, quadrants corresponds to several values of λ µ , λ σ and λ π . (bottom row) Probability distributions of the impacted parameters. From left to right: edge weights w ij = µ i -µ j 2 , variances of components σ 2 k , and mixing coe cients π k .

  Fig. 4.10. Application of the principal graph learning to the extraction of road network from vehicle positions. (top left) The 67 193 input datapoints of the Berlin dataset. (top right) Black lines are edges of the regularized graph of Algorithm (1) overplotted on the raw datapoints.The graph was obtained using the average graph topology with K = 7300 and σ 0 = 0.003, Y = (10/σ 2 0 , 5, 1). Red circles highlight features that could not be caught by a non-cycling topology. (bottom left) Same as for the top panel, but overplotted on the ground truth map extracted from OpenStreetMap. (bottom right) Probability distributions of variances for nodes in the corresponding circled coloured regions of the top panel.

"

  A quoi bon faire des maths appliquées si c'est pour ne pas les appliquer ?" V. B 5.1 Context and motivations . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Filamentary pattern detection . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 T-ReX: Tree-based Ridge eXtractor . . . . . . . . . . . . . . . . . . . 5.2.2 Filamentary pattern extraction from Illustris subhalos . . . . . . . . 5.2.3 Performance evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Identi cation of individual laments . . . . . . . . . . . . . . . . . . .5.3.1 A graph-based de nition for laments . . . . . . . . . . . . . . . . . 5.3.2 Characteristics of individual laments . . . . . . . . . . . . . . . . . 5.3.3 Association of galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Filaments characteristics in simulations . . . . . . . . . . . . . . . . . 5.4.1 Simulations and principal graphs . . . . . . . . . . . . . . . . . . . . 5.4.2 Comparison of laments characteristics . . . . . . . . . . . . . . . . 5.5 The impact of the cosmic web on cluster properties in simulations . 5.5.1 Data, lamentary pattern and connectivity . . . . . . . . . . . . . . . 5.5.2 Impact of connectivity on the growth and shapes of clusters . . . . . 5.5.3 Impact of cluster dynamical states on the connectivity . . . . . . . . 5.5.4 The in uence of mass growth history . . . . . . . . . . . . . . . . . . 5.6 Summary and perspectives . . . . . . . . . . . . . . . . . . . . . . . . .

Fig

  Fig. 5.1. Di erences between the classical MST (middle panel) and the optimised one obtained from the T-ReX algorithm (right panel) with λ µ = 5, λ σ = 0, λ π = 0, σ 0 = 1 Mpc/h and under a xed-variance scheme. Both algorithms are applied on the 2D subhalos distribution of the Illustris simulation shown in the left panel.

  A encodes the graph adjacency and λ µ is the strength of the prior. More precisely, λ µ regulates the trade-o between the data delity term representing how much the graph should pass close to the galaxies and how much it should be short and smooth. The T-ReX algorithm 1 exposed in Chapter 4 can be used to obtain the best possible 2 graph given an initialisation and a set of hyper-parameters λ µ , λ σ and λ π . The application of T-ReX is shown in right panel of Fig.5.1 based on the input datapoints shown in the left one. The obtained graph has a smooth adaptive behaviour, especially visible in the zoom regions when compared to the crude MST shown in the middle panel that is reaching all datapoints.

Fig. 5 . 2 .

 52 Fig. 5.2. (left) Probability map I obtained from subhalos displayed in the left panel of Fig. 5.1 with B = 100 and N b = 0.75. The resolution of the probability map is 250Kpc/h. (right) Superlevel set Γ 0.25 (I) (red squares) overplotted on the DM distribution together with subhalos (black dots).

3 .

 3 Fig. 5.3. Probability maps with increasing mass threshold M cut . From left to right, M cut = {0, 0.85, 1.35, 3.22, 11} × 10 10 M /h corresponding, respectively, to 100%, 83%, 60%, 31%, and 10% of the total subhalos in the slice.

Fig. 5

 5 Fig. 5.4. Cumulative distribution of distances {d Jx } between positions of the binary maps Γ 0.25 (I 100 ) obtained with increasing mass threshold M cut to the one with J% of the datapoints. M cut = {0.85, 1.35, 3.22, 11} × 10 10 M /h leading respectively to 83%, 60%, 31% and 10% of the total number of subhalos in the slice.

  of the detections provided by the unsmoothed version T-ReX us are contained in the Nexus+ skeleton and 81% of the Nexus+ detections are found by the smoothed version of T-ReX. This indicates that the smoothed version of T-ReX contains a large part of the Nexus+ skeleton but with a larger amount of the volume detected, explained by the smoothing

  Fig. 5.5. Identi cation results provided by four detection methods on a randomly chosen 2 Mpc/h depth slice of the full 3D detection for each method. Green pixels are walls, blue are laments, red are clusters and white are voids or unclassi ed regions.

  Fig. 5.6. Illustration of the de nition of individual laments on a toy dataset. (left) Black points are datapoints and coloured ones are those from the smooth graph structure of T-ReX. Filaments are de ned as the branches linking bifurcations and extremities in the graph as discussed in Sect. 5.3.1. (right) Illustration of associated datapoints to each individual lament (coloured datapoints) or to the detected uniform background (grey datapoints) depending on the graph (black line) and associated variances (not represented) as exposed in Sect. 5.3.3.

Fig. 5

 5 Fig. 5.7. Galaxy distribution of the EAGLE simulation coloured by their environments by T-ReX (left panel) and Nexus+ (right panel). Galaxies in green are those standing in laments while red ones are found in nodes. Note also that we only show the galaxies classi ed in these environments, showing only around 65% of the galaxies in both cases.

b

  Fig. 5.8. Probability distribution functions of several quantities extracted from the three catalogues of laments. Are shown the length L (top left), the curvature γ (top right), the overdensity ρ f /ρ b (bottom left) and the radii as de ned by Eq. 5.8 (bottom right) of laments.

  [2010];Codis et al. [2018]; Darragh Ford et al. [2019];

Fig. 5 .

 5 Fig. 5.10. Illustration of the ellipsoidal shapes and large-scale environments of four simulated galaxy clusters. The blue ellipsoids are computed by calculated the mass tensor of DM distribution, following [Jing & Suto, 2002]. The red lines represent cosmic laments reconstructed by the T-ReX algorithm from the galaxy distribution. Yellow spheres have a radius of 1.5R 200 .

  Fig. 5.11. Probability distribution function of groups and clusters connectivity, κ, for four mass bins.

Fig. 5 .

 5 Fig. 5.12. (left) Ellipticity as a function of the group mass for three bins of connectivity: κ = {1, 2}, κ = 3 and κ > 3. (right) Mean instantaneous mass growth dM/dt computed at z 0 as a function of the group mass for three same bins of connectivity. Error bars are the 68% errors on the mean, derived from bootstrap resampling.

Fig. 5

 5 Fig. 5.13. (left) Evolution of the connectivity as a function of mass for relaxed and unrelaxed clusters. (right) Evolution of the connectivity as a function of mass for relaxed and unrelaxed clusters for three subsamples with di erent mass assembly histories. Error bars are the 68% errors on the mean, derived from bootstrap resampling.

6. 3

 3 Fig. 6.1. A 2.77 Mpc/h depth slices showing the full eld δ in the top panel and the di erent elds {δ V , δ F , δ W , δ N } computed from the T-web classi cation of particles described in Sect. 6.2.2, all estimated with the PCS interpolation scheme. Environmental elds have been normalised such that δ = α f α δ α .

3 .

 3 Fig.6.3. Ratio of environmental mass fractions between di erent cosmologies and the ducial ones for an eigenvalue threshold of λ fid th . Points are centred on the average over the N deriv = 500 realisations for each cosmology and error bars show the ±3σ interval. The grey shaded area corresponds to the ±3σ interval of the ducial cosmology fractions obtained from the N fid = 7000 realisations.

  Fig. 6.5. Partial derivatives ∂P αα (k)/∂θ i for the di erent environmental auto-spectra and for each studied parameters {Ω m , Ω b , h, n s , σ 8 , M ν }. Dashed (resp. plain) lines correspond to negative (resp. positive) values of the derivative. The grey area depicts the range of k > k max excluded from the analysis.

  25 h/Mpc. The individually smaller errors obtained in the environments are not observed at all

Fig

  Fig.6.9. Evolution of the marginalised constraint σ θ i put on cosmological parameters {Ω m , Ω b , h, n s , σ 8 } and the sum of neutrino mass M ν in real space with the maximum scale used for the Fisher analysis, namely k max .

Fig. 6 .

 6 Fig.6.11. Monopoles in real (plain lines) and redshift (dashed lines) spaces averaged over the N fid realisations for the matter and the di erent environments. Monopoles of the environments are individually normalised by their mass fractions f α to see each contribution to the matter monopole.

  Fig. 6.14. (left) Correlation coe cients C ij for matter-matter power spectrum monopole and quadrupole. (right) Same for P s,αα (k) coe cients extracted from the several environments. Each sub-matrix goes from k = 0.1 h/Mpc to k = k max = 0.5 h/Mpc.

Fig. 6 .Fig

 6 Fig.6.15. 1σ con dence ellipses for all the pairs of cosmological (Ω m , Ω b , h, n s , σ 8 , M ν ) and nuisance (λ th ) parameters obtained from the monopoles and quadrupoles from the matter power spectrum, or the one from the di erent environments and their combination in redshift space.

  Fig.6.17. Evolution of the SNR with the maximum scale for the power spectra used in the Fisher analysis in redshift space, namely k max , for the monopoles P s =0 only (dashed lines) and monopoles plus quadrupoles P s =0 + P s =2 (plain lines).

  Fig. 6.18. Convergence analysis of the numerical precision matrix as a function of N fid with xed N deriv = 500 (left panel) and derivatives as a function of N deriv with xed N fid = 7000 (right panel) for the combined spectra statistics P comb in real space. The grey area shows the ±1% variations in the marginalised constraints.

Fig. 6 .

 6 Fig. 6.19. Convergence analysis of the numerical precision matrix as a function of N fid with xed N deriv = 500 (left panel) and derivatives as a function of N deriv with xed N fid = 7000 (right panel) for the combined spectra statistics P s,comb ={0,2} in redshift space. The grey area shows the ±1% variations in the marginalised constraints.

Titre:

  Les environnements de la toile cosmique : identification, caractérisation et quantification de l'information cosmologique Mots clés : Cosmologie: Structures grandes échelles de l'Univers, Toile cosmique; Méthodes: Méthodes statistiques, Analyse de motifs, Apprentissage automatique non supervisé, Modèles de mélange.

Title:

  Cosmic web environments: identification, characterisation, and quantification of cosmological information Keywords: Cosmology: Large-scale structure of Universe, Cosmic Web; Methodology: Statistical Methods, Pattern analysis, Unsupervised Machine Learning, Mixture models.

Table 1 .

 1 1. Proportion of the three main components of the ΛCDM model as measured by Planck Collaboration XIII et al. [2016].

	Component	Mass-Energy proportion
	Baryons	4.9
	Cold dark matter	26.8
	Dark energy	68.3
	alternatives have been proposed for instance introducing dark energy in a di erent way, to
	extend the model accounting for additional free parameters like the summed neutrino mass
	M

ν or to modify general relativity [see e.

  Large-scale structures in simulations . . . . . . . . . . . . . . . . . . . 21 2.1.1 First exhibitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.2 Dark matter only and hydrodynamical simulations . . . . . . . . . . 22 2.2 The cosmic web through galaxies . . . . . . . . . . . . . . . . . . . . . 23

	"Je préfère vivre dans le monde des simulations. "
	C. G
	2.1

2.2.1 Galaxy surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.2 Observational e ects . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.3 Galaxy bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Motivations for cosmic web classi cation . . . . . . . . . . . . . . . . 27 2.3.1 The limitations of statistical analyses . . . . . . . . . . . . . . . . . . 27 2.3.2 The cosmological sensitivity of environments . . . . . . . . . . . . . 29 2.3.3 The role of the environment in shaping galaxies and clusters . . . . 29 2.4 Challenges in detecting cosmic laments . . . . . . . . . . . . . . . . 30 2.4.1 Structural complexity of the pattern . . . . . . . . . . . . . . . . . . 31 2.4.2 Non-unicity of the de nition . . . . . . . . . . . . . . . . . . . . . . 31 2.5 Conclusions and perspectives for the thesis . . . . . . . . . . . . . . . 33

Table 2 .

 2 1. A non-exhaustive list of the existing procedures to extract cosmic laments. Table extended from Libeskind et al. [2017]. Algorithms whose names are tagged with an asterisk

	are those requiring a continuous input, usually involving a pre-processing step to estimate
	the density eld δ from the set of given particles.
	Type	Method	Filament de nition	References
		Adapted MST	Branches of a post-processed MST	Alpaslan et al. [2014a,b]
	Graphs	Semita	Branches of a MST smoothed with splines	Pereyra et al. [2020b]
		T-ReX	Branches of a principal graph	Bonnaire et al. [2020], Chapter 4 and 5
		Bisous	Connected and aligned cylinders	Stoica et al. [2007], Tempel et al. [2016]
	Point-based geometric	FINE	Lines linking two tracer nodes	González & Padilla [2010]
		SCMS	Ridges of the distribution	Chen et al. [2014, 2015] Duque et al. [2021]
	Supervised Machine Learning	U-net * Random Forest	Tag cells with a U-net trained on MMF outputs Tag particles with RF based on knn and PCA	Aragon-Calvo [2019] Buncher & Carrasco Kind [2020]
		T-web * ,	Cells with local	Hahn et al. [2007],
	Hessian	CLASSIC * MMF * ,	tidal anisotropies Cells with multi-scale	Kitaura & Angulo [2012] Aragon-Calvo et al. [2007],
		NEXUS+ *	local anisotropies of δ	Cautun et al. [2013]
	Topology	Spineweb * DisPerSE *	Intersections between 3 watershed basins Constant gradient lines between critical points of δ	Aragón-Calvo et al. [2010] Sousbie [2011] Sousbie et al. [2011]
	Phase-space	ORIGAMI MSWA	Particles that have undergone shell-crossings along 2 axes Regions with more than 17 streams	Falck et al. [2012] Ramachandra & Shandarin [2015]

*

  Clustering and its drawbacks . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Mixture models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.1 General formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.2 The Gaussian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3 Expectation-Maximisation algorithm . . . . . . . . . . . . . . . . . . . 46 3.3.1 Introduction through Mixture Models . . . . . . . . . . . . . . . . . 46 3.3.2 Iterative scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

	"De mon point de vue, si on arrive à comprendre,
	c'est toujours bien... "
	A. D
	3.1 Context and motivations . . . . . . . . . . . . . . . . . . . . . . . . . . 39
	3.1.1 Machine learning and physics . . . . . . . . . . . . . . . . . . . . . . 40
	3.1.2 Optimisation problems and regularisation . . . . . . . . . . . . . . . 41
	3.1.3 . 47
	3.3.3 The particular case of Gaussian mixtures . . . . . . . . . . . . . . . . 48
	3.4 Phase transitions in Gaussian mixtures . . . . . . . . . . . . . . . . . . 49
	3.4.1 Statistical physics formulation of clustering . . . . . . . . . . . . . . 49
	3.4.2 From paramagnetic to condensation phase . . . . . . . . . . . . . . . 50
	3.4.3 Hard annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
	3.4.4 Soft annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
	3.4.5 Graph-regularised mixture model . . . . . . . . . . . . . . . . . . . . 59
	3.5

  3.3 illustrates one iteration of the algorithm.

	log 𝑝(𝑿|𝚯 (𝑡+1) ) 𝐿( ො 𝑞, 𝚯 (t+1) )	𝐷 KL ( ො 𝑞, 𝑝)
	𝐿( ො 𝑞, 𝚯 (t) )
		log 𝑝 𝑿 𝚯)
		𝐿( ො 𝑞, 𝚯)
		𝚯 (𝑡+1) 𝚯 (𝑡)	𝚯

Fig. 3.3

. Schematic view of one iteration of the Expectation-Maximisation procedure. The grey dot represents an iteration t. By applying the E and M steps, one moves to the black dot, climbing the log-likelihood towards the local maximum. Starting from another point (red one) would however lead the optimisation to converge to a lower value of the log-likelihood. Details can be found in Sect.

3.3.2.

  Elements of graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Illustrative application: Road network . . . . . . . . . . . . . . . . . . 87 4.8 Summary and prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 This chapter is presenting the results from Bonnaire et al. [2021b] and partially those from Bonnaire et al.

.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.1.1 Spatially structured point-cloud data . . . . . . . . . . . . . . . . . . 66 4.1.2 Principal curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2 4.2.1 Introduction and de nitions . . . . . . . . . . . . . . . . . . . . . . . 68 4.2.2 Linear algebra representations . . . . . . . . . . . . . . . . . . . . . . 69 4.2.3 Some graph constructions . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3 Graph regularised mixture models . . . . . . . . . . . . . . . . . . . . 71 4.3.1 Full model and formalism . . . . . . . . . . . . . . . . . . . . . . . . 71 4.3.2 Algorithm and illustrative results . . . . . . . . . . . . . . . . . . . . 76 4.4 About graph priors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.4.1 Basic graph constructions . . . . . . . . . . . . . . . . . . . . . . . . 79 4.4.2 The average graph prior . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.5 Convergence and time complexity . . . . . . . . . . . . . . . . . . . . . 82 4.5.1 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5.2 Time complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.5.3 Runtimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.6 Hyper-parameters and initialisation . . . . . . . . . . . . . . . . . . . 84 4.6.1 The impact of parameters . . . . . . . . . . . . . . . . . . . . . . . . 84 4.6.2 Initialisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.7

  )where D is a K × K diagonal matrix with d ii = deg(i) and A is the adjacency matrix de ned above. This algebraic representation has led to numerous studies and is even at the origin of a sub eld of research called "spectral graph theory" [seeChung, 1999; Brouwer & Haemers, 2012, for reviews]. The Laplacian matrix is hence a central operator whose eigenvalues are closely related with the macroscopic properties of the graph. For instance, the smallest ei-These last examples are based on the established analogy between graphs and manifolds[Chung, 1999] that we will be exploiting in Sect. 4.3.1 to constrain the smoothness of the estimated principal graph in the proposed framework.

genvalue of L is 0 and its multiplicity is associated to the number of connected subgraphs in G

[Hagen & Kahng, 1992]

. Based on extensions of this result, many e cient clustering algorithms emerged to partition a graph [see e.g.

Alpert et al., 1999; Nascimento & De Carvalho, 2011; Saade et al., 2014]

or to build projection of high-dimensional dataset in lower dimension

[Belkin & Niyogi, 2001, 2003;[START_REF] Hein | [END_REF]

.

  .5)where|H i | denotes the cardinal of H i de ned as x 1 H i (x)=1 and |H 1 ∩ H 2 | is the cardinal of the intersection between H 1 and H 2 detections de ned as x 1 H 1 (x)=1 1 H 2 (x)=1. Hence, S(H 1 , H 2 ) measures the proportion of H 1 detections that are contained in H 2 and is thus asymmetric. In other words, if we consider H 2 as a reference, S(H 1 , H 2 ) represents the proportion of true detections provided by H 1 .

Table 5 .

 5 1. Similarity index S(H 1 , H 2 ) as de ned in eq. (5.5) between the considered methods applied on the entire 3D cube. T-ReX us refers to the unsmoothed version of the detection while T-ReX s refers to the smoothed one over the 26 neighboring voxels. The same tendency is observed concerning Bisous for which the detections are mostly contained in other skeletons (last row of Table

	H 1	2 H	T -R e X u s	T -R e X s	N e x u s +	D i s P e r S E	B i s o u s
	T-ReX us	1	1	0.85	0.62	0.37
	T-ReX s		0.48	1	0.62	0.62	0.24
	Nexus+	0.53	0.81	1	0.62	0.30
	DisPerSE	0.22	0.46	0.35	1	0.12
	Bisous		0.66	0.87	0.86	0.62	1
	leading to a thicker lamentary pattern.

Table 5 .

 5 2. Proportion of galaxies in the Nexus+ categories as identi ed in T-ReX as being in either nodes or laments depending on the density of the bifurcation considered as nodes. Filaments classi cation are obtained with n bif > 30n g .

	T-ReX	Nodes	Nodes	Nodes	Filaments
	Nexus+	n bif > 5n g	n bif > 30n g	n bif > 100n g
	Nodes	0.994	0.963	0.962	0.037
	Filaments	0.265	0.130	0.056	0.702
	Walls	0.035	0.004	0.001	0.343
	Voids	0.004	0.000	0.000	0.086

  6.1 Context and introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.1.1 The matter power spectrum as a cosmological probe . . . . . . . . . 119 6.1.2 The cosmic environments as an alternative probe . . . . . . . . . . . 120 6.2 Data & Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.2.1 The Quijote suite of simulations . . . . . . . . . . . . . . . . . . . . . 122 6.2.2 Cosmic web segmentation . . . . . . . . . . . . . . . . . . . . . . . . 123 6.3 Environments sensitivity to cosmology . . . . . . . . . . . . . . . . . . 124 6.3.1 Cosmic fractions as a function of cosmological parameters . . . . . . 124 6.3.2 Power spectra in cosmic environments . . . . . . . . . . . . . . . . . 126 6.4 Constraining power of cosmic environments . . . . . . . . . . . . . . 127 6.4.1 Fisher formalism for information content quanti cation . . . . . . . 127 6.4.2 Real-space auto-spectra . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.4.3 Redshift-space auto-spectra . . . . . . . . . . . . . . . . . . . . . . . 135 6.4.4 Stability and convergence analysis . . . . . . . . . . . . . . . . . . . 142 6.5 Conclusion and perspectives . . . . . . . . . . . . . . . . . . . . . . . . 143

Table 6 .

 6 1. Speci cation of the simulations from the Quijote suite with their denomination and the number of realisations. Underlined values are those changing from the ducial setup. 2LPT stands for 2 nd order Lagrangian Perturbation Theory and ZA for Zel'dovich approximation.

	Name	Ω m	Ω b	h	n s	σ 8	M ν ICs # of real.
	Fiducial 0.3175 0.049 0.6711 0.9624 0.834 0 2LPT	15000
	Ω + m	0.3275 0.049 0.6711 0.9624 0.834 0 2LPT	500
	Ω -m Ω + b Ω -b h +	0.3075 0.049 0.6711 0.9624 0.834 0 2LPT 0.3175 0.051 0.6711 0.9624 0.834 0 2LPT 0.3175 0.047 0.6711 0.9624 0.834 0 2LPT 0.3175 0.049 0.6911 0.9624 0.834 0 2LPT	500 500 500 500
	h -	0.3175 0.049 0.6511 0.9624 0.834 0 2LPT	500
	n + s	0.3175 0.049 0.6711 0.9824 0.834 0 2LPT	500
	n -s σ + 8 σ -8	0.3175 0.049 0.6711 0.9424 0.834 0 2LPT 0.3175 0.049 0.6711 0.9624 0.849 0 2LPT 0.3175 0.049 0.6711 0.9624 0.819 0 2LPT	500 500 500
	M 0 ν	0.3175 0.049 0.6711 0.9624 0.834 0	ZA	500
	M + ν	0.3175 0.049 0.6711 0.9624 0.834 0.1 ZA	500
	M ++ ν	0.3175 0.049 0.6711 0.9624 0.834 0.2 ZA	500
	M +++ ν	0.3175 0.049 0.6711 0.9624 0.834 0.4 ZA	500
	6.2.1 The Quijote suite of simulations
	Quijote [Villaescusa-Navarro

Table 6 .

 6 2. Cosmic web classi cation rules in the cell x depending on the eigenvalues λ1 ≤ λ 2 ≤ λ 3 of the tidal tensor. , λ 2 , λ 3 < λ th Wall λ 1 , λ 2 < λ th , λ 3 > λ th Filament λ 1 < λ th , λ 2 , λ 3 > λ th Node λ 1 , λ 2 , λ 3 > λ thenvironments to the overall matter density elds displayed in the top panel for a thin 2.77 Mpc/h depth slice. As expected, nodes describe a discrete set of dense objects found at the extremities of laments and intersection of walls while voids show large low-density areas covering most of the surface.

	Environment Condition
	Void	λ 1

  Fig.6.7. 1σ con dence ellipses for all the pairs of cosmological (Ω m , Ω b , h, n s , σ 8 , M ν ) and nuisance (λ th ) parameters obtained from the matter power spectrum, or the one from the di erent environments and their combination in real space. Fig.6.8. Zoomed con dence ellipses obtained with P mm , P VV , P WW , P FF , P NN and their combination, P comb , in the M ν -Ω m (left panel) and M ν -σ 8 (right panel) planes in real space. Table6.3. Marginalised 1-σ constraints obtained from the power spectrum in di erent environments for all cosmological parameters.

		2						2
		1						1
	M ν	2 1 0	0.2	Ω m	0.4	0.6	M ν	2 1 0	P FF P NN 0.80 0.82 0.84 0.86 σ 8 P WW P VV P mm
										P comb
	Statistics	σ Ωm		σ Ω b	σ h			σ ns	σ σ8	σ Mν
	P mm	0.0967	0.0412	0.5128		0.4998	0.013	0.8857
	P VV	0.0327 (3.0) 0.0184 (2.2) 0.1960 (2.6) 0.1244 (4.0) 0.0204 (0.6) 0.3695 (2.4)
	P WW	0.0392 (2.5) 0.0199 (2.1) 0.1800 (2.8) 0.0796 (6.3) 0.0258 (0.5) 0.9938 (0.9)
	P FF	0.0322 (3.0) 0.0190 (2.2) 0.2399 (2.1) 0.0223 (2.2) 0.0225 (0.6) 0.4392 (2.0)
	P NN	0.0872 (1.1) 0.0473 (0.9) 0.6117 (0.8) 0.6005 (0.8) 0.1208 (0.1) 0.5427 (1.6)
	P comb	0.0122 (8.0) 0.0091 (4.5) 0.0766 (6.7) 0.0292 (17.1) 0.0020 (6.5) 0.0592 (15.0)
									σ 8	0	1 M ν	0.28 0.30 0.32 λ th

Table 6 .

 6 4. Marginalised 1-σ constraints obtained from the analysis of power spectra monopoles and quadrupoles computed in the di erent environments for all cosmological parameters. Partial derivatives ∂P αα (k)/∂θ i for the di erent environmental auto-spectra and for each studied parameters {Ω m , Ω b , h, n s , σ 8 , M ν }. Dashed (resp. plain) lines correspond to negative (resp. positive) values of the derivative. The grey area depicts the range of k > k max excluded from the analysis.

	Statistics	σ Ωm	σ Ω b	σ h	σ ns	σ σ 8	σ Mν
	P s,mm =0	0.0079	0.0144	0.1485	0.0794	0.0123	0.3876
	P s,VV =0	0.0307 (0.3) 0.0192 (0.7) 0.1998 (0.7) 0.1128 (0.7) 0.0158 (0.8) 0.3289 (1.2)
	P s,WW =0	0.0260 (0.3) 0.0224 (0.6) 0.2164 (0.7) 0.1786 (0.4) 0.0332 (0.4) 0.9826 (0.4)
	P s,FF =0	0.0187 (0.4) 0.0182 (0.8) 0.2175 (0.7) 0.1740 (0.5) 0.0291 (0.4) 0.3401 (1.1)
	P s,NN =0	0.0340 (0.2) 0.0272 (0.5) 0.3596 (0.4) 0.3944 (0.2) 0.0963 (0.1) 1.1548 (0.3)
	P s,comb =0	0.0051 (1.6) 0.0104 (1.4) 0.0839 (1.8) 0.0348 (2.3) 0.0033 (3.8) 0.0709 (5.5)
	P s,mm ={0,2}	0.0048	0.0133	0.1391	0.0716	0.0020	0.0838
	P s,VV ={0,2}	0.021 (0.2) 0.0143 (0.9) 0.1316 (1.1) 0.0801 (0.9) 0.0093 (0.2) 0.0984 (0.9)
	P s,WW ={0,2}	0.0130 (0.4) 0.0213 (0.6) 0.1921 (0.7) 0.0766 (0.9) 0.0146 (0.1) 0.1449 (0.6)
	P s,FF ={0,2}	0.0093 (0.5) 0.0158 (0.8) 0.1844 (0.8) 0.1283 (0.6) 0.0084 (0.2) 0.1825 (0.5)
	P s,NN ={0,2}	0.0126 (0.4) 0.0210 (0.6) 0.2378 (0.6) 0.1553 (0.5) 0.0140 (0.1) 0.5322 (0.2)
	P s,comb ={0,2}	0.0034 (1.4) 0.0097 (1.4) 0.0761 (1.8) 0.0306 (2.3) 0.0020 (1.0) 0.0349 (2.4)

  al.: Persistent homology of the cosmic web . I : Hierarchical topology in Λ CDM cosmologies. arXiv e-prints 28 (2020), Nr. November, 1-28. https:// arxiv.org/ abs/ 2011.12851 29 [Woiselle et al. 2010] W , A., S , J. L., F , J.: 3D curvelet transforms and astronomical data restoration. Appl. Comput. Harmon. Anal. 28 (2010), Nr. 2, 171-188. http: // dx.doi.org/ 10.1016/ j.acha.2009.12.003 28 [Wu 1983] W , C. F. J.: On the Convergence Properties of the EM Algorithm. Ann. Stat. 11 (1983), 95--103. http:// dx.doi.org/ 10.1214/ aos/ 1176346060 47, 48 Testing homogeneity on large scales in the Sloan Digital Sky Survey Data Release One. Mon. Not. R. Astron. Soc. 364 (2005), Nr. 2, 601-606. http:// dx.doi.org/ 10.1111/ j.1365-2966. 2005.09578.x 13 [Yang et al. 2017] Y , Xiaohu, Z , Youcai, L , Wentao, et al.: Revealing the Cosmic Web-dependent Halo Bias. Astrophys. J. 848 (2017), 60. http:// dx.doi.org/ 10.3847/ 1538-4357/ aa8c7a 27 [York et al. 2000] Y , D. G., A , J., A , J.E., et al.: The Sloan Digital Sky Survey: Technical Summary. Astrophys. J. 120 (2000), 1579-1587. http:// dx.doi.org/ 10.1086/ 301513 5, 23, 94, 117 [Yoshida et al. 2000] Y , Naoki, S , Volker, W , Simon D., T , Giuseppe: Weakly Self-interacting Dark Matter and the Structure of Dark Halos. Astrophys. J. 544 (2000), Nr. 2, 87-90. http:// dx.doi.org/ 10.1086/ 317306 30, 94 [Yuille 1990] Y , Alan L.: Generalized Deformable Models, Statistical Physics, and Matching Problems. Neural Comput. 2 (1990), S. 1-24. http:// dx.doi.org/ 10.1162/ neco.1990.2.1.1 74 [Zdeborova & Krzakala 2016] Z , Lenka, K , Florent: Statistical physics of inference: Thresholds and algorithms. Adv. Phys. 65 (2016), S. 453-552. http:// dx.doi.org/ 10. 1080/ 00018732.2016.1211393 41 [Zel'dovich 1970] Z ' : Gravitational instability: an approximate theory for large density perturbations. Astron. Astrophys. 500 (1970), 13-18. https:// ui.adsabs.harvard.edu/ abs/ 1970A{&}A.....5...84Z 5, 15, 21, 123 [Zhang et al. 2016] Z , Chiyuan, B , Samy, H , Moritz, et al.: Understanding deep learning requires rethinking generalization. Commun. ACM 64 (2016), Nr. 3, S. 107-115. http:// dx.doi.org/ 10.1145/ 3446776 41 [Zhang et al. 2019] Z , Xinyue, W , Yanfang, Z , Wei, et al.: From Dark Matter to Galaxies with Convolutional Neural Networks. arXiv e-prints (2019). https:// arxiv.org/ abs/ 1902.05965 40

	[Yadav et al. 2005] Y	, Jaswant, B	, Somnath, P	, Biswajit, S	, T. R.:

That we sometimes unrigorously write a for convenience.

The parsec (pc) is a unit of distance used in astronomy and 1 pc =

3.0857 × 10 16 m.

For a complete review of the topic, we refer toBernardeau et al. [2002].

Which is a valid assumption for a wide range of timescales from the epoch of recombination(3.8 × 10

years after the Big Bang, z ∼ 1100), to z ∼ 2

Although simulations can help in computing ensemble averages by providing many realisations of a same universe, as we will see in the Chapter

Even if the eld is not centred, one can always study the random variable de ned subtracting the mean which is now centred.

Some "constrained simulations" propose to tune the initial conditions of the simulation to reproduce the observed local Universe in a Bayesian framework and studies the properties of the time evolving reconstructed matter density eld [see e.g.,Kolatt et al., 1996;[START_REF] Jasche | [END_REF]

2013; Sorce et al., 2016]. 2 https:// www.illustris-project.org/

L 1 regularisation is also very famous and known in the context of regression as the Lasso[Tibshirani, 1996]. This regularisation is used to build sparse representations of signals, forcing some coe cients to 0, a vast topic discussed in more details byStarck et al. [2010].

Similarly, if we use an L 1 regularisation, optimisation problem (3.3) can be seen as the maximisation of a Gaussian likelihood with a Laplace prior distribution.

In fact, some clustering approaches are not fully unsupervised and exploit partial pre-classi ed data to guide the overall clustering, making them semi-supervised [see e.g.Grira et al., 2004; Bair, 2013].

Stating that, for any random variable X and convex function f , f (E{X}) ≤ E{f (X)}[START_REF] Jensen | [END_REF].

The separation in three panels with di erent colours is for visualisation purposes. No prior knowledge was used in the analysis.

In our experiments, keeping λ σ ≈ O(1) did not change the results quantitatively.

See Gerber & Whitaker [2013] for a regularisation-free formulation of the principal curve in which an alternative of the mean-squared projection distance cost is proposed.

The rst term is computed through Eq. (4.5) and the second one is the prior from Eq. (4.10).

Available at https:// git.ias.u-psud.fr/ tbonnair/ t-rex.

https:// www.openstreetmap.org/

Which appears as a parameter in the xed-variance scheme, but as an initialisation when the variance is learnt during the optimisation.

http:// www.illustris-project.org/ data/

http:// data.aip.de/ tracingthecosmicweb/ doi:10.17876/ data/ 2017_1

https:// www.ascl.net/ 1512.008

Considering Nexus+ as a ground truth.

https:// quijote-simulations.readthedocs.io/ en/ latest/

Note that we express here the mass fractions in terms of number of particles since they all have the same mass in the N -body simulations.

Acknowledgements

 Soc. 375 (2007), Nr. 1, S. 337-347. http:// dx.doi.org/ 10.1111/ j.1365-2966[START_REF] Singer | [END_REF].11312.x 32, 68, 71, 95 [Colberg et al. 2005 1365-2966.2011.19820.x 114, 115 [Radford et al. 2016 , 1986, , 194-281 40, 62 [Smoot et al. 1992