J ' Aimerais

Alain D Arthur

Lionel

Anaïs, Sandrine Tuong-Huy Pierre

Keywords: Fréchet mean, group action, quotient space, consistency, Hilbert space, manifolds, registration, max-max

Je souhaiterais vivement remercier mes deux directeurs de thèse. Merci à Stéphanie pour avoir pensé à moi pour ce beau sujet de thèse ainsi que pour tes constants encouragements et conseils durant ces années. Je remercie également Xavier pour les intéressantes discussions que l'on a pu avoir et pour toutes les questions soulevées. Merci de m'avoir guidé tout en me laissant une grande liberté. Je n'oublie pas non plus les bons moments passés ensemble en conférences ainsi que dans la mine.

Merci également à l'équipe d'Asclepios avec laquelle j'ai toujours passé de bons moments que ce soit à Sophia ou Auron. Merci à Nicholas de si bien diriger cette équipe. Je souhaite également remercier Isabelle de m'avoir si souvent (trop souvent ?) aidé pour les missions et pour l'organisation de cette soutenance. Merci aux doctorants, ingénieurs et anciens doctorants de leur compagnie. Je souhaite en particulier remercier tous ceux qui m'ont prodigué leurs excellents conseils lors des répétitions. Merci aussi à Nina pour les discussions scientifiques que l'on a eues. Merci à tous mes amis et à ma famille pour votre constant soutien durant cette thèse.

Enfin, je voudrais remercier ma femme Maëlle, qui tous les jours me supporte. Faire ma thèse à ses côtés était la plus belle chose qui puisse m'arriver.

Merci à tous !

Abstracts

Consistance des statistiques dans les espaces quotient de dimension infinie En anatomie computationnel, on suppose que les formes d'organes sont issues de déformation d'un template commun. Les données peuvent être des images ou des surfaces d'organes, les déformations peuvent être des difféomorphismes. Pour estimer le template, on utilise souvent un algorithme, appelé «max-max», qui minimise parmi tous les template candidats, la somme des carrées des distances après recalage entre les données et le template candidat. Le recalage étant une étape dans l'algorithme qui trouve la meilleur déformation pour passer d'une forme à une autre.

Le but de cette thèse est d'étudier cet algorithme max-max d'un point de vue mathématique. En particulier, on prouve que cet algorithme est inconsistant à cause du bruit. Cela veut dire que même avec un nombre infini de données et avec un algorithme de minimisation parfait, on estime le template original avec une erreur. Pour prouver cette inconsistance, différentes hypothèses sont requises dans différent résultats de cette thèse. Nous devons donc expliquer ces hypothèses, et surtout produire des résultats avec les hypothèses les plus faibles possibles, pour s'approcher du cadre utilisé dans les applications.

Pour prouver l'inconsistance, on formalise mathématiquement l'estimation de template. On suppose que les déformations sont des éléments aléatoires d'un groupe qui agit sur l'espace des observations. De plus, l'algorithme étudié est interprété comme le calcul de la moyenne de Fréchet dans l'espace des observations quotienté par le groupe des déformations. Dans cette thèse, on prouve que l'inconsistance est dû à la contraction de la distance quotient par rapport à la distance dans l'espace des observations. Dans cette thèse, les observations appartiennent à des espaces comme les espaces de Hilbert ou les variétés Riemanniennes, l'inconsistance est obtenue pour presque tous les bruits.

Un autre but de cette thèse est de quantifier l'inconsistance. On estime l'erreur entre le template originel et le template estimé. Cela met en évidence les paramètres qui gouvernent l'inconsistance. On obtient un équivalent de biais de consistance en fonction du niveau de bruit. Ainsi, l'inconsistance est inévitable quand le niveau de bruit est suffisamment grand.

Mots clés: Moyenne de Fréchet, action de groupe, espace quotient, consistance, espace de Hilbert, variétés, recalage

Consistency of statistics in infinite dimensional quotient spaces

In computational anatomy, organ shapes are assumed to be deformation of a common template. The data can be organ images but also organ surfaces, and the deformations are often assumed to be diffeomorphisms. In order to estimate the template, one often uses the max-max algorithm which minimizes, among all the prospective templates, the sum of the squared distance after registration between the data and a prospective template. Registration is here the step of the algorithm which finds the best deformation between two shapes.

The goal of this thesis is to study this template estimation method from a mathematically point of view. We prove in particular that this algorithm is inconsistent due to the noise. This means that even with an infinite number of data, and with a perfect minimization algorithm, one estimates the original template with an error. In order to prove inconsistency, various hypotheses are required in different results in this thesis. We are committed to explain these hypotheses, and we aim at providing results with the weakest hypotheses possible, in order to approach the frameworks used for applications.

In order to prove inconsistency, we formalize the template estimation into a mathematical framework. Deformations are assumed to be random elements of a group which acts on the space of observations. Besides, the studied algorithm is interpreted as the computation of the Fréchet mean in the space of observations quotiented by the group of deformations. In this thesis, we prove that the inconsistency comes from the contraction of the distance in the quotient space with respect to the distance in the space of observations. As a result, we consider that observations belong to general spaces such as Hilbert spaces and Riemannian manifolds, in these spaces, the inconsistency appears for general noise.

Another goal of this thesis is to quantify this inconsistency. We estimate the error between the original template and the estimated template. This highlights the parameters which govern the inconsistency. We obtained a Taylor expansion of the consistency bias with respect to the noise level. As a consequence, the inconsistency is unavoidable when the noise level is high.

Notation

Les statistiques sont l'art d'utiliser toutes les lettres.

The following notations are commonly used in this thesis:

• "¨": group action of G on M , for x P M , g P G, g ¨x P M .

• x , y: dot product on M when M is a Hilbert space.

• } }: Hilbert norm on M when M is a Hilbert space.

• rms: orbit of m, rms " tg ¨m, g P Gu.

• Bpx, rq: open ball of center x and radius r.

• CB: consistency bias.

• Cpmq: cut locus of a point m belonging to a complete Riemannian manifold.

• Conept 0 q: the Voronoï Cone associated to the template t 0 defined as the set of points of M closer from t 0 than the other points of rt 0 s.

• d M : distance in the ambient space, often given by the euclidean norm in Hilbert space, or the Riemannian distance in complete Riemannian manifold.

• d Q (when d M is invariant under the group action): quotient (pseudo-)distance between two orbits, d Q pras, rbsq " inf gPG d M pa, g ¨bq.

• E: variance of Y (or X) in the ambient space M , Epmq " Epd 2 M pm, Y qq.

• e G : the identity element of G.

• E: expectation of a random variable in R or in a Hilbert space.

• ε: a noise in M with Epεq " 0 (sometimes Ep}ε} 2 q " 1).

• Exp: the Riemannian exponential map in complete Riemannian manifold.

• F : variance in the quotient space of Y (or X), F pmq " Epd 2 Q prms, rY sqq.

• FM: set of all the Fréchet means of a random variable in a metric space pX , d X q F M pZq " argmin xPX Epd 2 X px, Zqq.

• F n : empirical variance in the quotient space of Y , F n pmq " 1 n n ř i"1 d 2 Q prms, rY i sq.

• FixpM q: set of fixed point under the group action, Fixpmq " tm P M, @g P G g ¨m " mu.

• η: a small positive number.

• θpvq: registration score of the unit vector v of the noise: θpvq " E `sup gPG xv, g ¨εy ˘.

• G: a group acting on M .

• gpa, bq: an element of G which registers a to b.

• H: a subgroup of G

• Isopmq: the isotropy group of m P M , Isopmq " tg P G, g ¨m " mu.

• Int: IntpAq is the interior of a set A.

• J: an auxiliary map used to minimize F n : Jpm, g 1 , . . . , g n q "

1 n

n ÿ i"1
}m ´gi ¨Yi } 2 .

• Log the logarithm map in complete Riemannian manifold.

• L 2 pR{Zq or L 2 pr0, 1sq: set of measurable functions on r0, 1s or R{Z which are squared integrable.

• λpvq: registration score of the unit vector v, λpvq " E `sup gPG xv, g ¨Y y ˘.

• M : ambient space, most of the time a Hilbert space, sometimes M is a complete Riemannian manifold.

• m ‹ : an element which minimizes F .

• P: probability measure.

• π: projection in the quotient space πpxq " rxs.

• Q: quotient space of M by G, Q " M {G " trms, m P M u.

• Reg: regularization term on the group G.

• s: section of the quotient, s : Q Ñ M with π ˝s " Id.

• S: the image of the section S " spQq.

• S: the unit sphere in M .

• σ: the noise level.

• t 0 : the template.

• τ : time-shift. The visionary work of D'Arcy Thompson [Thompson 1942], in the beginning of the 20th century, consisted in the study of the form of animals. The main goal was to create a classification between two species. This raised the question of how to do one study of the difference of form between two animal drawings. His answer was to introduce a grid superimposed onto the image of the first animal, and deform this grid to match to the second one. The more the grid needs to be deformed the more different the two species are. Since this pioneering work, which was not completely written mathematically, computational anatomy has arisen [START_REF] Grenander | Computational Anatomy: An Emerging Discipline[END_REF]]. This recent discipline aims to build statistics on data with mathematical framework which can be numerically implemented.

There are several reasons for the development of this newly created discipline:

• The increasing use of medical imaging, with high resolution images.

• These images are digitized and are accessible across the world.

• The increasing of computational power which allows us to work with a large number of high resolutions images.

• The mathematical tools and theory have been developed in order to tackle medical issues, for instance shape space theory [Kendall 1989].

One use of computational anatomy would be to predict disease. For instance, one could imagine that we use this deformation model in order to compare a new 1.2. Statistics in computational anatomy patient to a known patient. However, as this known patient is singular, he may not be representative among the population, this could bias the study. Moreover, a doctor never compares a new patient to an old one. A doctor would rather want to compare a new patient to all the previous patients. A doctor had synthesized the common features of the previous patients who share the same illness. Computing the mean patient (in this thesis we use the word template) would allow us to mime this synthesis. This leads us to study statistics in computational anatomy, in particular, the template estimation.

Statistics in computational anatomy

Once one has data like organs images, one wants to perform statistics on this data. There are several levels we can think of:

• Given two elements (for instance two medical images), one can estimate the amount of deformations to match the first image (the source image) to the second one (the target image). This step is called registration and formalize the idea of D'Arcy Thompson [Thompson 1942]. One active field of research is to build admissible group of deformations in order to register elements [START_REF] Trouvé | An approach of pattern recognition through infinite dimensional group action[END_REF], Miller 2006].

• If we are able to perform the registration step, then we can estimate the mean of images. Due to the deformation, data do not belong to an Euclidean space, one can generalize the notion of mean with the Fréchet mean introduced by Fréchet [Fréchet 1948]. Then we obtain an element which is representative of our population images [Guimond 2000, Joshi 2004]. Many names are given for this element: template, prototype, virtual patient etc; in the following we will use the word template. This can be done for instance with the LDDMM framework [Beg 2005] or with the Demons algorithm [Thirion 1998[START_REF] Vercauteren | [END_REF][START_REF] Lombaert | Spectral Log-Demons: Diffeomorphic Image Registration with Very Large Deformations[END_REF].

Note that in practice, we obtain both the template and the deformations between the template and each images [Durrleman 2014, Durrleman 2013, Allassonnière 2015]. The set of the template and the deformations is called an atlas.

• The template estimation can be a useful step, but it does not explain the variability of data. Then Principal Geodesic Analysis [START_REF] Fletcher | [END_REF][START_REF] Fletcher | [END_REF], Sommer 2010[START_REF] Sommer | [END_REF]] (PGA) or Principal Composant Analysis [START_REF] Huckemann | [END_REF][START_REF] Seiler | [END_REF]] (PCA), barycentric subspaces [Pennec 2016[START_REF] Rohé | [END_REF]] etc. can be developed in order to estimate which directions best explain the data.

In this thesis we focus on the estimation of the template (the second level in the previous list). One goal of this thesis is to study the consistency of the minimization algorithm used to estimate the template. As a result, we prove that this method is Chapter 1. Introduction not consistent. This means that even with the whole distribution (and not only a sample), one does not find the original template, as soon as noise is added. Besides we also focus on finding the error between the original template and the estimated template.

Geometrical framework of template estimation

A brief overview of manifolds and Riemannian manifolds

The goal of this section is not to recall some formal definitions but to just make a brief recap on some notions used in this thesis. We refer to [do Carmo Valero 1992] for precise definitions and proofs. In this thesis, we prove several properties of the template estimation, where data belong to Hilbert spaces and Riemannian manifold. Besides, in this thesis, deformations leads us to consider a kind of spaces called quotient space which are not manifolds in general. However we will see in chapter 5 that someway the quotient spaces behave as manifolds. Making an analogy between quotient space and manifold will be a useful tool in to understand the property of the template estimation in chapter 5.

We say that a set M is a finite dimensional manifold of dimension n if M locally looks like an open set of R n : Definition 1.1. Let M be a topological space. We say that M is a differentiable manifold of dimension n if there exists a family of injective maps ϕ a : U a Ă R n Ñ M , where for a P I, U a is an open set of R n such that:

M " ď aPI ϕpU a q and for any a P I, b P I, with W " ϕ a pU a q X ϕ b pU b q ‰ H, the sets ϕ ´1pW qandϕ ´1pW q are open sets of R n and the map ϕ ´1 b ˝ϕa is differentiable.

With this definition, it is possible to define for every point of a manifold M has a tangent space.

In particular, at each point of a manifold m, the space can be approximated by its tangent plan noted T m M .

Moreover, we say that a manifold M is a Riemannian manifold if on each tangent plan m, we have a dot product x , y m , which depends continuously of the point m (we note } } m the associated norm).

This allows us to differentiate curve included in the manifold. As a consequence, we are able to define the length of a curve of the manifold. Then, when M is a path-connected, (this means that given two points a and b, it exists a continuous curve connecting a to b), we can define the geodesic distance between a and b as the infimum of the length of all the curves connecting a to b. One can prove that this provide actually a distance called the geodesic distance (or the Riemannian distance) on M noted d M . From now on, we assume that M is a connected path space.

Besides, by considering that the geodesics are critical point of the energy, one can prove that the geodesics are exactly the solution of a differential equation of order 2. Due to the theory of differential equation, there is always one local solution given one initial point m P M , and one initial vector speed v P T m M . As a result, there exists one geodesic curve γ :s ´η, ηrÑ M such that γp0q " m and 9

γp0q " v. Let m be a point in M , let v be a vector belonging to T m M . If a geodesic curve γ which satisfies γp0q " m and 9

γp0q " v is defined, on an interval I such that 1 P I, then we define the exponential map as Exp m pvq " γp1q P M . Besides, it is possible to prove that Exp m is a local diffeomorphism at 0 the null vector in T m M . We call Log m the inverse diffeomorphism. Besides we have: }Log m paq} m " d M pm, aq (see figure 1.2).

Next, we wonder if the geodesic curve can be extended globally to a map defined on R. The Hopf-Rinow theorem give the answer: Theorem 1.1 (Hopf-Rinow Theorem). Let M be a connected finite dimensional Riemannian manifold. Then the following statements are equivalent:

1. pM, d M q is a complete metric space where d M is the geodesic distance.

2. M is geodesically complete this means that for every m P M , the exponential map is defined on the entire tangent space T m M .

3. The closed and bounded subsets of M are compact;

From now on, we assume that M is a complete connected Riemannian manifold. Let γ be such a geodesic curve, then it is possible that γ stops to be a minimal geodesic at the time t 0 , in this case, we call γpt 0 q a cut point of m. And we call cut locus of m, noted Cpmq the set of all cut point of m.

Finally we give a useful result, which allows to differentiate the square distance in manifold: Proposition 1.1. Let M be a complete Riemannian manifolld, let x 0 , y P M . We assume that y R Cpx 0 q. Then x Þ Ñ d M px, yq 2 is differentiable at x 0 . Besides, ∇ 1 d 2 M px 0 , yq " ´2Log x 0 pyq, where ∇ 1 is the gradient of x Þ Ñ d M px, yq 2 .

This extends the well known result of the differentiation of the square Euclidean norm in Hilbert space x Þ Ñ }x ´y} 2 is differentiable at x 0 , and ∇ 1 }x 0 ´y} 2 " ´2py´x 0 q. This leads us to say that the Log map in complete Riemannian manifolds is like the subtraction in Hilbert spaces.

Deformations and quotient space

Since we assume that there are some deformations of our images. We need to provide a mathematical framework of this idea. In this thesis, we assume that the deformations come from a group action defined below: Definition 1.2. We say that a gropup G acts on our ambient space M if there is map G ˆM Ñ M , pg, mq Þ Ñ g ¨m s.t. e G ¨x " x and g ¨pg 1 ¨xq " pgg 1 q ¨x for every g, g P G, x P M , where e G is the identity element of G. For a point m P M we call orbit of m, the set rms " tg ¨m, g P Gu. The orbits of M forms a partition of M . Besides, we call quotient of M by G, noted Q " M {G the set of all orbits.

Due to the deformations induced by the group action, statistics cannot be performed in the ambient M , however its can be performed in the quotient space.

We want to insist on this point: the quotient space is not a manifold in general. The differential structure of the quotient space is an orbifold (see [START_REF] Thurston | [END_REF]] for a definition of orbifold). However, we do not use the formal definition of an orbifold in this thesis. One interesting idea about the differential structure of quotient spaces or orbifolds is that the local dimension of the structure is not constant. This may make the analysis more difficult since it prevents us to apply any statistical theorems involving linear spaces or manifolds.

What are the data?

• 1D/2D/3D images: f : R D Ñ R at the point/pixel/voxel p, f ppq is the intensity of the images. For instance with D " 1, we obtain some signals with D points, the study of this signal is important in ECG [Hitziger 2013, Bigot 2013] • Landmarks: set of points of interest, for instance, an organ can be encoded numerically, by selecting some characteristic points of the surface of the organ [Bookstein 1986, Joshi 2000]. This selection can been done manually or automatically. One drawback of landmarks is that it is hard to compare two shapes, when these two shapes are encoded with a different number of points.

This means that all the data do not belong to the same space, which make difficult the comparison between two shapes.

• Currents [START_REF] Vaillant | [END_REF]] and varifolds [Charon 2013] circumvent this obstacle: these two concepts embed surfaces into a Hilbert space. Then it is possible to do statistical analysis in these frameworks [Durrleman 2014, Durrleman 2009].

1.3.4 What are the actions?

• Diffeomorphisms on images: if f : R D Ñ R is an image (in dimension D), and ϕ a diffeomorphism, ϕ ¨f " f ˝ϕ deforms the image.

• Diffeomorphisms on landmarks: ϕ ¨px 1 , . . . , x n q " pϕpx 1 q, . . . , ϕpx n qq. Note that there is the particular case of rigid transformations when ϕ is a combination of rotations and translations.

• Diffeomorphisms on currents/varifolds.

As we will see it in this thesis, there are many actions which can be studied. Each property of the action can be used in order to prove the inconsistency. One important class of action, in this thesis, is the isometric actions in Hilbert spaces (the action is linear, and the Hilbert norm is conserved), for instance:

Example 1.1 (horizontal translation). The action of continuous translation of functions defined on a torus: we take G " pR{Zq D acting on M " L 2 ppR{Zq D , Rq with: @τ P G @f P M pτ ¨f q : t Þ Ñ f pt `τ q

Then f Þ Ñ τ ¨f is linear, besides }τ ¨f } " }f } for } } the norm in M given by }f } 2 " ş pR{Zq D f 2 pxqdx.

Remark 1.1. In this thesis, translation is used in two different context, one is the translation of the variable in a function: f Þ Ñ f p `τ q. This translation is linear with respect with f . We call this translation, «horizontal translation» (because we translate the variable). On the contrary there is the translation of a point x by a vector v: x Þ Ñ x `v. This map is no longer linear but only affine. We call this translation «vertical translation.

Remark 1.2. In all this thesis, we use the expression isometric action for the action on a Hilbert space with a linear action which lets the norm invariant. We are aware that this is not the standard definition which would be "an action is isometric if d M pm, m 1 q " d M pg ¨m, g ¨m1 q for every m, m 1 P M , for every g P G". We prefer using the word isometric for the linear isometries in linear spaces. And we say that the distance d M is invariant if d M pm, m 1 q " d M pg ¨m, g ¨m1 q for every m, m 1 P M , for every g P G.

Distance in quotient space

We recall the definition of pseudo distance and distance on any set X .

Chapter 1. Introduction Definition 1.3. Let X be a set, and d X : X ˆX Ñ R `be a map. We say that d X is a distance in X (respectively a pseudo-distance) if the following properties are satisfied for every x, y, z P X :

• x " y ðñ d X px, yq " 0 (respectively x " y ùñ d X px, yq " 0)

• d X px, yq " d X py, xq symmetry

• d X px, yq ď d X px, zq `dX pz, yq triangular inequality

One assumes that the distance d M in the ambient space M is invariant with respect to in the group action, this means that: @g P G @m, n P M d M pg ¨m, g ¨nq " d M pm, nq

In this case, we can define d Q in the quotient space:

d Q pras, rbsq " inf gPG d M pa, g ¨bq Proposition 1.2. The map d Q is a pseudo-distance in Q.
Proof. First d Q is a well defined map, indeed, due to the invariant distance, one can verify that @h, j P G inf Namely, that the definition of d Q does not depend on the chosen element in the orbit of ras or rbs.

Secondly, let ras, rbs, rcs three points of Q:

• d Q pras, rasq ď d M pa, e g ¨aq " d M pa, aq " 0 (because d M is a distance). Therefore d Q pras, rasq " 0.

• d M pa, g ¨bq " d M pg ´1 ¨a, bq " d M pb, g ´1 ¨aq (because d M is symmetric). Therefore d Q pras, rbsq " d Q prbs, rasq by taking the infimum in the previous inequality and by using the fact that g Þ Ñ g ´1 is a bijective map in G.

• d M pa, g ¨bq ď d M pa, h ¨cq `dM ph ¨c, g ¨bq for every g, h P G by using the triangular inequality in M . Besides we have:

d Q pras, rbsq ď d M pa, g ¨bq ď d M pa, h ¨cq `dM pc, ph ´1gq ¨bq.
By taking the infimum over g, we get:

d Q pras, rbsq ď d M pa, h ¨cq `dQ prcs, rbsq.
Finally, by taking the infimum over h we get: Proof. All we have to verify is if d Q pras, rbsq " 0 implies ras " rbs. Let us assume that d Q pras, rbsq " 0, then for every n P N it exists g n P G such that d M pa, g n ¨bq Ñ 0. This means in particular that g n ¨b is a convergent sequence. Moreover this sequence converges to a. As a consequence a is the limit of elements which are all in the orbit of b. Therefore a is in the closeness of rbs. Then, as rbs is a closed set (by assumption) of M , we conclude that a P rbs and then ras " rbs.

d Q pras, rbsq ď d Q pras,
The orbits are automatically closed if the group G is compact and acts continuously on the ambient space M . In this thesis, when the distance is invariant under a group action, we have a pseudo-distance in the quotient and not necessarily a distance, however this is enough for the analysis of statistics we make. Besides, we may call d Q quotient distance even if it is only a pseudo-distance.

Remark 1.3. For every a, b P M , d Q pras, rbsq ď d M pa, bq. Moreover, when a and b are in generic position we have:

1 d Q pras, rbsq ă d M pa, bq.
In this thesis, we use several concepts which are similar, therefore we make a brief recap to avoid any confusion:

• We say that the action is isometric if M is an Hilbert space and if x Þ Ñ g ¨x is a linear map which leaves the norm invariant: }g ¨x} " }x} for all g P G and x P M .

• For M a metric space, we say that the distance d M is invariant under the group action G if: @x, y P M @g P G d M pg ¨x, g ¨yq " d M px, yq.

• Let pX , d X q and pY, d Y q be two sets with discrepancy measures d X and d Y , we only assume that d

X : X Ñ X Ñ R `same for d Y . Let Ψ : X Ñ Y be a map, we say that Ψ is congruent if: @px, x 1 q P X d X px, x 1 q " d Y pΨpxq, Ψpx 1 qq.
Chapter 1. Introduction

Statistical estimation method

Since we are in a quotient space, the estimation of the mean by the empirical sum does not apply. However one can use another approach known as the Fréchet mean which can be define in quotient spaces. Indeed, as soon as we have a measure of discrepancy between orbits, one can define and minimize a variance (if the discrepancy is a distance) or a pre-variance (when one of the property of the quotient distance is not fulfilled).

Fréchet mean in Hilbert space

Let M be a Hilbert space. We consider Z a random variable in M , such that Ep}Z} 2 q ă `8. We consider the variance as the map E : M Ñ R `defined as Epmq " Ep}Z ´m} 2 q. We call Fréchet mean of Z the set of global minimizers of E.

The empirical Fréchet means are defined as the global minimizer of the empirical variance

E n : M Ñ R `with E n pmq " 1 n n ř i"1 }Z i ´m} 2 for Z 1 , . . . , Z n n observations of Z.
The following proposition states that the Fréchet mean of Z and the expected value of Z are the same element:

Proposition 1.4. Let M be a Hilbert space, and d M the distance given by the euclidean norm. Starting from Z a random variable such that Ep}Z} 2 q ă `8, then m Þ Ñ Epmq " Ep}Z ´m} 2 q is well defined and m minimizes E if and only if m " EpZq.

Since the square norm is differentiable, it is easy to verify that EpZq is the only critical point of the map E, but this is not enough to show the property of being a global minimum.

Proof. First it is easy to expand Epmq:

Epmq " }m} 2 ´2 xm, EpZqy `Ep}Z} 2 q, in particular we can compute the variance at the point EpZq:

EpEpZqq " ´}EpZq} 2 `Ep}Z} 2 q.

For all m P M , one gets:

Epmq ě }m} 2 ´2}m} ˆ}EpZq} `Ep}Z} 2 q (1.1)
ě ´}EpZq} 2 `Ep}Z} 2 q " EpEpZqq.

(1.2)

The inequality (1.1) is the use of the Cauchy-Schwarz inequality. Besides, and there is an equality in the Cauchy-Schwarz inequality if and only if m and EpZq are positively dependent (this means that m " λEpZq or EpZq " λm for λ ě 0). Moreover x Þ Ñ x 2 ´2x}EpZq} reaches this minimum at an unique point x " }EpZq}. This implies inequality (1.2). This shows that EpZq minimizes the map E, and that any other minimizer should be positively dependant to EpZq and having the same norm that EpZq, this proves the uniqueness. ˝

We get the same result with empirical Fréchet mean in a Hilbert space, the empirical Fréchet mean of Z being

1 n n ř i"1 Z i .
As we have said, we want to perform statistics of data which are not in linear spaces, but only in metric spaces. It requires to generalize the notion of Fréchet mean in metric spaces.

Fréchet mean in metric spaces

Definition 1.4. Let pM, d M q be a metric space. If Z is a random variable in M , such that for every m P M Epd M pZ, mq 2 q ă `8, then we can define the variance of Z at any point m P M by:

Epmq " Epd 2 M pZ, mqq.
We say that m is a Fréchet mean of Z if m minimizes globally E. Likewise, if we have Z 1 , . . . , Z n a sample of Z (independent and identically distributed), we define the empirical variance of Z at the point m by:

E n pmq " 1 n n ÿ i"1 Epd 2 M pZ i , mqq,
and we say that m n is an empirical Fréchet mean of Z if m n minimizes globally E n .

If an element minimizes locally E (respectively E n ), we call it Karcher mean (respectively empirical Karcher mean).

Proposition 1.5. Let pM, d M q be a metric space. If Z is a random variable in M , such that there exists m 0 P M such that Epd M pZ, m 0 q 2 q ă `8 then for every m P M , Epd M pZ, mq 2 q ă `8. Therefore we can define the variance E, besides the variance is a continuous map.

Proof. First, by the triangular inequality we have for all m P M :

d 2 M pZ, mq ď pd M pZ, m 0 q `dM pm 0 , mqq 2 ď d M pZ, m 0 q 2 `2d M pm 0 , mqd M pZ, m 0 q `dM pm 0 , mq 2 ď d M pZ, m 0 q 2 `2d M pm 0 , mqpd M pZ, m 0 q 2 `1q `dM pm 0 , mq 2 .
In the left hand side, every term has a finite expected value.

Secondly for m and m 1 we have, by the triangular inequality:

|Epmq ´Epm 1 q| ď Ep|d M pZ, mq 2 ´dM pZ, m 1 q 2 | ď Ep|d M pZ, mq ´dM pZ, m 1 q| ˆpd M pZ, mq `dM pZ, m 1 qqq ď d M pm, m 1 q ˆEpd M pZ, mq `dM pZ, m 1 qq,
proving the continuity of the map E. ˝

In all this thesis, we always consider random variable Z such that there exists m 0 P M satisfying Epd M pZ, m 0 q 2 q ă `8.

In order to find the global/local minima of the variance, one can show that, when M is a complete Riemannian manifold and if the cut locus is a null set for the probability measure of Z, then the variance is a differentiable map, and one can compute its gradient (see [Pennec 2006] for instance, it will be also proven in lemma 3.9). As global/local minimum of the variance, the Fréchet/Karcher means are critical point of the variance. This leads to the following definition taken from [Émery 1991] (see also figure 1.3): Definition 1.5. Let M be a complete Riemannian manifold. Let Z be a random variable in M , such that Epd M pZ, m 0 q 2 q ă `8 for some m 0 P M . We say that m is an exponentiel barycenter if PpZ P Cpmqq " 0 and if the differential of the variance at the point m is 0, namely if:

∇Epmq " ´2EpLog m pZqq " 0 M T m M ' Z ' Log m Z ' m ' Log m Z ' Z Figure 1.3: If m is the Fréchet mean, then m is an exponentiel barycenter:
EpLog m pZqq " 0. The length of the blue curve is equal to the length of the blue segment (same with red).

In this thesis, we will deal with the Fréchet mean in the quotient space. This means that we will consider the quotient distance d Q previously defined in order to define the Fréchet mean in quotient space.

One issue is that in metric spaces (including manifolds), the existence or the uniqueness is not guarantied. Therefore existence or uniqueness of the Fréchet mean is commonly studied with interesting results [Karcher 1977, Kendall 1990, Afsari 2011, Arnaudon 2005, Arnaudon 2014, Charlier 2013] among many others. These results are mainly proved for data living in manifolds. Besides, one the existence is ensured, methods are developed in order to estimate this Fréchet mean [START_REF] Le | Estimation of Riemannian barycentres[END_REF]].

Likewise the issue of convergence of Fréchet empirical mean to Fréchet mean is studied [Ziezold 1977] (for data living in metric spaces). Besides, it is also possible to get an evaluation of the speed of convergence provided in [Bhattacharya 2008, Bhattacharya 2003, Bhattacharya 2005] with central limit theorem (for data living in manifolds). It is also possible to study the speed of convergence for data living in quotient space [Huckemann 2011].

In this thesis, we deal with quotient spaces of finite or infinite dimension, which do not fall into the previous frameworks of manifold preventing us from most of these results (except [Ziezold 1977], for instance).

Geometrical and statistical sufficient conditions for inconsistency

In this thesis, we prove, in several different contexts, the inconsistency of the template estimation with the Fréchet mean in quotient space. In most of situations, Theorems can be summarized in two steps:

1. First, due to the geometry, we can define a geometrical set of points in M .

2. Secondly, we require that the support our random variable Y , X or ε is not included in this particular set.

This structure of proof is also common, for instance in the study of stochastic algoritms [START_REF] Delyon | Stochastic approximation with decreasing gain: Convergence and asymptotic theory[END_REF]]. Let us review some results of this thesis when this sketch of proof applies:

• Theorems 3.1 and 3.4, we define the cone of the template as the set of points closer from the template than any other points in the orbit of the template.

As soon as the support of X is not included in this cone, there is inconsistency.

• Theorems 3.6 and 3.7 (when the template is a fixed point under the action of G), the particular set is the fixed points. We require that the support of the noisy template X is not included in this set, then there is inconsistency.

• Theorem 4.1, we require that the support of the noise ε is not included in the fixed point under the action of G, then there is inconsistency as soon as the noise is large enough.

• Theorem 5.2, we require that the observable Y takes value, with a non zero probability, in the set of points which have a isotropy group reduced to te G u.

To this end, we recall now the support a measure (or a random variable):

Definition 1.6. Let P be a (probability) measure on pM, d M q a metric space, we define the support of P as:

supppPq " tx P M, @r ą 0 PpBpx, rqq ą 0u,

where Bpx, rq is the open ball for the distance d M centred at x with a radius r. Let be X a random variable with a law P we define the support of X as the support of P.

Noise and probability in Hilbert Spaces

Generally, there is no difficulty to generate noise. For instance in R, the Gaussian noise is commonly used, and other noise can be considered. Gaussian noise can easily be generalized in finite dimensional space through the choice a covariance matrix. Moreover it is also possible to generalize in finite dimensional manifolds [Pennec 2006].

Let us talk about the noise in Hilbert spaces, since we will use often in this thesis a noise in the Hilbert space.

Let us begin by a simple example: L 2 pr0, 1s, Rq the set of square integrable real functions defined on r0, 1s. If we want to simulate a noise in this space, one could think that it is enough to chose for all t P r0, 1s f ptq randomly. However there is no reason a priori for the resulting function f to be measurable. Therefore f does not belong to L 2 pr0, 1s, Rq.

One way to make it works, would be for instance to take the Fourier basis of L 2 which is a one to one application form L 2 to l 2 and define the noise on the coefficient. Let pγ n q nPZ a random sequence, where γ n P C. We choose the law of γ n in order to have ř |γ n | 2 ă `8 almost surely. We consider the random function:

X : t Þ Ñ ÿ nPZ γ n e 2πinx
This function has a sense as soon as ř n |γ n | 2 ă `8. Moreover f will be a real function as soon as γ n " γ ´n for all n P Z. The expected value will be (by linearity), the function:

x Þ Ñ

ÿ nPZ Epγ n qe 2πinx .
Likewise one can compute the variance of its random variable by

ř n E `|γ n | 2 ˘Ép|γ n |q 2 q
Note that we do not need to simulate an infinity number of random variables. Indeed, one can first to choose the number of frequencies N randomly (a Poisson distribution for instance), then random numbers: p 1 , . . . , p N (the frequencies), simulate γ 1 , . . . , γ N (the amplitude of each frequencies) and the random function would be

X Þ Ñ N ÿ i"1 γ n e 2πipnx .
This kind of noise can be immediately generalized in any separable Hilbert space, since it suffices to choose pe n q nPN , an orthonormal Hilbert basis, and to choose γ n randomly as previously, then the random point would be ř n γ n e n 1.4. Statistical estimation method 29 1.4.5 Backward/forward model, backward/forward estimation

In this thesis, but also in the literature, there are two ways of considering deformed objects:

• On the one hand, one can assume that the noise is added after deformation, namely that the observable variable is Φ ¨t0 `ε, where t 0 is the template, Φ the unknown and random deformation and ε the additive noise. This is the model introduced by [START_REF] Grenander | Computational Anatomy: An Emerging Discipline[END_REF]] and often considered in computational anatomy. Under this assumption the noise ε is a noise in measurement. In chapter 4, we study the consistency of the template estimation with the Fréchet mean in quotient spaces when observations are created by this model. This model is call forward model in opposition to the backward model defined below.

• On the other hand, one can assume that the noise added before deformation, namely that the observable variable is Φ ¨pt 0 `εq. In this case the noise ε is rather a variability in the shape. Once this variability has been added, the deformation operates on the sum. This is the backward model. In chapter 3, we study the consistency of the template estimation with Fréchet mean in quotient spaces when observations are sampled by this model .

In fact, the model which is the most realistic is probably a mixture of both models. This leads to assume that the observations are Y " Φ ¨pt 0 `εq `ε1 , we will prove the inconsistency of the template estimation with the Fréchet mean when observations are generated by this model in chapter 5.

Once we have the observations, we have to choose between two methods of estimation:

• On the one hand, the forward estimation method: one defines and minimizes the variance

F pmq " E ˆinf gPG d M pg ¨m, Y q 2 ˙.
In other words, ones tries to fit the template to the data. This estimation method will be studied, for non isometric action in chapter 4.

• On the other hand, the backward estimation method: we define and minimize the variance

F pmq " E ˆinf gPG d M pm, g ¨Y q 2
˙.

In other words, ones tries to fit the data to the template. This estimation method will be studied, for non isometric action in chapter 5.

When the distance d M is invariant under the group action the two methods of estimation are equivalent, because d Q is symmetric (since it is a pseudo-distance as we have see it in proposition 1.2). Therefore in this case, we do not have a choice to make. When d M is no longer invariant, the forward estimation method may seem more realistic, for example if you have computed a template, it may seem more realistic to deform the template to the data, than the opposite if you have assumed a forward generative model. But the backward estimation method is also easier to implement.

All these questions are about points of view, which can be discussed, for instance, in [Durrleman 2009]. One could think that the forward model is the true instead of the backward (or vice versa). But at the end of the day, every statistical models are wrong [START_REF] Box | [END_REF]]. Therefore, to the best of our knowledge, this kind of choice are rather philosophical than mathematical: there is no obvious choice because there is no mathematical statements proving than one is better than the other.

Template estimation in this thesis and in related works

The consistency was first studied in the particular case of Procustes means: Procustes mean is the mean of data when rotations, translations (and sometimes scaling) have been removed [Lele 1993, Kent 1997[START_REF] Le | On the consistency of procrustean mean shapes[END_REF], Huckemann 2011]. Note that Procustes means are related to Fréchet means as noticed in [START_REF] Le | On the consistency of procrustean mean shapes[END_REF]]. However, there is no contradiction between this work and, for instance, the article of [Kent 1997]. Indeed, [Kent 1997] showed the consistency when the scaling parameters were taken into account. In most of our work, we deal with isometric actions which exclude this scaling effect. Bigot and Charlier [Bigot 2011] studied the question of the template estimation with a finite sample in the case of translated signals or images by providing a lower bound of the consistency bias. This lower bound was unfortunately not so informative as it is converging to zero asymptotically when the dimension of the space tends to infinity.

Miolane [START_REF] Miolane | [END_REF]] provided a general explanation of why the template is badly estimated for a isometric group action thanks to a geometric interpretation. She showed that the external curvature of the orbit of the template is responsible for the inconsistency. This result was quantified with Gaussian noise for general manifolds.

In this thesis, we study the template estimation by computing the Fréchet mean in quotient space of the observable variable. Our thesis is that this estimation is not consistent, this means that the template is generally not a Fréchet mean of the observable variable mapped in the quotient space. We call consistency bias the distance between the template and the Fréchet means in the quotient space of the observable variable. Contrarily to [Lele 1993, Kent 1997[START_REF] Le | On the consistency of procrustean mean shapes[END_REF], Huckemann 2011, Bigot 2011[START_REF] Huckemann | On the meaning of mean shape: manifold stability, locus and the two sample test[END_REF], we study the template estimation with an abstract action. Therefore, we use a general framework described, for instance, in [START_REF] Huckemann | [END_REF]].

Heuristic of inconsistency

In this thesis, we want to prove that the template estimation is not consistent with the Fréchet mean in the quotient space. Before giving any proofs, we can give an heuristic: Let us take M a Hilbert space, G a group acting, such that d M is invariant under the group action, we note π : M Ñ Q " M {G the canonical projection in the quotient space: πpmq " rms. For Y " Φ ¨pt 0 `σεq a random deformation of the template added to the noise. We define X " t 0 `σε, therefore Y " Φ ¨X. Thus, there is consistency if and only if:

πpt 0 q P FMpπpY qq.
As t 0 " EpXq (because we add a centred noise), and πpY q " πpXq, there is consistency if and only if πpEpXqq P FMpπpXqq. As we have seen it, the Fréchet mean of X is reduced to EpXq because X lives in M a Hilbert space. Therefore we can state that: πpFMpXqq P FMpπpXqq if and only if there is consistency.

Then, the question of consistency is reduced to know if the projection into the quotient space commutes with the Fréchet mean. There are, at least, two trivial cases where this is true:

• If σ " 0, then X " t 0 almost surely, and πpXq " πpt 0 q is a constant variable, then FMpπpXqq " πpEpXqq, and there is consistency.

• If Q is a linear space, the notion of Fréchet mean is just the expectation, therefore there is consistency if and only if πpEpXqq " EpπpXqq. However, it is a well know fact that in general f pEpXqq ‰ Epf pXqq (then in this case why we would have πpFMpXqq P FMpπpXqq? . One remarkable exception is for f a affine map. As a result, if Q is a linear space and if π the canonical projection is an affine, there is consistency.

Therefore when σ ‰ 0 and when the quotient space is not a linear space, there is no reason a priori for πpFMpXqq P FMpπpXqq. And this thesis aims to provide proofs of this heuristic.

Manuscript overview

• Chapter 3 establishes theorems which proves inconsistency in the case of isometric action in a Hilbert space. This works has been partially presented in the Mathematical Foundations of Computational Anatomy workshop [Allassonnière 2015], then published in the SIAM imaging science journal [Devilliers 2017c].

• Chapter 4 establishes an asymptotic behaviour of the consistency bias in the case of isometric action in a Hilbert space. Moreover the used method paves the way to the proof of inconsistency when the action is no longer isometric. We prove in this chapter that the inconsistency appears as soon as the noise is large enough for non isometric actions in the foreward estimation. It has been partially published in the Information Processing in Medical Imaging conference [Devilliers 2017a] and extended in the Entropy journal [Devilliers 2017b].

• Chapter 5 is an opening to generalization. We find an implicit formula for the Fréchet mean in quotient space used to provide new proofs of inconsistency.

In this chapter, we restrict ourselves in the case of a backward estimation. In particular, we also prove inconsistency in complete Riemannian manifold. As a result, we prove that the Fréchet mean in the quotient space is noisier than the original template. Ce travail n'avait pas été écrit complètement mathématiquement, mais a inspiré une nouvelle discipline : l'anatomie computationnelle [START_REF] Grenander | Computational Anatomy: An Emerging Discipline[END_REF]] Cette discipline récente cherche à construire des statistiques sur des données avec un cadre mathématique qui peuvent être implantées sur un ordinateur.

Introduction (français)

Il y a plusieurs raisons qui ont conduit au développement de cette nouvelle discipline:

• L'augmentation de l'utilisation des images médicales en haute résolution.

• Le fait que ces images sont numérisées et puissent être transmises dans le monde entier.

• L'augmentation de la capacité de calcul des ordinateurs.

• Le développement d'outils et de théories mathématiques, par exemple la théorie des espaces de formes [Kendall 1989].

L'anatomie computationelle pourrait être utilisée pour prédire l'apparition de maladies. Par exemple, on peut imaginer utiliser ce modèle de déformation pour comparer un nouveau patient avec un patient connu. Cependant, ce patient connu 2.2. Statistique en anatomie computationnelle est singulier, il peut ne pas être représentatif de la population. Ceci pourrait biaiser l'étude. De plus, un docteur ne compare jamais un nouveau patient avec un ancien. Il compare plutôt ce patient avec tous les patients précédents qui partagent la même maladie. Calculer le patient moyen (dans cette thèse on utilisera le mot template) nous permettrait de faire mimer le processus mental qu'effectue le médecin. Ceci nous conduit à étudier les statistiques en anatomie computationnelle, et en particulier l'estimation de ce template.

Statistique en anatomie computationnelle

Lorsqu'on a des données comme des images d'organes, on veut effectuer des statistiques sur ces données. On peut séparer cette analyse en plusieurs niveaux :

• Étant donné deux éléments (par exemple deux images médicales), on peut estimer la déformation qui permet de faire coller la première image à la seconde. Cette étape, appelée «recalage», formalise l'idée de D'Arcy Thompson. L'un des champs de recherche est de construire des groupes de déformations pour recaler ces éléments [START_REF] Trouvé | An approach of pattern recognition through infinite dimensional group action[END_REF], Miller 2006].

• Si on est capable d'effectuer la première étape, alors on peut estimer la moyenne des images. À cause des déformations, les données n'appartiennent plus à un espace euclidien, cependant on peut généraliser la notion de moyenne avec la moyenne de Fréchet [Fréchet 1948]. Ainsi, on obtient un élément qui est représentatif de notre population [Guimond 2000, Joshi 2004]. Cet élément peut être appelé de différentes façons : template, prototype, patient virtuel etc. Dans toute cette thèse, on utilisera le mot template. Cette estimation peut être effectuée avec la méthode LDDMM [Beg 2005] ou avec l'algorithme des Démons [Thirion 1998[START_REF] Vercauteren | [END_REF][START_REF] Lombaert | Spectral Log-Demons: Diffeomorphic Image Registration with Very Large Deformations[END_REF]].

• Dans la pratique, on obtient le template, et les déformations entre le template et chacune des images [Durrleman 2014, Durrleman 2013, Allassonnière 2015]. L'ensemble formé par le template et ces déformations est appelé atlas.

• L'estimation de template peut être une étape utile, mais elle n'explique pas la variabilité des données. Ainsi, Principal Composant Analysis [START_REF] Huckemann | [END_REF][START_REF] Seiler | [END_REF]] (PCA) ou Principal Geodesic Analysis [START_REF] Fletcher | [END_REF][START_REF] Fletcher | [END_REF], Sommer 2010[START_REF] Sommer | [END_REF]] (PGA), les espaces barycentriques [Pennec 2016[START_REF] Rohé | [END_REF] etc. peuvent être développés pour estimer les directions qui expliquent le mieux les données.

Dans cette thèse, nous nous concentrons sur l'estimation de template (le second niveau dans la liste ci-dessus). Un des buts de cette thèse est d'étudier la consistance de l'algorithme de minimisation utilisé pour estimer le template. On prouve que cette méthode n'est pas consistante. Cela veut dire que, même avec la distribution entière (et non un échantillon), on ne trouve pas le template originel dès que du bruit a été ajouté. De plus, on cherche aussi à estimer l'erreur entre le template originel et le template estimé.

2.3 Cadre géométrique pour l'estimation de template

Distance dans le quotient

Soit pM, d M q un espace métrique, on suppose que G est un groupe qui agit sur M . On suppose que la distance d M est invariante par l'action de groupe, cela veut dire que :

@g P G @m, n P M d M pg ¨m, g ¨nq " d M pm, nq
Dans ce cas, on peut définir une pseudo distance dans le quotient Q " M {G : Lorsqu'on est dans un quotient, l'estimation de la moyenne par la moyenne empirique ne s'applique pas. Cependant on peut utiliser la moyenne de Fréchet.

d Q pras, rbsq " inf

Moyenne de Fréchet dans les espaces de Hilbert

Soit M un espace de Hilbert et soit Z une variable aléatoire dans M telle que Ep}Z} 2 q ă `8. On définit la variance comme l'application E : M Ñ R `définie par Epmq " Ep}Z ´m} 2 . On appelle moyenne de Fréchet l'ensemble des minimiseurs globaux de E. La moyenne de Fréchet empirique est définie comme les minimiseurs globaux de la variance empirique

E n : M Ñ R `avec E n pmq " 1 n n ř i"1
}Z i ´m} 2 pour Z 1 , . . . , Z n n observations de Z. La proposition suivante établit que la moyenne de Fréchet de Z est l'espérance de Z : Proposition 2.1. Soit M un espace de Hilbert, et d M la distance donnée par la norme euclidienne. Soit Z une variable aléatoire telle que Ep}Z} 2 q ă `8, alors m Þ Ñ Epmq " Ep}Z ´m} 2 q est bien définie et m minimise E si et seulement si m " EpZq.

On obtient le même résultat avec les moyennes de Fréchet dans les espaces de Hilbert, la moyenne de Fréchet empirique de Z étant

1 n n ř i"1 Z i .
Comme dit précédemment, nous voulons effectuer des statistiques sur des données qui ne vivent pas dans un espace vectoriel mais plutôt dans un espace métrique. On généralise donc la notion de moyenne de Fréchet dans les espaces métriques.

Moyenne de Fréchet dans les espaces métriques

Definition 2.1. Soit pM, d M q un espace métrique. Soit Z une variable aléatoire dans M , telle que pour tout m P M Epd M pZ, mq 2 q ă `8, alors on peut définir la variance de Z en tout point m P M par :

Epmq " Epd 2 M pZ, mqq
On dit que m est une moyenne de Fréchet de Z si m minimise E. De même, pour un échantillon Z 1 , . . . , Z n un échantillon de Z, on définit la variance empirique de Z au point m par :

E n pmq " 1 n n ÿ i"1 Epd 2 M pZ i , mqq
On dit que m n est une moyenne de Fréchet empirique de Z si m n minimise E n . Un élément minimisant localement E (respectivement E n ) est appelé moyenne de Karcher (respectivement moyenne de Karcher locale).

Proposition 2.2. Soit pM, d M q un espaces métrique. Si Z est une variable aléatoire dans M telle qu'il existe m N P M telle que Epd M pZ, m 0 q 2 q ă `8, alors pour tout m, la variance au point m -Epmq " Epd M pZ, mq 2 q est bien définie, de plus E est une application continue.

Pour trouver les minima locaux/globaux de la variance, on peut montrer que lorsque M est une variété Riemannienne complète et si le lieu de coupure est un ensemble de mesure nulle, alors la variance est différentiable et on peut calculer son gradient (voir [Pennec 2006] par exemple, ce résultat sera aussi prouvé au lemme 3.9). Comme ce sont minima globaux/locaux les moyennes de Fréchet/Karcher sont les points critiques de la variance. Cela conduit, dans [Émery 1991], à la définition suivante : Definition 2.2. Soit M une variété Riemannienne complète. Soit Z une variable aléatoire dans M telle que Epd M pZ, m 0 q 2 q ă `8 pour un certain m 0 P M . On dit que m est barycentre exponentiel si PpZ P Cpmqq " 0 et si : ∇Epmq " ´2EpLog m pZqq " 0 Dans cette thèse, on va étudier la moyenne de Fréchet dans les espaces quotients. Cela signifie que l'on va utiliser la distance d Q définie précédemment pour définir la moyenne de Fréchet dans les espaces quotients.

Dans les espaces métriques (y compris les variétés), l'existence ou l'unicité n'est pas garantie. Par conséquent, l'existence ou l'unicité de la moyenne de Fréchet est souvient étudiée avec des résultats intéressants par exemple [Karcher 1977, Kendall 1990, Afsari 2011, Arnaudon 2005, Arnaudon 2014, Charlier 2013]. Ces résultats sont souvent prouvés dans le cas des variétés. De plus, une fois que l'existence est assurée, on peut développer des méthodes pour estimer la moyenne de Fréchet [START_REF] Le | Estimation of Riemannian barycentres[END_REF]].

De même, on peut étudier la convergence de la moyenne de Fréchet empirique [Ziezold 1977] (dans le cas des espaces métriques). On peut aussi évaluer la vitesse de convergence [Bhattacharya 2008, Bhattacharya 2003, Bhattacharya 2005] avec des théorèmes central limite dans le cas des variétés ou dans le cas des espaces quotients [Huckemann 2011].

Dans toute cette thèse, on s'intéresse aux espaces quotients de dimension finie ou infinie, ainsi on ne peut pas utiliser la plupart de ces résultats (excepté un résultat comme celui de [Ziezold 1977] par exemple).

Estimation de template dans cette thèse et dans d'autres travaux

La consistance a d'abord été étudiée dans le cas particulier des moyennes de Procuste : La moyenne de Procuste est la moyenne des données quand les rotations, translations (et parfois les homothéties) ont été enlevées [Lele 1993, Kent 1997[START_REF] Le | On the consistency of procrustean mean shapes[END_REF], Huckemann 2011]. Le [START_REF] Le | On the consistency of procrustean mean shapes[END_REF]] a remarqué que les moyennes de Procuste sont reliées aux moyennes de Fréchet. Cependant il n'y a pas de contradiction entre ce travail et par exemple l'article [Kent 1997]. En effet, [Kent 1997] montre la consistance quand le paramètre d'homothétie a été pris en compte. Dans ce travail, on étudiera souvent les actions isométriques, ce qui exclut les homothéties.

Bigot et Charlier [Bigot 2011] ont étudié la question de l'estimation de template avec un échantillon de taille fini dans les cas des signaux/images translatés en fournissant une minoration du biais de consistance. Cette minoration n'était malheureusement pas assez informative car elle convergeait vers zéro quand la dimension de l'espace tend vers plus l'infini.

Miolane [START_REF] Miolane | [END_REF]] a expliqué de façon générale pourquoi le template était aussi mal estimé lorsque la distance est invariante sous l'action de groupe grâce à une interprétation géométrique. Elle a montré que la courbure externe de l'orbite du template cause l'inconsistance. Ce résulat a été quantifié avec un bruit gaussien dans des variétés.

Dans cette thèse, on étudie l'estimation de template par le calcul de la moyenne de Fréchet dans l'espace quotient de la variable aléatoire observable. Notre thèse est que cette estimation n'est pas consistante. Cela veut dire que le template n'est généralement pas une moyenne de Fréchet de la variable aléatoire observable projetée dans l'espace quotient. On appelle biais de consistance la distance entre le template et les moyennes de Fréchet. Contrairement à [Lele 1993, Kent 1997[START_REF] Le | On the consistency of procrustean mean shapes[END_REF], Huckemann 2011, Bigot 2011[START_REF] Huckemann | On the meaning of mean shape: manifold stability, locus and the two sample test[END_REF], on étudie l'estimation de template avec une action de groupe abstraite. Par conséquent on utilise un cadre général, décrit par exemple dans [START_REF] Huckemann | [END_REF]]. A short version of this chapter has been presented in a workshop [Allassonnière 2015], then has been extended in the SIAM imaging sciences journal [Devilliers 2017c]. Note that, comparing to this journal paper, we add section 3.4.7 proving that the variance in the quotient space is not differentiable everywhere.

Organisation du manuscrit

Abstract: In this chapter, we study the consistency of the template estimation with the Fréchet mean in quotient spaces. The Fréchet mean in quotient spaces is often used when the observations are deformed or transformed by a group action. We show that in most cases this estimator is actually inconsistent. We exhibit a sufficient condition for this inconsistency, which amounts to the folding of the distribution of the noisy template when it is projected to the quotient space. This condition appears to be fulfilled as soon as the support of the noise is large enough. To quantify this inconsistency we provide lower and upper bounds of the bias as a function of the variability (the noise level). This shows that the consistency bias cannot be neglected when the variability increases.

Introduction

In Kendall's shape space theory [Kendall 1989], in computational anatomy [START_REF] Grenander | Computational Anatomy: An Emerging Discipline[END_REF]], in statistics on signals, or in image analysis, one often aims at estimating a template. A template stands for a prototype of the data. The data can be the shape of an organ studied in a population [Durrleman 2014] or an aircraft [Lefebvre 2012], an electrical signal of the human body, a MR image, etc. To analyze the observations, one assumes that these data follow a statistical model. One often models observations as random deformations of the template with additional noise. This deformable template model proposed in [START_REF] Grenander | Computational Anatomy: An Emerging Discipline[END_REF]] is commonly used in computational anatomy. The concept of deformation introduces the notion of group action: the deformations we consider are elements of a group which acts on the space of observations, called here the ambient space (also called top space). Since the deformations are unknown, one usually considers equivalent classes of observations under the group action. In other words, one considers the quotient space of the ambient space (or top space) by the group. In this particular setting, the template estimation is most of the time based on the minimization of the empirical variance in the quotient space (for instance, [Kurtek 2011b, Joshi 2004, Sabuncu 2008] among many others). The points that minimize the empirical variance are called the empirical Fréchet mean. The Fréchet means introduced in [Fréchet 1948] is comprised of the elements minimizing the variance. This generalizes the notion of expected value in nonlinear spaces. Note that the existence or uniqueness of Fréchet mean is not ensured. But sufficient conditions may be given in order to reach existence and uniqueness (for instance, [Karcher 1977, Kendall 1990]).

Several group actions are used in practice: some signals can be shifted in time compared to other signals (action of horizontal translations [Hitziger 2013]), landmarks can be transformed rigidly [Kendall 1989], shapes can be deformed by diffeomorphisms [Durrleman 2014], etc. In this paper we restrict to transformation which leads the norm unchanged. Rotations, for instance, leave the norm unchanged, but it may seem restrictive. In fact, the square root trick detailed in section 3.5, allows one to build norms which are unchanged, for instance, by reparametrization of curves with a diffeomorphism, where our work can be applied. We raise several issues concerning the estimation of the template.

1. Is the Fréchet mean in the quotient space equal to the original template projected in the quotient space? In other words, is the template estimation with the Fréchet mean in quotient space consistent?

2. If there is an inconsistency, how large is the consistency bias? Indeed, we may expect the consistency bias to be negligible in many practicable cases.

3. If one gets only a finite sample, one can only estimate the empirical Fréchet mean. How far is the empirical Fréchet mean from the original template?

These issues originated from an example exhibited by Allassonnière, Amit, and Trouvé [START_REF] Stéphanie Allassonnière | Towards a coherent statistical framework for dense deformable template estimation[END_REF]]: they took a step function as a template and added some noise and shifted in time this function. By repeating this process they created a data sample from this template. With this data sample, they tried to estimate the template with the empirical Fréchet mean in the quotient space. In this example, minimizing the empirical variance did not succeed in estimating well the template when the noise added to the template increases, even with a large sample size.

One solution to ensure convergence to the template is to replace this estimation method with a Bayesian paradigm ( [START_REF] Stéphanie Allassonnière | Construction of Bayesian deformable models via a stochastic approximation algorithm: a convergence study[END_REF], Bontemps 2014] or [Zhang 2013]). But there is a need to have a better understanding of the failure of the template estimation with the Fréchet mean. One can studied the inconsistency of the template estimation. [START_REF] Bigot | On the consistency of Fréchet means in deformable models for curve and image analysis[END_REF]] first studied the question of the template estimation with a finite sample in the case of translated signals or images by providing a lower bound of the consistency bias. This lower bound was unfortunately not so informative as it is converging to zero asymptotically when the dimension of the space tends to infinity. Miolane and coauthors [Miolane 2015[START_REF] Miolane | [END_REF]] later provided a more general explanation of why the template is badly estimated for a general group action thanks to a geometric interpretation. They showed that the external curvature of the orbits is responsible for the inconsistency. This result was further quantified with Gaussian noise. In this chapter, we provide sufficient conditions on the noise for which inconsistency appears, and we quantify the consistency bias in the general (non necessarily Gaussian) case. Moreover, we mostly consider a vector space (possibly infinite dimensional) as the ambient space while the article of Miolane and co-authors is restricted to finite dimensional manifolds.

This chapter is organized as follows. Section 3.2 details the mathematical terms that we use and the generative model. In sections 3.3 and 3.4, we exhibit a sufficient condition that lead to an inconsistency when the template is not a fixed point under the group action. This sufficient condition can be roughly understand as follows: with a non zero probability, the projection of the random variable on the orbit of the template is different from the template itself. This condition is actually quite general. In particular, this condition it is always fulfilled with the Gaussian noise or with any noise whose support is the whole space. Moreover we quantify the consistency bias with lower and upper bounds. We restrict our study to Hilbert spaces and isometric actions. This means that the space is linear, the group acts linearly and leaves the norm (or the dot product) unchanged. Section 3.3 is dedicated to finite groups. Then we generalise our result in section 3.4 to non-finite groups.

To complete this study, we extend in section 3.5 the result when the template is a fixed point under the group action and when the ambient space is a manifold. As a result we show that the inconsistency exists for almost all noises. Although the bias can be neglected when the noise level is sufficiently small, its linear asymptotic behaviour with respect to the noise level show that it becomes unavoidable for large noises.

Definitions, notation and generative model

We denote by M the ambient space, which is the image/shape space, and G the group acting on M . The action is a map:

G ˆM Ñ M pg, mq Þ Ñ g ¨m,
satisfying the following properties: for all g, g 1 P G, m P M pgg 1 q ¨m " g ¨pg 1 ¨mq and e G ¨m " m where e G is the neutral element of G. For m P M we note by rms the orbit of m (or the class of m). This is the set of points reachable from m under the group action: rms " tg ¨m, g P Gu. Note that if we take two orbits rms and rns there are two possibilities:

1. The orbits are equal: rms " rns i.e. Dg P G s.t. n " g ¨m.

2. The orbits have an empty intersection: rms X rns " H.

We call quotient of M by the group G the set all orbits. This quotient is noted by:

Q " M {G " trms, m P M u.
The orbit of an element m P M can be seen as the subset of M of all elements g ¨m for g P G or as a point in the quotient space. In this chapter we use these two ways. We project an element m of the ambient space M into the quotient by taking rms.

Now we are interested in adding a structure on the quotient from an existing structure in the ambient space: take M a metric space, with d M its distance. Suppose that d M is invariant under the group action which means that:

@g P G, @a, b P M d M pa, bq " d M pg ¨a, g ¨bq.
Then we obtain a pseudo-distance on Q defined by:

d Q pras, rbsq " inf gPG d M pg ¨a, bq. (3.1)
We remind that a distance on M is a map d M : M ˆM Þ Ñ R `such that for all m, n, p P M the following hold:

1. d M pm, nq " d M pn, mq (symmetry).

2. d M pm, nq ď d M pm, pq `dM pp, nq (triangular inequality).

3. d M pm, mq " 0.

4. d M pm, nq " 0 ðñ m " n.

A pseudo-distance satisfies only the first three conditions. If we suppose that all the orbits are closed sets of M , then one can show that d Q is a distance. In this chapter, we assume that d Q is always a distance, even if a pseudo-distance would be sufficient. d Q pras, rbsq can be interpreted as the distance between the shapes a and b, once one has removed the parametrisation by the group G. In other words, a and b have been registered. In this chapter, except in section 3.5, we suppose that the the group acts isometrically on a Hilbert space, this means that the map x Þ Ñ g ¨x is linear, and that the norm associated to the dot product is conserved: }g ¨x} " }x}.

Then d M pa, bq " }a ´b} is a particular case of invariant distance.

We now introduce the generative model used in this chapter for M a vector space. Let us take a template t 0 P M to which we add a unbiased noise ε: X " t 0 `ε. Finally we transform X with a random shift Φ of G. We assume that this variable Φ is independent of X and the only observed variable is:

Y " Φ ¨X " Φ ¨pt 0 `εq, with Epεq " 0, (3.2)
while Φ, X and ε are hidden variables. Note that it is not the generative model defined by Grenander and often used in computational anatomy. Where the observed variable is rather Y 1 " Φ ¨t0 `ε1 . But when the noise is isotropic and the action is isometric, one can show that the two models have the same law, since Φ ¨ε and ε have the same probability distribution. As a consequence, the inconsistency of the template estimation with the Fréchet mean in quotient space with one model implies the inconsistency with the other model. Because the former model (3.2) leads to simpler computation we consider only this model.

We can now set the inverse problem: given the observation Y , how to estimate the template t 0 in M ? This is an ill-posed problem. Indeed for some element group g P G, the template t 0 can be replaced by the translated g ¨t0 , the shift Φ by Φg ´1 and the noise ε by g ¨ε, which leads to the same observation Y . So instead of estimating the template t 0 , we estimate its orbit rt 0 s. By projecting the observation Y in the quotient space we obtain rY s. Although the observation Y " Φ ¨X and the noisy template X are different random variables in the ambient space, their projections on the quotient space lead to the same random orbit rY s " rXs. That is why we consider the generative model (3.2): the projection in the quotient space remove the transformation of the group G. From now on, we use the random orbit rXs in lieu of the random orbit of the observation rY s.

The variance of the random orbit rXs (sometimes called the Fréchet functional or the energy function) at the quotient point rms P Q is the expected value of the square distance between rms and the random orbit rXs, namely:

Q Q rms Þ Ñ Epd Q prms, rXsq 2 q (3.3)
Q Q rms Þ Ñ 1 n n ÿ i"1 d Q prms, rY i sq 2 " 1 n n ÿ i"1 inf g i PG }m ´gi ¨Yi } 2 . (3.4)
Thanks to the equality of the quotient variables rXs and rY s, an element which minimises this map is an empirical Fréchet mean of rXs.

In order to minimize the empirical quotient variance (3.4), the max-max algorithm1 alternatively minimizes the function Jpm, pg i q i q " 1 n n ř i"1 }m ´gi ¨Yi } 2 over a point m of the orbit rms and over the hidden transformation pg i q 1ďiďn P G n . With these notations we can reformulate our questions as follows:

1. Is the orbit of the template rt 0 s a minimiser of the quotient variance defined in (3.3)? If not, the Fréchet mean in quotient space is an inconsistent estimator of rt 0 s.

2. In this last case, can we quantify the quotient distance between rt 0 s and a Fréchet mean of rXs?

3. Can we quantify the distance between rt 0 s and an empirical Fréchet mean of a n-sample?

This chapter shows that the answer to the first question is usually "no" in the framework of a Hilbert space M on which a group G acts linearly and isometrically.

The only exception is theorem 3.6 where the ambient space M is a manifold. In order to prove inconsistency, an important notion in this framework is the isotropy group of a point m in the ambient space. This is the subgroup which leaves this point unchanged:

Isopmq " tg P G, g ¨m " mu.

We start in section 3.3 with the simple example where the group is finite and the isotropy group of the template is reduced to the identity element (Isopt 0 q " te G u, in this case t 0 is called a regular point). We turn in section 3.4 to the case of a general group and an isotropy group of the template which does not cover the whole group (Isopt 0 q ‰ G) i.e t 0 is not a fixed point under the group action. To complete the analysis, we assume in section 3.5 that the template t 0 is a fixed point which means that Isopt 0 q " G.

In sections 3.3 and 3.4 we show lower and upper bounds of the consistency bias which we define as the quotient distance between the template orbit and the Fréchet mean in quotient space. These results give an answer to the second question. In section 3.4, we show a lower bound for the case of the empirical Fréchet mean which answers to the third question.

As we deal with different notions whose name or definition may seem similar, we use the following vocabulary:

1. The variance of the noisy template X in the ambient space is the function

E : m P M Þ Ñ Ep}m´X} 2 q.
The unique element which minimises this function is the Fréchet mean of X in the ambient space. With our assumptions it is the template t 0 itself.

2. We call variability (or noise level) of the template the value of the variance at this minimum: σ 2 " Ep}t 0 ´X} 2 q " Ept 0 q.

3. The variance of the random orbit rXs in the quotient space is the function

F : m Þ Ñ Epd Q prms, rXsq 2 q.
Notice that we define this function from the ambient space and not from the quotient space. With this definition, an orbit rm ‹ s is a Fréchet mean of rXs if the point m ‹ is a global minimiser of F .

In sections 3.3 and 3.4, we exhibit a sufficient condition for the inconsistency, which is: the noisy template X takes value with a non zero probability in the set of points which are strictly closer to g ¨t0 for some g P G than the template t 0 itself. This is linked to the folding of the distribution of the noisy template when it is projected to the quotient space. The points for which the distance to the template orbit in the quotient space is equal to the distance to the template in the ambient space are projected without being folded. If the support of the distribution of the noisy template contains folded points (we only assume that the probability measure of X, noted P, is a regular measure), then there is inconsistency. The support of the noisy template X is defined by the set of points x such that PpX P Bpx, rqq ą 0 for all r ą 0. For different geometries of the orbit of the template, we show that this condition is fulfilled as soon as the support of the noise is large enough.

The recent article of Cleveland et al. [START_REF] Cleveland | Norm-preserving constraint in the Fisher-Rao registration and its application in signal estimation[END_REF]] may seem contradictory with our current work. Indeed the consistency of the template estimation with the Fréchet mean in quotient space is proved under hypotheses which seem to satisfy our framework: the norm is unchanged under their group action (isometric action) and a noise is present in their generative model. However we believe that the noise they consider might actually not be measurable. Indeed, their ambient space is:

L 2 pr0, 1sq " " f : r0, 1s Ñ R such that f is measurable and ż 1 0 f 2 ptqdt ă `8* .
The noise e is supposed to be in L 2 pr0, 1sq such that for all t, s P r0, 1s, Epeptqq " 0 and Epeptqepsqq " σ 2 1 s"t , for σ ą 0. This means that eptq and epsq are chosen without correlation as soon as s ‰ t. In this case, it is not clear for us that the resulting function e is measurable, and thus that its Lebesgue integration makes sense. Thus, the existence of such a random process should be established before we can fairly compare the results of both works. Furthermore, no discrete version of their theorem is given.

Study of consistency for finite group

In this section, we consider a finite group G acting isometrically and effectively on M " R n a finite dimensional space equipped with the euclidean norm } }, associated to the dot product x , y.

We say that the action is effective if x Þ Ñ g ¨x is the identity map if and only if g " e G . Note that if the action is not effective, we can define a new effective action by simply quotienting G by the subgroup of the element g P G such that x Þ Ñ g ¨x is the identity map.

The template is assumed to be a regular point which means that the isotropy group of the template is reduced to the neutral element of G. Note that the measure of singular points (the points which are not regular) is a null set for the Lebesgue measure (see item 1 in section 3.3.2).

Example 3.1. The action of horizontal translation: this action is a simplified setting for image registration, where images can be obtained by the translation of one scan to another due to different poses. More precisely, we take the vector space M " R T where G " T " pZ{N Zq D is the finite torus in D-dimension. An element of R T is seen as a function m : T Ñ R, where mpτ q is the grey value at pixel τ . When D " 1, m can be seen like a discretised signal with N points, when D " 2, we can see m like an image with N ˆN pixels etc. We then define the group action of T on R T by:

τ P T, m P R T τ ¨m : x Þ Ñ mpx `τ q.
This group acts isometrically and effectively on M " R T .

In this setting, if Ep}X} 2 q ă `8 then the variance of rXs is well defined:

F : m P M Þ Ñ Epd Q prXs, rmsq 2 q. (3.5)
In this framework, F is non-negative and continuous. Then we can prove the existence of the Fréchet mean in the quotient space:

Proposition 3.1. Let G be a group acting isometrically on M an Euclidean space, then F has a minimizer.

Proof. Thanks to Cauchy-Schwarz inequality we have:

lim }m}Ñ8 F pmq ě lim }m}Ñ8 }m} 2 ´2}m}Ep}X}q `Ep}X} 2 q " `8.
Thus for some R ą 0 we have: for all m P M if }m} ą R then F pmq ě F p0q `1.

The closed ball Bp0, Rq is a compact set (because M is a finite vector space) then F restricted to this ball reaches its minimum at some point m ‹ (since F is continuous).

Then for all m P M , we have:

If }m} ď R then F pmq ě F pm ‹ q If }m} ą R then F pmq ě F p0q `1 ą F p0q ě F pm ‹ q
Therefore rm ‹ s is a Fréchet mean of rXs in the quotient Q " M {G. Note that this ensure the existence of the Fréchet mean in quotient spaces but not its uniqueness.

In this section, we show that as soon as the support of the distribution of X is big enough, the orbit of the template is not a Fréchet mean of rXs. We provide an upper bound of the consistency bias depending on the variability of X and an example of computation of this consistency bias.

Presence of inconsistency

0 t 0 g ¨t0 g 1 ¨t0 Conept 0 q Figure 3
.1: Planar representation of a part of the orbit of the template t 0 . The lines are the hyperplanes whose points are equally distant of two distinct elements of the orbit of t 0 , Conept 0 q represented in points is the set of points closer from t 0 than any other points in the orbit of t 0 . Theorem 3.1 states that if the support (the dotted disk) of the random variable X is not included in this cone, then there is an inconsistency.

The following theorem gives a sufficient condition on the random variable X for an inconsistency: Theorem 3.1. Let G be a finite group acting on M " R n isometrically and effectively. Assume that the random variable X is absolutely continuous with respect to the Lebesgue's measure, with Ep}X} 2 q ă `8. We assume that t 0 " EpXq is a regular point.

We define Conept 0 q as the set of points closer from t 0 than any other points of the orbit rt 0 s, see figure 3.1 or item 6 in section 3.3.2 for a formal definition.

In other words, Conept 0 q is defined as the set of points already registered with t 0 . Suppose that:

P pX R Conept 0 qq ą 0, (3.6)
then rt 0 s is not a Fréchet mean of rXs.

Because the action is isometric, this set is really a cone (this point is proved in section 3.3.2), this justifies the name. Note that is the Voronoï cell associated to the template. The Voronoï cells have been defined and used by Voronoï [START_REF] Voronoi | Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les parallélloèdres primitifs[END_REF]] and Dirichlet [START_REF] Lejeune | Über die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen[END_REF]].

The proof of theorem 3.1 is based on two steps: first, differentiating the variance F of rXs. Second, showing that the gradient at the template is not zero, therefore the template can not be a minimum of F . Theorem 3.2 makes the first step.

Theorem 3.2. The variance F of rXs is differentiable at any regular points. For m 0 a regular point, we define gpx, m 0 q as the almost unique g P G minimizing }m 0 ´g ¨x} (in other words, gpx, m 0 q ¨x P Conepm 0 q). This allows us to compute the gradient of F at m 0 : ∇F pm 0 q " 2pm 0 ´EpgpX, m 0 q ¨Xqq.

(3.7)

Remark 3.1. It may seem that it is not obvious that gpX, m 0 q ¨X is measurable. This is a natural requirement before taking the expected value. However, x Þ Ñ m 0 ǵpX, m 0 q ¨X is nothing else that the gradient of a measurable and differentiable function which is m 0 Þ Ñ d 2 Q prXs, rm 0 sq (if we admit theorem 3.2). Therefore its measurability is obvious. In chapter 5, the group will be no longer finite, and the measurability will become an issue, which will be discussed.

This Theorem is proved in section 3.3.2. Then we show that the gradient of F at t 0 is not zero. To ensure that F is differentiable at t 0 we suppose in the assumptions of theorem 3.1 that t 0 " EpXq is a regular point. Thanks to theorem 3.2 we have:

∇F pt 0 q " 2pt 0 ´EpgpX, t 0 q ¨Xqq.
Therefore ∇F pt 0 q{2 is the difference between two terms, which are represented on figure 3.2: on figure 3.2a there is a mass under the two hyperplanes outside Conept 0 q, so this mass is nearer from g ¨t0 for some g P G than from t 0 . In the following expression Z " EpgpX, t 0 q ¨Xq, for X R Conept 0 q, gpX, t 0 qX P Conept 0 q such points are represented in red (grid-line) on figure 3.2, (in this case, we say that X is folded). This suggests that the point Z " EpgpX, t 0 q¨Xq which is the mean of points in Conept 0 q is further away from 0 than t 0 . Then ∇F pt 0 q{2 " t 0 ´Z should be not zero, and t 0 " EpXq is not a critical point of the variance of rXs. As a conclusion rt 0 s is not a Fréchet mean of rXs. This is turned into a rigorous proof in section 3.3.3.

0 t 0 g ¨t0 g 1 ¨t0
Conept 0 q (a) Graphic representation of the template t0 " EpXq mean of points of the support of X.

0 t 0 Z g ¨t0 g 1 ¨t0 Conept 0 q (b) Graphic representation of Z "
EpgpX, t0q ¨Xq. The points X which were outside Conept0q are now in Conept0q (thanks to gpX, t0q). This part, in grid-line, represents the points which have been folded.

Figure 3.2: Z is the mean of points in Conept 0 q where Conept 0 q is the set of points closer from t 0 than g ¨t0 for g P Gze G . Therefore it seems that Z is higher that t 0 , therefore ∇F pt 0 q " 2pt 0 ´Zq ‰ 0.

Note also that theorem 3.2 gives a criteria of the Fréchet means of rXs; if rm ‹ s is a Fréchet mean of rXs there are two cases: m ‹ is not a regular point or m ‹ " EpgpX, m ‹ q ¨Xq.

(3.8)

In the proof of theorem 3.1, we took M an Euclidean space and we work with the Lebesgue's measure in order to have PpX P Hq " 0 for every hyperplane H. Therefore the proof of theorem 3.1 can be extended immediately to any Hilbert space M , if we make now the assumption that PpX P Hq " 0 for every hyperplane H, as long as we keep a finite group acting isometrically and effectively on M . Figure 3.2 illustrates the condition of theorem 3.1: if there is no mass beyond the hyperplanes, then the two terms in ∇F pt 0 q are equal (because almost surely gpX, t 0 q ¨X " X). Therefore in this case we have ∇F pt 0 q " 0. This does not prove necessarily that there is no inconsistency, just that the template t 0 is a critical point of F . Moreover this figure can give us an intuition on what the consistency bias (the distance between rt 0 s and the set of all Fréchet mean in the quotient space) depends: for t 0 a fixed regular point, when the variability of X (defined by Ep}X ´t0 } 2 q) increases the mass beyond the hyperplanes on figure 3.2 also increases, the distance between EpgpX, t 0 q ¨Xq and t 0 (i.e. the norm of ∇F pt 0 q) augments. Therefore q the Fréchet mean should be further from t 0 , (because at this point one should have ∇F pqq " 0 or q is a singular point). Therefore the consistency bias appears to increase with the variability of X. By establishing a lower and upper bound of the consistency bias and by computing the consistency bias in a very simple case, sections 3.3. 5, 3.3.6, 3.4.3 and 3.4.4 investigate how far this hypothesis is true.

Proof of theorem 3.2: differentiation of the variance in the quotient space

In order to show theorem 3.2 we proceed in three steps. First we see some following properties and definitions which will be used. Most of these properties are the consequences of the fact that the group G is finite. Then we show that the integrand of F is differentiable. Finally we show that we can permute gradient and integral signs.

1. The set of singular points in R n , is a null set (for the Lebesgue's measure), since it is equal to:

ď g‰e G kerpx Þ Ñ g ¨x ´xq,
a finite union of proper linear subspaces of R n thanks to the linearity and effectively of the action and to the finite group.

2. If m is regular, then for g, g 1 two different elements of G, we pose:

Hpg ¨m, g 1 ¨mq " tx P R n , }x ´g ¨m} " }x ´g1 ¨m}u.

Moreover Hpg ¨m, g 1 ¨mq " pg ¨m ´g1 ¨mq K is an hyperplane.

3. For m a regular point we define the set of points which are equally distant from two different points of the orbit of m:

A m " ď g‰g 1
Hpg ¨m, g 1 ¨mq.

Then A m is a null set. For m regular and x R A m the minimum in the definition of the quotient distance :

d Q prms, rxsq " min gPG }m ´g ¨x}, (3.9)
is reached at a unique g P G, we call gpx, mq this unique element.

4. By expansion of the squared norm: g minimises }m ´g ¨x} if and only if g maximizes xm, g ¨xy.

5. If m is regular and x R A m then: @g P Gztgpx, mqu, }m ´gpx, mq ¨x} ă }m ´g ¨x}, by continuity of the norm and by the fact that G is a finite group, we can find α ą 0, such that for µ P Bpm, αq and y P Bpx, αq: @g P Gztgpx, mqu }µ ´gpx, mq ¨y} ă }µ ´g ¨y}.

Therefore for such y and µ we have:

gpx, mq " gpy, µq.

6. For m a regular point, we define Conepmq the convex cone of R n :

Conepmq " tx P R n { @g P G }x ´m} ď }x ´g ¨m}u (3.10)

" tx P R n { @g P G xm, xy ě xgm, xyu.
This is the intersection of |G| ´1 half-spaces (some of them could be equal): each half space is delimited by Hpm, gmq for g ‰ e G (see figure 3.1). Conepmq is the set of points whose projection on rms is m, (where the projection of one point p on rms is one point g ¨m which minimizes the set t}p ´g ¨m}, g P Gu).

7. Taking a regular point m allows us to see the quotient. For every point x P R n we have: rxs

Ş Conepmq ‰ H, cardprxs Ş Conepmqq ě 2 if and only if x P A m .
The borders of the cone are ConepmqzIntpConepmqq " Conepmq X A m (we denote by IntpAq the interior of a part A). Therefore Q " R n {G can be seen like Conepmq whose borders have been glued together.

The proof of theorem 3.2 is the consequence of the following lemmas. The first lemma studies the differentiability of the integrand, and the second allows us to permute gradient and integral sign. Let us denote by f the integrand of F : @ m, x P M f px, mq " min gPG }m ´g ¨x} 2 .

(3.11)

Thus we have: F pmq " Epf pX, mqq. The min of differentiable functions is not necessarily differentiable, however we prove the following result:

Lemma 3.1. Let m 0 be a regular point, if x R A m 0 then m Þ Ñ f px, mq is differen- tiable at m 0 , besides we have: Bf Bm px, m 0 q " 2pm 0 ´gpx, m 0 q ¨xq (3.12)
Proof. If m 0 is regular and x R A m 0 then we know from the item 5 of the section 3.3.2 that gpx, m 0 q is locally constant. Therefore around m 0 , we have:

f px, mq " }m ´gpx, m 0 q ¨x} 2 ,
which can differentiate with respect to m at m 0 . This proves the lemma 3.1. Now we want to prove that we can permute the integral and the gradient sign. The following lemma provides us a sufficient condition to permute integral and differentiation signs thanks to the dominated convergence theorem: Lemma 3.2. For every m 0 P M " R n , we have the existence of an integrable function Ψ : M Ñ R `(for Pthe law of X) such that: @m P Bpm 0 , 1q, @x P M |f px, m 0 q ´f px, mq| ď }m ´m0 }Ψpxq, (3.13)

where Ψpxq " 2}m 0 } `1 `2}x} + Proof. For all g P G, m P M we have:

}g ¨x ´m0 } 2 ´}g ¨x ´m} 2 " xm ´m0 , 2g ¨x ´pm 0 `mqy ď }m ´m0 } ˆp}m 0 `m} `}2x}q min gPG }g ¨x ´m0 } 2 ď }m ´m0 } p}m 0 `m} `}2x}q `}g ¨x ´m} 2 min gPG }g ¨x ´m0 } 2 ď }m ´m0 } p}m 0 `m} `}2x}q `min gPG }g ¨x ´m} 2 min gPG }g ¨x ´m0 } 2 ´min gPG }g ¨x ´m} 2 ď }m ´m0 } p2}m 0 } `}m ´m0 } `}2x}q
By symmetry we get also the same control of f px, mq ´f px, m 0 q, then:

|f px, m 0 q ´f px, mq| ď }m 0 ´m} p2}m 0 } `}m ´m0 } `}2x}q (3.14)
The function Ψ should depend on x or m 0 , but not on m. That is why we take only m P Bpm 0 , 1q, then we replace }m ´m0 } by 1 in (3.14), which concludes.

3.3.3

Proof of theorem 3.1: the gradient is not zero at the template

To prove it, we suppose that ∇F pt 0 q " 0, and we take the dot product with t 0 :

x∇F pt 0 q, t 0 y " 2EpxX, t 0 y ´xgpX, t 0 q ¨X, t 0 yq " 0.

(3.15)

The item 4 of px, mq Þ Ñ gpx, mq seen at section 3.3.2 leads to:

xX, t 0 y ´xgpX, t 0 q ¨X, t 0 y ď 0 almost surely.

So the expected value of a non-positive random variable is null. Then xX, t 0 y ´xgpX, t 0 q ¨X, t 0 y " 0 almost surely xX, t 0 y " xgpX, t 0 q ¨X, t 0 y almost surely.

Then g " e G maximizes the dot product almost surely. Therefore (as we know that gpX, t 0 q is unique almost surely, since t 0 is regular):

gpX, t 0 q " e G almost surely,
which is a contradiction with Equation (3.6).

54

Chapter 3. Inconsistency in Hilbert space for isometric action

t 0 g ¨t0 g 1 ¨t0 Conept 0 q O y Conepyq Figure 3.3: y Þ Ñ Conepyq is continuous.
When the support of the X is bounded and included in the interior of Conept 0 q the hatched cone. For y sufficiently close to the template t 0 , the support of the X (the ball in red) is still included in Conepyq (in grey), then F pyq " pEp}X ´y} 2 ). Therefore in this case, rt 0 s is at least a Karcher mean of rXs.

Study of consistency when the support of X is included in the cone of the template

We can also wonder if the converse of theorem 3.1 is true: if the support is included in Conept 0 q, is there consistency? We do not have a general answer to that. In the simple example section 3.3.6 it happens that condition (3.6) is necessary and sufficient. More generally the following proposition provides a partial converse:

Proposition 3.2. If the support of X is a compact set included in the interior of Conept 0 q, then the orbit of the template rt 0 s is at least a Karcher mean of rXs (a Karcher mean is a local minimum of the variance).

Sketch of the proof: If the support of X is a compact set included in the interior of Conept 0 q then we know that X-almost surely: d Q prXs, rt 0 sq " }X ´t0 }. Thus the variance at t 0 in the quotient space is equal to the variance at t 0 in the ambient space. Now if we assume a sort of continuity of the cone (see figure 3.3) for y in a small neighbourhood of t 0 , the support of X is still included in the interior of Conepyq. We still have d Q prXs, rysq " }X ´y} X-almost surely. In other words, locally around t 0 , the variance in the quotient space is equal to the variance in the ambient space. Moreover we know that t 0 " EpXq is the only global minimiser of the variance of X: m Þ Ñ Ep}m ´X} 2 q " Epmq. Therefore t 0 is a local minimum of F the variance in the quotient space (since the two variances are locally equal). Therefore rt 0 s is at least a Karcher mean of rXs in this case.

However, our notion of continuity of the cone is too vague, therefore we give a more rigorous proof based on the same idea:

Proof. We know that we have SupportpXq Ă Conept 0 q and we want to show that: Dη ą 0s.t.@y P Bpt 0 , rq SupportpXq Ă Conepyq (3.16)

Let us assume that equation (3.16) does not hold. In this case we have:

@n P N Dy n P Bpt 0 , 1 n q Dz n P SupportpXq and z n R Conepy n q

We have that py n q n converges to t 0 . As the support of ε is assumed to be compact, by extraction, without loss of generality we can assume that z n converges to z. First z P SupportpXq (because SupportpXq is a closed set. Now as z n R Conepy n q, there exists g n P G such that xz n , y n ´gn ¨yn y ă 0. As G is a finite group, by extraction, without loss of generality, we can assume that pg n q n is a constant sequence. Then we have xz n , y n ´gy n y ă 0. By taking the limit, we have that xz, t 0 ´gt 0 y ď 0. As a consequence z is not in the interior of the cone of the template. This is absurd, since z P SupportpXq Ă IntpConept 0 qq. Therefore we have proved that equation (3.16) holds. As a consequence, @y P Bpt 0 , ηq, F pyq " Ep}X ´y} 2 q ě Ep}X ´t0 } 2 q " F pt 0 q.

Lastly for η sufficiently small we have Bpt 0 , ηq Ă Conept 0 q. This proves that rt 0 s is a minimum of the variance restricted to Bprt 0 s, ηq (the open ball in Q of center rt 0 s and radius η). This proves that rt 0 s is a Karcher mean of rXs. Can the template be a Fréchet mean instead of being only a Karcher mean? The following simple example show that the template can be Karcher mean without being a Fréchet mean.

Example 3.2. Let us take M " R 2 the euclidean plane. G " tI 2 , ´I2 u acts isometrically on M . On this example, the quotient distance is d Q pras, rbsq " minp}a ´b}, }a `b}q. Let us take A and B two points, let X a random variable such that PpX " Aq " PpX " Bq " 1 2 . Then X " t 0 `ε, where t 0 " A`B 2 , and ε is a noise such that Epεq " 0.

Now, all we have to do is to chose wisely A and B. We propose A " p1, 2.7q and B " p1, 0.7q then t 0 " p1, 1q, A, B are inside the interior of Conept 0 q, (here Conept 0 q " tx P R 2 , s.t. xx, t 0 y ě 0u). The variance in the quotient space is defined by:

F pmq " 1 2 `minp}m ´A} 2 , }m `A} 2 q `minp}m ´B} 2 , }m `B} 2 q Ȏn
this example it easy to verify that 2 " F p A´B 2 q ă F pt 0 q " 2.89. And the Fréchet mean of rXs is exactly rm ‹ s " r A´B 2 s. Therefore it is possible that the template estimation is inconsistent even if the random variable X is included into the cone of the template.

' A ' B ' ´A ' ´B ' t 0 ' m ‹ ' 0 Conept 0 q Figure 3.4:
Representation of the template t 0 , A, B, in gray the cone of t 0 .

Upper bound of the consistency bias

In this Subsection we show an explicit upper bound of the consistency bias.

Theorem 3.3. When G is a finite group acting isometrically on M " R n , we denote |G| the cardinal of the group G. If X is Gaussian vector: X " N pt 0 , w 2 Id R n q, and m ‹ P argmin F , then we have the upper bound of the consistency bias:

d Q prt 0 s, rm ‹ sq ď w a 8 lnp|G|q.
(3.17)

When X " N pt 0 , w 2 Id n q the variability of X is σ 2 " Ep||X ´t0 || 2 q " nw 2 and we can write the upper bound of the bias: d Q prt 0 s, rm ‹ sq ď σ ? n a 8 ln |G|. This Theorem shows that the consistency bias is low when the variability of X is small, which tends to confirm our hypothesis in section 3.3.1. It is important to notice that this upper bound explodes very slowly when the cardinal of the group tends to infinity.

In order to show this Theorem, we use the following lemma:

Lemma 3.3. We write X " t 0 `ε where Epεq " 0 and we make the assumption that the noise ε is a subgaussian random variable. This means that it exists c ą 0 and w ą 0 such that:

@m P M " R n , Epexppxε, myqq ď c exp ˆw2 }m} 2 2 ˙. (3.18)
For m P M we note ρ " d Q prms, rt 0 sq " inf gPG }g ¨m ´t0 }. If we have:

ρ ě w a 2 lnpc|G|q, (3.19)
then we have: ρ2 ´ρw a 8 lnpc|G|q ď F pmq ´Ep}ε} 2 q.

(3.20)

Study of consistency for finite group

57

-2 3 2 -1 1 0 1 2 m->f(m) 0 t0 mstar -1 2 3 4 5 6 7 8 -2 -3 Figure 3.5: Variation of the variance m Þ Ñ F pmq.
Proof of lemma 3.3. First we expand the right member of the inequality (3.20):

Ep}ε} 2 q ´F pmq " E ˆmax gPG p}X ´t0 } 2 ´}X ´g ¨m} 2 q Ẇe use the formula }A} 2 ´}A `B} 2 " ´2 xA, By ´}B} 2 with A " X ´t0 and B " t 0 ´g ¨m:

Ep}ε} 2 q ´F pmq " E " max gPG `´2 xX ´t0 , t 0 ´g ¨my ´}t 0 ´gm} 2 ˘ " Epmax gPG η g q,
(3.21) with η g " ´}t 0 ´g ¨m} 2 `2 xε, gm ´t0 y. Our goal is to find a lower bound of F pmq ´Ep}ε} 2 q, that is why we search an upper bound of Epmax gPG η g q with the Jensen's inequality. We take x ą 0 and we get by using the assumption (3.18):

exppxEpmax gPG η g qq ď Epexppmax gPG xη g qq ď E ˜ÿ gPG exppxη g q ḑ ÿ gPG expp´x}t 0 ´gm} 2 qEpexppxε, 2xpgm ´t0 qyq ď c ÿ gPG expp´x}t 0 ´gm} 2 q expp2w 2 x 2 }gm ´t0 } 2 q ď c ÿ gPG expp}gm ´t0 } 2 p´x `2x 2 w 2 qq (3.22)
Now if p´x `2w 2 x 2 q ă 0, we can take an upper bound of the sum sign in (3.22) by taking the smallest value in the sum sign, which is reached when g minimizes }g ¨m ´t0 } multiplied by the number of elements summed. Moreover:

p´x `2x 2 w 2 q ă 0 ðñ 0 ă x ă 1 2w 2 .
Then we have:

exppxEpmax gPG η g qq ď c|G| exppρ 2 p´x `2x 2 w 2 qq as soon as 0 ă x ă 1 2w 2 .
Then by taking the logarithm map:

Epmax gPG η g q ď ln c|G| x `p2xw 2 ´1qρ 2 . (3.23)
Now we find the x which optimizes inequality (3.23). By differentiation, the right member of inequality (3.23) is minimal for x ‹ " a ln c|G|{2{pw ρq which is a valid choice because x ‹ P p0, 1 2w 2 q by using the assumption (3.19). With the equations (3.21) and (3.23) and x ‹ we get the result.

Proof of theorem 3.3. We take m ‹ P argmin F , ρ " d Q prm ‹ s, rt 0 sq, and ε " X ´t0 . We have: F pm ‹ q ď F pt 0 q ď Ep}ε} 2 q then F pm ‹ q ´Ep}ε} 2 q ď 0. If ρ ą w a 2 lnp|G|q then we can apply lemma 3.3 with c " 1. Thus: ρ2 ´ρw a 8 lnp|G|q ď 2F pm ‹ q ´Ep}ε} 2 q ď 0, which yields to ρ ď w a 8 lnp|G|q. If ρ ď w a 2 lnp|G|q, we have nothing to prove. Note that the proof of this upper bound does not use the fact that the action is isometric, therefore this upper bound is true for every finite group action. More precisely, when the action is not isometric, if m ‹ minimizes m Þ Ñ F pmq " Epinf gPG }Xǵ ¨m} 2 q then inf gPG }t 0 ´g ¨m‹ } ď w a 8 ln |G|.

Study of the consistency bias in a simple example

In this Subsection, we take a particular case of example 3.1: the action of horizontal translation with T " Z{2Z. We identify R T with R 2 and we note by pu, vq T an element of R T . In this setting, one can completely describe the action of T on R T : 0 ¨pu, vq T " pu, vq T and 1 ¨pu, vq T " pv, uq T . The set of singularities is the line L " tpu, uq T , u P Ru. We note HP A " tpu, vq T , v ą uu the half-plane above L and HP B the half-plane below L.

This simple example will allow us to provide necessary and sufficient condition for an inconsistency at regular and singular points. Moreover we can compute exactly the consistency bias, and exhibit which parameters govern the bias. We can then find an equivalent of the consistency bias when the noise tends to zero or infinity. More precisely, we have the following theorem: 3.3. Study of consistency for finite group 59 Proposition 3.3. Let X be a random variable such that Ep}X} 2 q ă `8 and t 0 " EpXq.

1. If t 0 P L, there is no inconsistency if and only if the support of X is included in the line L " tpu, uq, u P Ru. If t 0 P HP A (respectively in HP B ), there is no inconsistency if and only if the support of X is included in

HP A Y L (respectively in HP B Y L).
2. If X is Gaussian: X " N pt 0 , w 2 Id 2 q, then the Fréchet mean of rXs exists and is unique. This Fréchet mean rm ‹ s is on the line passing through EpXq and perpendicular to L and the consistency bias ρ " d Q prt 0 s, rm ‹ sq is the function of s and d " distpt 0 , Lq given by:

ρpd, sq " w 2 π ż `8 d w r 2 exp ˆ´r 2 2 ˙κ ˆd rw ˙dr, (3.24)
where κ is a non-negative function on r0, 1s defined by κpxq " sinparccospxqq x arccospxq. (c) w Þ Ñ ρp0, wq is linear with respect to w (for d " 0 the template is a fixed point).

Remark 3.2. Here, contrarily to the case of the action of rotation in [START_REF] Miolane | [END_REF]], it is not the ratio }EpXq} over the noise which matters to estimate the consistency bias. Rather the ratio distpEpXq, Lq over the noise. However in both cases we measure the distance between the signal and the singularities which was t0u in [START_REF] Miolane | [END_REF]] for the action of rotations, L in this case.

Proof. We suppose that EpXq P HP A Y L. In this setting we call τ px, mq one of element of the group G " T which minimizes }τ ¨x ´m} see (3.9) instead of gpx, mq.

The variance in the quotient space at the point m is:

F pmq " E ˆmin τ PZ{2Z }τ ¨X ´m} 2 ˙" Ep}τ pX, mq ¨X ´m} 2 q.
As we want to minimize F and F p1¨mq " F pmq, we can suppose that m P HP A YL.

We can find the value of τ px, mq for x P M :

• If x P HP A Y L we can set τ px, mq " 0 (because in this case x, m are on the same half plane delimited by L the perpendicular bisector of m and ´m).

• If x P HP B then we can set τ px, mq " 1 (because in this case x, m are not on the same half plane delimited by L the perpendicular bisector of m and ´m).

The map rxs Þ Ñ gpx, mq ¨x will be called an congruent section in section 5.4.1. This allows use to write the variance at the point m P HP A :

F pmq " `E `}X ´m} 2 1 tXPHP A YLu ˘`E `}1 ¨X ´m} 2 1 tXPHP B u ˘Then
we define the random variable Z by: Z " X1 XPHP A YL `1 ¨X1 XPHP B , such that for m P HP A we have: F pmq " Ep}Z ´m} 2 q and F pmq " F p1 ¨mq. Thus if m ‹ is a global minimiser of F , then m ‹ " EpZq or m ‹ " 1 ¨EpZq. So the Fréchet mean of rXs is rEpZqs. Here instead of using theorem 3.1, we can work explicitly: Indeed there is no inconsistency if and only if EpZq " EpXq, (EpZq " 1 ¨EpXq would be another possibility, but by assumption EpZq, EpXq P HP A ), by writing X " X1 XPHP A `X1 XPHP B YL , we have:

EpZq " EpXq ðñ Ep1 ¨X1 XPHP B YL q " EpX1 XPHP B YL q ðñ 1 ¨EpX1 XPHP B YL q " EpX1 XPHP B YL q ðñ EpX1 XPHP B YL q P L ðñ PpX P HP B q " 0,
Therefore there is an inconsistency if and only if PpX P HP B q ą 0 (we remind that we made the assumption that EpXq P HP A Y L). If EpXq is regular (i.e.

EpXq R L), then there is an inconsistency if and only if X takes values in HP B , (this is exactly the condition of theorem 3.1, but in this particular case, this is a necessarily and sufficient condition). This proves point 1.

Now we make the assumption that X follows a Gaussian noise in order compute

EpZq (note that we could take another noise, as long as we are able to compute EpZq). For that we convert to polar coordinates: we write pu, vq T , a vector of R Z{2Z , under the form:

pu, vq T " EpXq `pr cos γ, r sin γq T , where r ą 0 and γ P r0, 2πs. We also define: d " distpEpXq, Lq, EpXq is a regular point if and only if d ą 0. We still suppose that EpXq " pα, βq T P HP A Y L. First we parametrise in function of pr, γq the points which are in HP B :

v ă u ðñ β `r sin γ ă α `r cos γ ðñ β ´α r ă ? 2 cospγ `π 4 q ðñ d r ă cospγ `π 4 q ðñ γ P " ´π 4 ´arccospd{rq, ´π 4
`arccospd{rq ı and d ă r 3.4. Inconsistency for finite and infinite group when the template is not a fixed point 61

Then we compute EpZq:

EpZq "EpX1 XPHP A q `Ep1 ¨X1 XPHP B q EpZq " ż d 0 ż 2π 0 ˆα `r cos γ β `r sin γ ˙exp ´´r 2 2w 2 2πw 2 rdγdr `ż `8 d ż 2π´π 4 ´arccosp d r q arccosp d r q´π 4 ˆα `r cos γ β `r sin γ ˙exp ´´r 2 2w 2 2πw 2 rdrdγ `ż `8 d ż ´π 4 `arccosp d r q ´π 4 ´arccosp d r q ˆβ `r sin γ α `r cos γ ˙exp ´´r 2 2w 2 2πw 2 rdrdγ "EpXq `ż `8 d r 2 expp´r 2 2w 2 q πw 2 ? 2κ ˆd r ˙dr ˆp´1, 1q T ,
We compute ρ " d Q prEpXqs, rEpZqsq where d Q is the distance in the quotient space defined in (3.1). As we know that EpXq, EpZq are in the same half-plane delimited by L, we have: ρ " d Q prEpZqs, rEpXqsq " }EpZq ´EpXq}. This proves equation (3.24), note that items 2a to 2c are the direct consequence of equation (3.24) and basic analysis.

3.4

Inconsistency for finite and infinite group when the template is not a fixed point

In the previous section, we prove the inconsistency when the group was finite. Being a finite group was a restriction to applications, therefore we now extend to non finite group. However, we still assume isometric action. In section 3.3 we exhibited sufficient condition to have an inconsistency, restricted to the case of finite group acting on an Euclidean space. We now generalize this analysis to Hilbert spaces of any dimension included infinite. Let M be such a Hilbert space with its dot product noted by x , y and its associated norm } }. In this section, we do not anymore suppose that the group G is finite. In the following, we prove that there is an inconsistency in a large number of situations, and we quantify the consistency bias with lower and upper bounds.

Example 3.3. The action of continuous horizontal translation: We take G " pR{Zq D acting on M " L 2 ppR{Zq D , Rq with:

@τ P G @f P M pτ ¨f q : t Þ Ñ f pt `τ q
This isometric action is the continuous version of the example 3.1: the elements of M are now continuous images in dimension D.

Presence of an inconsistency

We state here a generalization of theorem 3.1:

Theorem 3.4. Let G be a group acting isometrically on M a Hilbert space, and X a random variable in M , Ep}X} 2 q ă `8 and EpXq " t 0 ‰ 0. If:

P pd Q prt 0 s, rXsq ă }t 0 ´X}q ą 0, (3.26)
or equivalently:

P ˜sup gPG xg ¨X, t 0 y ą xX, t 0 y ¸ą 0. (3.27) Then rt 0 s is not a Fréchet mean of rXs in Q " M {G.
The condition of this Theorem is the same condition of theorem 3.1: the support of the law of X contains points closer from g ¨t0 for some g than t 0 . Thus the condition (3.27) is equivalent to Epd Q prXs, rt 0 sq 2 q ă Ep}X ´t0 } 2 q. In other words, the variance in the quotient space at t 0 is strictly smaller than the variance in the ambient space at t 0 .

Proof. First the two conditions are equivalent by definition of the quotient distance and by expansion of the square norm of }t 0 ´X} and of }t 0 ´gX} for g P G.

As above, we define the variance of rXs by:

F pmq " E ˆinf gPG }g ¨X ´m} 2 ˙.
In order to prove this Theorem, we find a point m such that F pmq ă F pt 0 q, which directly implies that rt 0 s is not be a Fréchet mean of rXs.

In the proof of theorem 3.1, we showed that under condition (3.6) we had x∇F pt 0 q, t 0 y ă 0. This leads us to study F restricted to R `t0 , we define for λ P R `:

f pλq " F pλt 0 q " E ˆinf gPG }g ¨X ´λt 0 } 2 ˙.
Thanks to the isometric action we can expand f pλq by:

f pλq " λ 2 }t 0 } 2 ´2λE ˜sup gPG xg ¨X, t 0 y ¸`Ep}X} 2 q, (3.28)
and explicit the unique element of R `which minimizes f :

λpt 0 q " E ˜sup gPG xg ¨X, t 0 y }t 0 } 2 . ( 3
q Ȩ ˜sup gPG xg ¨X, t 0 y 1 XRConept 0 q "E ´xX, t 0 y 1 XPConept 0 q Ē ˜sup gPG xg ¨X, t 0 y 1 XRConept 0 q ąEpxX, t 0 yq " xEpXq, t 0 y " }t 0 } 2 , (3.30)
which implies λpt 0 q ą 1. Then F pλpt 0 qt 0 q ă F pt 0 q. Note that }t 0 } 2 pλpt 0 q´1q " E `sup gPG xg ¨X, t 0 y ˘´EpxX, t 0 yq (which is positive) is exactly ´x∇F pt 0 q, t 0 y {2 in the case of finite group, see Equation (3.15). Here we find the same expression without having to differentiate the variance F , which may be not possible in the current setting.

Analysis of the condition in theorem 3.4

We now look for general cases when we are sure that Equation (3.27) holds which implies the presence of inconsistency. We saw in section 3.3 that when the group is finite, it is possible to have no inconsistency only if the support of the law is included in a cone delimited by some hyperplanes. The hyperplanes were defined as the set of points equally distant of the template t 0 and g ¨t0 for g P G. Therefore if the cardinal of the group becomes more and more important, one could think that in order to have no inconsistency the space where X should takes value becomes smaller and smaller. At the limit it leaves only at most an hyperplane. In the following, we formalise this idea to make it rigorous. We show that the cases where theorem 3.4 cannot be applied are not generic cases.

First we can notice that it is not possible to have the condition (3.27) if t 0 is a fixed point under the action of G. Indeed in this case xg ¨X, t 0 y " @ X, g ´1t 0 D " xX, t 0 y). So from now, we suppose that t 0 is not a fixed point. Now let us see some settings when we have the condition (3.26) and thus condition (3.27). First, let us recall the definition of the Voronoï cell of a point t 0 : Conept 0 q " tx P M, @g P G, }x ´t0 } ď }x ´g ¨t0 }u

The structure of this paragraph will be always the same, first we give, in some lemmas, a property of this cone, then we deduce a case where we ensure inconsistency.

Lemma 3.4. Let G be a group acting isometrically on a Hilbert space M . Let t 0 be a point in M . Assume that t 0 is a limit point of rt 0 s, this means that t 0 can be a limit of a sequence belonging to rt 0 sztt 0 u. In this case we have:

t 0 R IntpConept 0 qq,
where IntpAq is the interior of A.

Proof. Let η be a positive number, we have to prove that Bpt 0 , ηq, the open ball centred at t 0 with a radius η, is not included in the cone of t 0 . By density, one takes g ¨t0 P Bpt 0 , ηqztt 0 u for some g P G, now if we take r such that r ă minp}g ¨t0 ´t0 }{2, η ´}g ¨t0 ´t0 }q, then Bpg ¨t0 , rq Ă Bpt 0 , ηq. Besides, for every x P Bpg ¨t0 , rq we have }x ´g ¨t0 } ă }x ´t0 }. Then x R Conept 0 q. Proposition 3.4. Let G be a group acting isometrically on a Hilbert space M , and X a random variable in M , with Ep}X} 2 q ă `8 and EpXq " t 0 ‰ 0. If:

1. The template t 0 is a limit point in rt 0 s.

2. There exists η ą 0 such that the support of X contains a ball Bpt 0 , ηq.

Then condition (3.27) holds, and the estimator is inconsistent according to theorem 3.4.

rt 0 s t 0 Bpt 0 , ηq g ¨t0 O Figure 3.6:
The smallest disk is included in the support of X and the points in that disk is closer from g ¨t0 than from t 0 . According to theorem 3.4 there is an inconsistency.

Proof. Thanks to lemma 3.4, the template t 0 is not in the interior of Conept 0 q. Therefore Bpt 0 , ηq is not included in Conept 0 q. Then we verify condition (3.27), and we can apply theorem 3.4. ˝ 3.4. Inconsistency for finite and infinite group when the template is not a fixed point 65

Proposition 3.4 proves that there is a large number of cases where we can ensure the presence of an inconsistency. For instance when M is a finite dimensional vector space and the random variable X has a continuous positive density (for the Lebesgue's measure) at t 0 , condition 2 of proposition 3.4 is fulfilled. Unfortunately this proposition do not cover the case where there is no mass at the expected value t 0 " EpXq. This situation could appear if X has two modes for instance. The following proposition deals with this situation: Lemma 3.5. Let G be a group acting isometrically on M . If we assume that the orbit of a point t 0 contains a differential curve:

Dϕ s.t. ϕ : p´a, aq Ñ rt 0 s is C 1 with ϕp0q " t 0 , ϕ 1 p0q " v ‰ 0.
In this case we have (we note v K , the set of points orthogonal to v).

Conept 0 q Ă v K .
Proof. In order to prove this lemma, we take a point not in v K and we prove that this point is not in Conept 0 q: Let y R v K . We have also, thanks to the isometric action: xt 0 , vy " 0. We make a Taylor expansion of the following square distance (see also figure 3.7) at 0: }ϕpxq ´y} 2 " }t 0 `xv `opxq ´y} 2 " }t 0 ´y} 2 ´2x xy, vy `opxq.

Then: Dx ‹ P p´a, aq s.t. }x ‹ } ă a, x xy, vy ą 0 and }ϕpx ‹ q ´y} ă }t 0 ´y}. For some g P G, ϕpx ‹ q " g ¨t0 . By continuity of the norm we have: Dr ą 0 s.t. @z P Bpy, rq }g ¨t0 ´z} ă }t 0 ´z}. Proposition 3.5. Let G be a group acting isometrically on M . Let X be a random variable in M , such that Ep}X} 2 q ă `8 and EpXq " t 0 ‰ 0. If:

1. Dϕ s.t. ϕ : p´a, aq Ñ rt 0 s is C 1 with ϕp0q " t 0 , ϕ 1 p0q " v ‰ 0. 2. The support of X is not included in the hyperplane v K : PpX R v K q ą 0.
Then condition (3.27) is fulfilled, which leads to an inconsistency thanks to theorem 3.4.

Proof. Thanks to lemma 3.5, we have: PpX R Conept 0 qq ě PpX R v K q ą 0. Therefore theorem 3.4 applies. Proposition 3.5 was a sufficient condition on inconsistency in the case of an orbit which contains a curve. This brings us to extend this result for orbits which are manifolds:

Lemma 3.6. Let G be a group acting isometrically on a Hilbert space M . If the orbit of a point t 0 is a manifold, then we have: Conept 0 q Ă T t 0 rt 0 s K where T t 0 rt 0 s the linear tangent space of rt 0 s at t 0 .

Proof. Once again, let us prove that a point which is not in T t 0 rt 0 s K is not in Conept 0 q: Let y R T t 0 rt 0 s K . Let us take v P T t 0 rt 0 s such that xy, vy ‰ 0 and choose ϕ a C 1 curve in rt 0 s, such that ϕp0q " t 0 and ϕ 1 p0q " v. Applying lemma 3.5, we get that y R Conept 0 q.

˝.

Proposition 3.6. Let G be a group acting isometrically on a Hilbert space M , X a random variable in M , with Ep}X} 2 q ă `8. Assume X " t 0 `σε, where t 0 ‰ 0 and Epεq " 0, and Ep}ε}q " 1. We suppose that rt 0 s is a sub-manifold of M and write T t 0 rt 0 s the linear tangent space of rt 0 s at t 0 . If:

PpX R T t 0 rt 0 s K q ą 0, (3.31)
which is equivalent to:

Ppε R T t 0 rt 0 s K q ą 0, (3.32)
then there is an inconsistency.

t 0 rt 0 s T t 0 rt 0 s T t 0 rt 0 s K y g ¨t0 O Figure 3.7: y R T t 0 rt 0 s K
therefore y is closer from g ¨t0 for some g P G than t 0 itself. In conclusion, if y is in the support of X, there is an inconsistency.

Proof. First equations (3.31) and (3.32) are equivalent, because X " t 0 `ε and t 0 P T t 0 rt 0 s K . Secondly, thanks to lemma 3.6 and equation (3.31) we get that PpX R Conept 0 qq ą 0. Therefore, we have proved the inconsistency by applying theorem 3.4. Note that Condition (3.31) is very weak, because T t 0 rt 0 s is a proper linear subspace of M .

Lower bound of the consistency bias

Under the assumption of theorem 3.4, we have an element λpt 0 qt 0 such that F pλpt 0 qt 0 q ă F pt 0 q where F is the variance of rXs. From this element, we deduce lower bounds of the consistency bias: 3.4. Inconsistency for finite and infinite group when the template is not a fixed point 67 Theorem 3.5. Let δ be the unique positive solution of the following equation:

δ 2 `2δ p}t 0 } `E}X}q ´}t 0 } 2 pλpt 0 q ´1q 2 " 0. (3.33)
Let δ ‹ be the unique positive solution of the following equation:

δ 2 `2δ}t 0 } ´1 `a1 `σ2 {}t 0 } 2 ¯´}t 0 } 2 pλpt 0 q ´1q 2 " 0, (3.34)
where σ 2 " Ep}X ´t0 } 2 q is the variability of X. Then δ and δ ‹ are two lower bounds of the consistency bias.

Proof. In order to prove this Theorem, we exhibit a ball around t 0 such that the points on this ball have a variance bigger than the variance at the point λpt 0 qt 0 , where λpt 0 q was defined in Equation (3.29): thanks to the expansion of the function f we did in (3.28) we get :

F pt 0 q ´F pλpt 0 qt 0 q " }t 0 } 2 pλpt 0 q ´1q 2 ą 0, (3.35)

Moreover we can show (exactly like equation (3.14)) that for all x P M :

|F pt 0 q ´F pxq| ď E ˆˇˇˇi nf gPG }g ¨X ´t0 } 2 ´inf gPG }g ¨X ´x} 2 ˇˇˇď }x ´t0 } p2}t 0 } `}x ´t0 } `Ep}2X}qq . (3.36)
With Equations (3.35) and (3.36), for all x P Bpt 0 , δq we have F pxq ą F pλpt 0 qt 0 q. No point in that ball mapped in the quotient space is a Fréchet mean of rXs. So δ is a lower bound of the consistency bias. Now by using the fact that Ep}X}q ď a }t 0 } 2 `σ2 , we get:

2|F pt 0 q ´F pxq| ď 2}x ´t0 } ˆ}t 0 } ´1 `a1 `σ2 {}t 0 } 2 ¯`}x ´t0 } 2
This proves that δ ‹ is also a lower bound of the consistency bias. δ‹ is smaller than δ, but the variability of X intervenes in δ ‹ . Therefore we propose to study the asymptotic behaviour of δ ‹ when the variability tends to infinity. We have the following proposition: Proposition 3.7. Under the hypotheses of theorem 3.5, we write X " t 0 `σε, with Epεq " 0, and Ep}ε} 2 q " 1 and note θpt 0 q " Epsup gPG xg ¨ε, t 0 {}t 0 }yq P p0, 1s, we have that:

δ ‹ " σÑ`8 σp a 1 `θpt 0 q 2 ´1q,
In particular, the consistency bias explodes when the variability of X tends to infinity. First let us proove the following lema which states that θpt 0 q P p0, 1s.

Lemma 3.7. Thanks to condition (3.27), we have θpt 0 q P p0, 1s.

Chapter 3. Inconsistency in Hilbert space for isometric action Proof of lemma 3.7. We have θpt 0 q ě Epxε, t 0 {}t 0 }y " 0. By a reductio ad absurdum: if θpt 0 q " 0, then sup gPG xg ¨ε, t 0 y " xε, t 0 y almost surely. We have then almost surely:

xX, t 0 y ď sup gPG xgX, t 0 y ď }t 0 } 2 `sup gPG σ xg ¨ε, t 0 y " }t 0 } 2 `σ xε, t 0 y ď xX, t 0 y ,
which is in contradiction with (3.27). Besides θpt 0 q ď Ep}ε}q ď a E}ε} 2 " 1. Proof of proposition 3.7. We exhibit equivalent of the terms in equation (3.34) when σ Ñ `8:

2}t 0 } ´1 `a1 `σ2 {}t 0 } 2 ¯" 2σ. (3.37)
Now by definition of λpt 0 q in Equation (3.29) and the decomposition of X " t 0 `σε we get:

}t 0 }pλpt 0 q ´1q " 1 }t 0 } E ˜sup gPG pxg ¨t0 , t 0 y `xg ¨σε, t 0 yq ¸´}t 0 } }t 0 }pλpt 0 q ´1q ď 1 }t 0 } E ˜sup gPG xg ¨σε, t 0 y ¸" σθpt 0 q (3.38) }t 0 }pλpt 0 q ´1q ě 1 }t 0 } E ˜sup gPG xg ¨σε, t 0 y ¸´2}t 0 } " σθpt 0 q ´2}t 0 }, (3.39)
The lower bound and the upper bound of }t 0 }pλpt 0 q ´1q found in (3.38) and (3.39) are both equivalent to σθpt 0 q, when σ Ñ `8. Then the constant term of the quadratic Equation (3.34) has an equivalent:

´}t 0 } 2 pλpt 0 q ´1q 2 " ´σ2 θpt 0 q 2 . (3.40)
Finallye if we solve the quadratic Equation (3.34), we write δ ‹ as a function of the coefficients of the quadratic equation (3.34). We use the equivalent of each of these terms thanks to equation (3.37) and (3.40), this proves proposition 3.7. Remark 3.3. Thanks to inequality (3.39), if }t 0 } σ ă θpt 0 q 2 , then }t 0 } 2 p1 ´λpt 0 qq 2 ě pσθpt 0 q´2}t 0 }q 2 , then we write δ ‹ as a function of the coefficients of Equation (3.34), we obtain a lower bound of the inconsistency bias as a function of }t 0 }, σ and θpt 0 q for σ ą 2}t 0 }{θpt 0 q:

δ ‹ }t 0 } ě ´p1 `a1 `σ2 {}t 0 } 2 q `bp1 `a1 `σ2 {}t 0 } 2 q 2 `pσθpt 0 q{}t 0 } ´2q 2 .
Although the constant θpt 0 q intervenes in this lower bound, it is not an explicit term. We now explicit its behaviour depending on t 0 . We remind that:

θpt 0 q " 1 }t 0 } E ˜sup gPG xg ¨ε, t 0 y ¸.

Inconsistency for finite and infinite group when the template is not a fixed point 69

To this end, we first note that the set of fixed points under the action of G is a closed linear space, (because we can write it as an intersection of the kernel of the continuous and linear functions: x Þ Ñ g ¨x ´x for all g P G). We denote by p the orthogonal projection on the set of fixed points FixpM q. Then for x P M , we have: distpx, FixpM qq " }x ´ppxq}. Which yields:

xg ¨ε, t 0 y " xg ¨ε, t 0 ´ppt 0 qy `xε, ppt 0 qy .

(3.41)

The right hand side of Equation (3.41) does not depend on g as ppt 0 q P FixpM q.

Then:

}t 0 }θpt 0 q " E ˜sup gPG xgε, t 0 ´ppt 0 qy ¸`xEpεq, ppt 0 qy .
Applying the Cauchy-Schwarz inequality and using Epεq " 0, we can conclude that:

θpt 0 q ď 1 }t 0 } distpt 0 , FixpM qqEp}ε}q " distpt 0 {}t 0 }, FixpM qqEp}ε}q. (3.42)
This leads to the following comment: our lower bound of the consistency bias is smaller when our normalized template t 0 {}t 0 } is closer to the set of fixed points.

Upper bound of the consistency bias

In this section, we find an upper bound of the consistency bias. More precisely we have the following Theorem:

Proposition 3.8. Let X be a random variable in M , such that X " t 0 `σε where σ ą 0, Epεq " 0 and Ep||ε|| 2 q " 1. We suppose that rm ‹ s is a Fréchet mean of rXs.

Then we have the following upper bound of the quotient distance between the orbit of the template t 0 and the Fréchet mean of rXs:

d Q prm ‹ s, rt 0 sq ď σθpm ‹ q `aσ 2 θpm ‹ q 2 `2}t 0 }σθpm ‹ q, (3.43)
It is also possible to improve this inequality:

d Q prm ‹ s, rt 0 sq ď σθpm ‹ ´m0 q `aσ 2 θpm ‹ ´m0 q 2 `2distpt 0 , FixpM qqσθpm ‹ ´m0 q,
(3.44) where we have noted θpmq " Epsup gPG xg ¨ε, m{}m}yq P r0, 1s if m ‰ 0 and θp0q " 0, and m 0 the orthogonal projection of t 0 on FixpM q.

Note that we made no hypothesis on the template in this proposition. We deduce from Equation (3.44) that d Q prm ‹ s, rt 0 sq ď σ `aσ 2 `2σdistpt 0 , FixpM qq is a Opσq when σ Ñ 8, but a Op ? σq when σ Ñ 0, in particular the consistency bias can be neglected when σ is small.

In order to prove equation (3.44), we give a useful lemma:

Lemma 3.8. Let m 0 be a fixed point under the action of G, then the translation map

T : rxs Þ Ñ rx ´m0 s, (3.45)
is well defined on the quotient. Moreover, this map is a congruent map of Q:

@pa, bq P Q 2 d Q pT paq, T pbqq " d Q pa, bq.
Proof of lemma 3.8. First, we need to prove that this map is well defined: let us assume that x and y are in the same orbit, we need to prove that rx´m 0 s " ry´m 0 s. We know that y " g ¨x for some g P G, therefore: y ´m0 " g ¨x ´gm 0 " gpx ´m0 q, indeed m 0 is a fixed point, and G acts linearly on M . Therefore we have ry ´m0 s " rx ´m0 s.

Secondly, we need to prove that for every x and y P M , we have d Q prxs, rysq " d Q prx ´m0 s, ry ´m0 sq. This is the consequence of: @g P G px ´m0 q ´g ¨py ´m0 q " x ´m0 `m0 ´g ¨y " x ´g ¨y, once again, this equation is true because g acts linearly, and because m 0 is a fixed point. By taking the infimum over g P G, we prove that the map is congruent.

Proof of proposition 3.8. First we have:

F pm ‹ q ď F pt 0 q " Epinf gPG ||t 0 ´gpt 0 `σεq|| 2 q ď Ep||σε|| 2 q " σ 2 . ( 3
d Q prm ‹ s, rt 0 sq 2 ď 2Epsup gPG xσε, g ¨m‹ yq " 2σθpm ‹ q||m ‹ ||, (3.48)
note that at this point, if m ‹ " 0 then Epsup gPG xσε, g ¨m‹ yq " 0 and θpm ‹ q " 0 although equation (3.48) is still true even if m ‹ " 0. Moreover with the triangular inequality applied at rm ‹ s, r0s and rt 0 s, one gets: }m ‹ } ď }t 0 } `dQ prm ‹ s, rt 0 sq and then:

d Q prm ‹ s, rt 0 sq 2 ď 2σθpm ‹ qpd Q prm ‹ s, rt 0 sq `}t 0 }q. (3.49)
We can solve inequality (3.49) and we get:

d Q prm ‹ s, rt 0 sq ď σθpm ‹ q `aσ 2 θpm ‹ q 2 `2}t 0 }σθpm ‹ q, (3.50)
We note by F X instead of F the variance in the quotient space of rXs, and we want to apply inequality (3.43) to X ´m0 . As m 0 is a fixed point, we have thanks to lemma 3.8:

F X pmq " E `dQ prXs, rmsq 2 ˘" Ep `dQ prX ´m0 s, rm ´m0s 2 q ˘" F X´m 0 pm ´m0 q 3.4. Inconsistency for finite and infinite group when the template is not a fixed point 71

Then: m ‹ P argmin F X ðñ m ‹ ´m0 P argmin F X´m 0 .

We apply Equation (3.43) to X ´m0 , with EpX ´m0 q " t 0 ´m0 and rm ‹ ´m0 s a Fréchet mean of rX ´m0 s. We get: d Q prm ‹ ´m0 s, rt 0 ´m0 sq ď σθpm ‹ ´m0 q`aσ 2 θpm ‹ ´m0 q 2 `2}t 0 ´m0 }σθpm ‹ ´m0 q.

Moreover d Q prm ‹ s, rt 0 sq " d Q prm ‹ ´m0 s, rt 0 ´m0 sq (see lemma 3.8), which concludes the proof.

3.4.5

Empirical Fréchet mean

In practice, we never compute the Fréchet mean in quotient space, only the empirical Fréchet mean in quotient space when the size of a sample is supposed to be large enough. If the empirical Fréchet in the quotient space means converges to the Fréchet mean in the quotient space then we can not use these empirical Fréchet mean in order to estimate the template. In [Bhattacharya 2008], it has been proved that the empirical Fréchet mean converges to the Fréchet mean with a 1 ? n convergence speed, however the law of the random variable is supposed to be included in a ball whose radius depends on the geometry on the manifold. Here we are not in a manifold, indeed the quotient space contains singularities, moreover we do not suppose that the law is necessarily bounded. However in [Ziezold 1977] the empirical Fréchet means is proved to converge to the Fréchet means but no convergence rate is provided.

We propose now to prove that the quotient distance between the template and the empirical Fréchet mean in quotient space have an lower bound which is the asymptotic of the one lower bound of the consistency bias found in (3.33). Take X, X 1 , . . . , X n independent and identically distributed (with t 0 " EpXq not a fixed point). We define the empirical variance of rXs by:

m P M Þ Ñ F n pmq " 1 n n ÿ i"1 d Q prms, rX i sq 2 " 1 n n ÿ i"1 inf gPG }m ´g ¨Xi } 2 ,
and we say that rm n‹ s is a empirical Fréchet mean of rXs if m n‹ is a global minimiser of F n .

Proposition 3.9. Let X, X 1 , . . . , X n independent and identically distributed random variables, with t 0 " EpXq. Let be rm n‹ s be an empirical Fréchet mean of rXs.

Then δ n is a lower bound of the quotient distance between the orbit of the template and rm n‹ s, where δ n is the unique positive solution of:

δ 2 `2 ˜||t 0 || `1 n n ÿ i"1 }X i } ¸δ ´}t 0 } 2 pλ n pt 0 q ´1q 2 " 0.
λ n pt 0 q is defined like λpt 0 q in section 3.4.1 by:

λ n pt 0 q " 1 n n ř i"1 sup gPG xg ¨Xi , t 0 y }t 0 } 2 .

Examples

In this Subsection, we discuss, in some examples, the application of theorem 3.4 and see the behaviour of the constant θpt 0 q. This constant intervened in lower bound of the consistency bias.

Action of horizontal translation on L 2 pR{Zq

We take an orbit O " rf 0 s, where f 0 P C 2 pR{Zq, non constant. We show easily that O is a manifold of dimension 1 and the tangent space at f 0 is 2 Rf 1 0 . Therefore a sufficient condition on X such that EpXq " f 0 to have an inconsistency is: PpX R f 1K 0 q ą 0 according to proposition 3.6. Now if we denote by 1 the constant function on R{Z equal to 1. We have in this setting: that the set of fixed points under the action of G is the set of constant functions: FixpM q " R1 and: distpf 0 , FixpM qq " }f 0 ´xf 0 , 1y 1} "

d ż 1 0 ˆf0 ptq ´ż 1 0 f 0 psqds ˙2 dt.
This distance to the fixed points is used in the upper bound of the constant θpt 0 q in Equation (3.42). Note that if f 0 is not differentiable, then rf 0 s is not necessarily a manifold, and (3.6) does not apply. However proposition 3.4 does: if f 0 is not a constant function, then rf 0 sztf 0 u is dense in rf 0 s. Therefore as soon as the support of X contains a ball around f 0 , there is an inconsistency.

Action of discrete horizontal translation on R Z{NZ

We come back on example 3.1, with D " 1 (discretised signals). For some signal t 0 , θpt 0 q previously defined is:

θpt 0 q " 1 }t 0 } E ˆmax τ PZ{NZ xε, τ ¨t0 y ˙.
Therefore if we have a sample of size I of ε iid, then:

θpt 0 q " 1 }t 0 } lim IÑ`8 1 I I ÿ i"1 max τ i PZ{N Z xε i , τ i ¨t0 y , 2 Indeed ϕ : s ´1 2 , 1 2 r Ñ O t Þ Ñ f0p.
´tq is a local parametrisation of O: f0 " ϕp0q, and we check that: lim xÑ0 }ϕpxq´ϕp0q´xf 1 0 } L 2 " 0 with Taylor-Lagrange inequality at the order 2. As a conclusion ϕ is differentiable at 0, and it is an immersion (since f 1 0 ‰ 0), and D0ϕ : x Þ Ñ xf 1 0 , then O is a manifold of dimension 1 and the tangent space of O at f0 is:

T f 0 O " D0ϕpRq " Rf 1 0 .
3.4. Inconsistency for finite and infinite group when the template is not a fixed point 73

By an exhaustive research, we can find the τ i 's which maximise the dot product, then with this sample and t 0 we can approximate θpt 0 q. We have done this approximation for several signals t 0 on figure 3.8. According the previous results, the bigger θpt 0 q is, the more important the lower bound of the consistency bias is. We remark that the θpt 0 q estimated is small, θpt 0 q ! 1 for different signals. . ε is here a Gaussian noise in R Z{100Z , such that Epεq " 0 and Ep}ε} 2 q " 1.

For instance the blue signal is a signal defined randomly, and when we approximate the θpt 0 q which corresponds to that t 0 we find » 0.25.

Action of rotations on R n

Now we consider the action of rotations on R n with a Gaussian noise. Take X " N pt 0 , s 2 Id n q then the variability of X is ns 2 , then X has a decomposition: X " t 0 `?nsε with Epεq " 0 and Ep}ε} 2 q " 1. According to proposition 3.7 we have by noting δ ‹ the lower bound of the consistency bias when s Ñ 8:

δ ‹ s Ñ ?
np´1 `a1 `θpt 0 q 2 q.

Now θpt 0 q " Epsup gPG xg ¨ε, t 0 qy {}t 0 } " Ep}ε}q Ñ 1 when n tends to infinity (expected value of the Chi distribution) we have that for n large enough:

lim sÑ8 δ ‹ s » ? np ? 2 ´1q.
We compare this result with the exact computation of the consistency bias (noted here CB) made by Miolane et al. [START_REF] Miolane | [END_REF]], which writes with our current notations:

lim sÑ8 CB s " ? 2 Γppn `1q{2q Γpn{2q .
Using a standard Taylor expansion on the Gamma function, we have that for n large enough:

lim sÑ8 CB s » ? n.
As a conclusion, when the dimension of the space is large enough our lower bound and the exact computation of the bias have the same asymptotic behaviour. It differs only by the constant ? 2 ´1 » 0.4 in our lower bound, 1 in the work of Miolane et al. [Miolane 2015].

Differentiability of the variance in the quotient space

In section 3.3.1, we study the consistency of the template estimation with the Fréchet mean in quotient space, by assuming that the group is finite. In order to prove the inconsistency we establish that the variance is differentiable at some points. Then, one natural question we ask is the following: is the variance always differentiable?

In this section, we show that the variance in the quotient space is not differentiable at 0 when the ambient space is an Hilbert spaces and for isometric action: F pmq " Epinf gPG }Y ´g ¨m} 2 q is not differentiable at 0. This question matters, since one may want to compute the Fréchet mean in quotient space with a gradient descent, but this requires, at least, that the variance is differentiable. For instance what if the Fréchet mean of rY s is rm ‹ s and that the variance F is not differentiable at m ‹ ?

First, we would to insist on this point: it is a well known fact that, for all y P M , m Þ Ñ }m ´y} 2 is differentiable with a gradient equal to m ´y. Then m Þ Ñ d Q prms, rysq " inf gPG }m ´g ¨y} 2 is defined as the infimum of differentiable functions. There is no guarantee that the resulting function is differentiable. x Þ Ñ }x} " ´minpx, ´xq is a toy example of the infimum of differentiable functions which is not differentiable.

One could think that we could deal with sub-differentiability, however the variance is probably not convex nor concave.

If the group is compact then, we have an element which reaches this infimum, d Q prms, rysq " }m ´g‹ y} 2 for some g ‹ P G. In this case, one could think that m Þ Ñ d Q prms, rysq is differentiable with a gradient equal to 2pm ´g‹ yq. However this is more complicated than that. Indeed g ‹ depends on m and y. We should call this element gpm, yq rather than g ‹ . One gets: d Q prms, rysq 2 " }m ´gpm, yq ¨y} 2 . Now, gpm, yq¨y depends on m, and it is more difficult to show the differentiability or to compute the gradient, since we have the existence of the element gpm, yq but we have not an explicit formula of this element. If one is not able to prove the differentiability of the squared distance, then it will be harder to prove the differentiablity of the variance (defined as the expectation of the squared distance). In the proof of theorem 3.1, we proved that for finite group gpm, yq was locally 3.4. Inconsistency for finite and infinite group when the template is not a fixed point 75 constant, that is why we could be able to differentiate the squared quotient distance and the variance.

Example 3.4. Example of the action of rotation Taking M an Euclidean space, and G the group of rotations of M . Then the quotient distance is given by: d Q pras, rbsq " |}a} ´}b}|. Therefore, the variance in the quotient space is equal to:

F pmq " Epp}Y } ´}m}q 2 q,
by expansion one gets:

F pmq " Ep}Y } 2 q ´2}m}Ep}Y }q `}m} 2 , now m Þ Ñ }m} 2 is differentiable everywhere, however m Þ Ñ }m} is not differentiable at 0. We can conclude that F is not differentiable at 0.

This example can be generalized to any isometric group action:

Proposition 3.10. Let G acting isometrically on M a Hilbert space. We do not assume that G is a finite group. Let us take Y a random variable in M such that Ep}Y } 2 q ă `8. Let us assume that it exists a point p P M such that Ppd Q prps, rY sq ă }p ´Y }q ą 0, then the variance m Þ Ñ F pmq " Epd 2 Q prms, rY sqq is not differentiable at 0. Note that it is easy to fulfill proposition 3.10, for instance with p " t 0 , as we have seen it in theorem 3.1.

Remark 3.4. We can prove at the same time that the squared distance is also not differentiable: m Þ Ñ d 2 Q prms, rysq is not differentiable at 0, if y is not a fixed point. Indeed, let us take Y a random variable equal to y, then F pmq " Epd 2 prms, rY sqq " d 2 prms, rysq is not differentiable at 0 according to proposition 3.10.

Proof of proposition 3.10. We first expand the variance:

F pmq " }m} 2 `E ˜}Y } 2 q ´2Epsup gPG xm, g ¨Y y ¸, then if this function is differentiable so is the function f : m Þ Ñ Epsup gPG xm, g ¨Y yq,
now by linearity of the action we remark that we have f pλmq " λf pmq for all m P M and λ ě 0, then if f was differentiable, f would be a linear function. Let us assume that f is linear. In this case we have for m, m 1 P M :

f pm `m1 q " Ep A hkkkkkkkkkkkkikkkkkkkkkkkkj sup gPG @ m `m1 , g ¨Y D q ď Epsup gPG xm, g ¨Y y `sup gPG @ m 1 , g ¨Y D loooooooooooooooooooomoooooooooooooooooooon B q ď f pmq `f pm 1 q ď f pm `m1 q,
where the inequality comes from the fact that the supremum of the sum of two terms is smaller than the sum of the two supremum. Then we have two random variables A, B such that A ď B and EpAq " EpBq, when we can conclude that A " B almost surely. It comes that: Then for any m P M we have Y -almost surely:

sup gPG @ m `m1 , g ¨Y D q " sup gPG xm, g ¨Y yq `sup gPG @ m 1 , g ¨Y D q Y ´a
@g P G xm, Y y " xg ¨m, Y y .
Then we have Y -almost surely }m ´Y } " d Q prms, rY sq. Taking m " p shows the contradiction with our hypothesis. In order to prove the remark remark 3.4, we can take Y a random variable which is constantly equal to y a non fixed point, as y is not a fixed point it exists g P G such that g ¨y ‰ y. We can take m 0 " g ¨y, this m 0 fulfills proposition 3.10. Remark 3.5. 0 is one fixed point among all the other, thanks to lemma 3.8, we can also conclude that the variance is not differentiable at any other fixed points.

Proof. Let m 0 be a fixed point. We can consider two random variables X and X ´m0 , if we note F X the variance of X, thanks to lemma 3.8, we have:

F X pmq " Epd 2
Q prms, rXsqq " Epd 2 Q prm ´m0 s, rX ´m0 sqq " F X´m 0 pm ´m0 q Now we know that F X´m 0 is not differentiable at 0 (proposition 3.10), therefore m Þ Ñ F X´m 0 pm ´m0 q is not differentiable at m 0 . We conclude that F X is not differentiable at m 0 . Őne remaining question is: can we study the differentiability of the square distance for non fixed points?

3. 5 Fréchet means ambient and quotient spaces are not consistent when the template is a fixed point

In this section, we do not assume that the ambient space M is a vector space, but rather a manifold. We need then to rewrite the generative model likewise: let t 0 P M , and X any random variable of M such as t 0 is a Fréchet mean of X. Then Y " Φ ¨X is the observed variable where Φ is a random variable whose value are in G. In this section we make the assumption that the template t 0 is a fixed point under the action of G.

3.5. Fréchet means ambient and quotient spaces are not consistent when the template is a fixed point 77

Result

Let X be a random variable on M and define the variance of X as:

Epmq " Epd M pm, Xq 2 q.

We say that t 0 is a Fréchet mean of X if t 0 is a global minimiser of the variance E.

We prove the following result:

Theorem 3.6. Assume that M is a complete finite dimensional Riemannian manifold and that d M is the geodesic distance on M . We suppose that d M is invariant under the group action. Let X be a random variable on M , with Epdpx, Xq 2 q ă `8 for some x P M . We assume that t 0 is a fixed point and a Fréchet mean of X and that PpX P Cpt 0 qq " 0 where Cpt 0 q is the cut locus of t 0 . Suppose that there exists a point in the support of X which is not a fixed point nor in the cut locus of t 0 . Then rt 0 s is not a Fréchet mean of rXs.

The previous result is finite dimensional and does not cover interesting infinite dimensional setting concerning curves for instance. However, a simple extension to the previous result can be stated when M is a Hilbert vector space since then the space is flat and some technical problems like the presence of cut locus point do not occur.

Theorem 3.7. Assume that M is a Hilbert space and that d M is given by the Hilbert norm on M . We suppose that d M is invariant under the group action. Let X be a random variable on M , with Ep}X} 2 q ă `8. We assume that t 0 " EpXq. Suppose that there exists a point in the support of the law of X that is not a fixed point for the action of G. Then rt 0 s is not a Fréchet mean of rXs.

Note that the reciprocal is true: if all the points in the support of the law of X are fixed points, then almost surely, for all m P M and for all g P G we have:

d M pX, mq " d M pg ¨X, mq " d Q prXs, rmsq.
Up to the projection on the quotient, we have that the variance of X is equal to the variance of rXs in M {G, therefore rt 0 s is a Fréchet mean of rXs if and only if t 0 is a Fréchet mean of X. There is no inconsistency in that case.

Example 3.5. Theorem 3.7 covers the interesting case of the Fisher Rao metric on functions:

F " tf : r0, 1s Ñ R | f is absolutely continuousu.

Then considering for G the group of smooth diffeomorphisms ϕ on r0, 1s such that ϕp0q " 0 and ϕp1q " 1, we have a right group action GˆF Ñ F given by ϕ¨f " f ˝ϕ.

The Fisher Rao metric is built as a pull back metric of the L 2 pr0, 1s, Rq space through the map Q : F Ñ L 2 given by: Qpf

q " 9 f { b | 9 f |.
This square root trick is often used, see for instance [Kurtek 2011b]. Note that in this case, Q is a bijective mapping with inverse given by q Þ Ñ f with f ptq " ş t 0 qpsq|qpsq|ds. We can define a group action on M " L 2 as: ϕ ¨q " q ˝ϕ? 9 ϕ, for which one can check easily by a change of variable that:

}ϕ ¨q ´ϕ ¨q1 } 2 " }q ˝ϕa 9 ϕ ´q1 ˝ϕa 9 ϕ} 2 " }q ´q1 } 2 .

So up to the mapping Q, the Fisher Rao metric on curve corresponds to the situation M where theorem 3.7 applies. Note that in this case the set of fixed points under the action of G corresponds in the space F to constant functions.

We can also provide an computation of the consistency bias in this setting:

Proposition 3.11. Under the assumptions of theorem 3.7, we write X " t 0 `σε where t 0 is a fixed point, σ ą 0, Epεq " 0 and Ep}ε} 2 q " 1. Furthermore, we assume that the group G acts isometrically on M . If there is a Fréchet mean of rXs, then the consistency bias is linear with respect to σ and it is equal to:

σ sup }v}"1 Epsup gPG xv, g ¨εyq.
Proof. For λ ą 0 and }v} " 1, we compute the variance F in the quotient space of rXs at the point t 0 `λv. Since t 0 is a fixed point we get:

F pt 0 `λvq " Epinf gPG }t 0 `λv´gX} 2 q " Ep}X} 2 q´}t 0 } 2 ´2λEpsup gPG xv, gpX ´t0 qyq`λ 2 .
Then we minimise F with respect to λ, and after we minimise with respect to v (with }v} " 1). Which concludes.

3.5.2

Proofs of these theorems 3.5.2.1 Proof of theorem 3.6

We start with the following simple result, which aims to differentiate the variance of X. This classical result (see [Pennec 2006] for instance) is proved again here to be the more self-contained as possible:

Lemma 3.9. Let X a random variable on M such that Epdpx, Xq 2 q ă `8 for some x P M . Then the variance m Þ Ñ Epmq " Epd M pm, Xq 2 q is a continuous function which is differentiable at any point m P M such that PpX P Cpmqq " 0 where Cpmq is the cut locus of m. Moreover at such point one has:

∇Epmq " ´2EpLog m pXqq,

where Log m : M zCpmq Ñ T m M is defined for any x P M zCpmq as the unique u P T m M such that Exp m puq " x and }u} m " d M px, mq.

Proof of lemma 3.9. By triangle inequality it is easy to show that E is finite and continuous everywhere. Moreover, it is a well known fact that We are now ready to prove theorem 3.6.

x Þ Ñ d M px,
Proof. (of theorem 3.6) Let m 0 be a point in the support of M which is not a fixed point and not in the cut locus of t 0 . Then there exists g 0 P G such that m 1 " g 0 m 0 ‰ m 0 . Note that as the distance is equivariant under the action of G, we have that m 1 " g 0 ¨m0 R Cpg 0 ¨t0 q " Cpt 0 q (t 0 is a fixed point under the action of G). Let v 0 " Log t 0 pm 0 q and v 1 " Log t 0 pm 1 q. We have v 0 ‰ v 1 and since Cpt 0 q is closed and the Log t 0 is a continuous application on M zCpt 0 q we have:

lim ηÑ0 1
PpX P Bpm 0 , ηqq Ep1 XPBpm 0 ,ηq Log t 0 pXqq " v 0 .

(we use here the fact that since m 0 is in the support of the law of X, PpX P Bpm 0 , ηqq ą 0 for any η ą 0 so that the denominator does not vanish and the fact that since M is a complete manifold, it is a locally compact space (the closed balls are compacts) and Log t 0 is locally bounded). Similarly:

lim ηÑ0 1
PpX P Bpm 0 , ηqq Ep1 XPBpm 0 ,ηq Log t 0 pg 0 ¨Xqq " v 1 .

Thus for sufficiently small η ą 0 we have (since v 0 ‰ v 1 ):

EpLog t 0 pXq1 XPBpm 0 ,ηq q ‰ EpLog t 0 pg 0 ¨Xq1 XPBpm 0 ,ηq q.

(3.51)

By using using a reductio ad absurdum, we suppose that rt 0 s is a Fréchet mean of rXs and we want to find a contradiction with (3.51). In order to do that we introduce simple functions as the function x Þ Ñ 1 xPBpm 0 ,ηq which intervenes in Equation (3.51).

Let s : M Ñ G be a simple function (i.e. a measurable function with finite number of values in G). Then x Þ Ñ hpxq " spxq ¨x is a measurable function 3 . Now, let

3 Indeed if: s " n ř i"1
gi1A i where pAiq1ďiďn is a partition of M (such that the sum is always defined). Then for any Borel set B Ă M we have:

h ´1pBq " n Ť i"1 g ´1 i pBq X Ai is a measurable set since x Þ Ñ gix is a measurable function.
E s pxq " Epdpx, spXq ¨Xq 2 q be the variance of the variable spXq ¨X. Note that (and this is the main point):

@g P G d M pt 0 , xq " d M pg ¨t0 , g ¨xq " d M pt 0 , g ¨xq " d Q prt 0 s, rxsq,
we have: E s pt 0 q " Ept 0 q. Assume now that rt 0 s a Fréchet mean for rXs on the quotient space and let us show that E s has a global minimum at t 0 . Indeed for any m, we have:

E s pmq " Epd M pm, spXq ¨Xq 2 q ě Epd Q prms, rXsq 2 q ě Epd Q prt 0 s, rXsq 2 q " E s pt 0 q.
Now, we want to apply lemma 3.9 to the random variables spXq ¨X and X at the point t 0 . Since we assume that X R Cpt 0 q almost surely and X R Cpt 0 q implies spXq ¨X R Cpt 0 q we get PpspXq ¨X P Cpt 0 qq " 0 and the lemma 3.9 applies. As t 0 is a minimum, we already know that the differential of E s (respectively E) at t 0 should be zero. We get:

EpLog t 0 pXqq " EpLog t 0 pspXq ¨Xqq " 0.

(3.52)

Now we apply Equation (3.52) to a particular simple function defined by spxq " g 0 1 xPBpm 0 ,ηq `eG 1 xRBpm 0 ,ηq . We split the two expected values in (3.52) into two parts:

EpLog t 0 pXq1 XPBpm 0 ,ηq q `EpLog t 0 pXq1 XRBpm 0 ,ηq q " 0, (3.53)

EpLog t 0 pg 0 ¨Xq1 XPBpm 0 ,ηq q `EpLog t 0 pXq1 XRBpm 0 ,ηq q " 0.

(3.54)

By substrating (3.53) from (3.54), one gets:

EpLog t 0 pXq1 XPBpm 0 ,ηq q " EpLog t 0 pg 0 ¨Xq1 XPBpm 0 ,ηq q, which is a contradiction with (3.51). Which concludes.

3.5.2.2

Proof of theorem 3.7

Proof. The extension to theorem 3.7 is quite straightforward. In this setting many things are now explicit since dpx, yq " }x´y} , ∇ x dpx, yq 2 " 2px´yq, Log x pyq " y´x and the cut locus is always empty. It is then sufficient to go along the previous proof and to change the quantity accordingly. Note that the local compactness of the space is not true in infinite dimension. However this was only used to prove that the log was locally bounded but this last result is trivial in this setting.

3.6

Conclusion and discussion

In this chapter, we exhibit conditions which imply that the template estimation with the Fréchet mean in quotient space is inconsistent. These conditions are rather generic. As a result, without any more information, a priori there is inconsistency. The behaviour of the consistency bias is summarized in table 3.1. Surely future works could improve these lower and upper bounds.

In a more general case: when we take an infinite-dimensional vector space quotiented by a non isometric group action, is there always an inconsistency? An important example of such action is the action of diffeomorphisms. Can we estimate the consistency bias? In this setting, one estimates the template (or an atlas), but does not exactly compute the Fréchet mean in quotient space, because a regularization term is added. In this setting, can we ensure that the consistency bias will be small enough to estimate the original template? Otherwise, one has to reconsider the template estimation with stochastic algorithms as in [START_REF] Stéphanie Allassonnière | Construction of Bayesian deformable models via a stochastic approximation algorithm: a convergence study[END_REF] or develop new methods [START_REF] Kühnel | [END_REF].

A short version of this chapter has been published in a conference [Devilliers 2017a], then has been extend in the Entropy journal [Devilliers 2017b]. Compared to this journal paper, section 4.2.5 has been added.

Abstract:

We tackle the problem of template estimation when data have been randomly deformed under a group action in the presence of noise. In order to estimate the template, one often minimizes the variance when the influence of the transformations have been removed (computation of the Fréchet mean in the quotient space). The consistency bias is defined as the distance (possibly zero) between the orbit of the template and the orbit of one element which minimizes the variance. In a first part, we restrict ourselves to isometric group action, in this case the Hilbert distance is invariant under the group action. We establish an asymptotic behavior of the consistency bias which is linear with respect to the noise level. As a result the inconsistency is unavoidable as soon as the noise is large enough. In practice, the template estimation with a finite sample is often done with an algorithm called "max-max". In a second part, also in the case of isometric group action, we show the convergence of this algorithm to an empirical Karcher mean. Our numerical experiments show that the bias observed in practice can not be attributed to the small sample size or to a convergence problem but is indeed due to the previously studied inconsistency. In a third part, we also present some insights of the case of a non invariant distance with respect to the group action. We will see that the inconsistency still holds as soon as the noise level is large enough. Moreover we prove the inconsistency even when a regularization term is added.

Introduction

General Introduction

The template estimation is a well known issue in different fields such as statistics on signals [Kurtek 2011b], shape theory, computational anatomy [Guimond 2000, Joshi 2004, Cootes 2004] etc. In these fields, the template (which can be viewed as the prototype of our data) can be (according to different vocabulary) shifted, transformed, wrapped or deformed due to different groups acting on data. Moreover, due to a limited precision in the measurement, the presence of noise is almost always unavoidable. These mixed effects on data lead us to study the consistency of algorithms which claim to compute the template. A popular algorithm consists in the minimization of the variance, in other words, the computation of the Fréchet mean in quotient space. This method has been already proved to be inconsistent [Bigot 2011[START_REF] Miolane | [END_REF], Devilliers 2017c]. In [Bigot 2011] the authors proves the inconsistency with a lower bound of the expectation of the error between the original template and the estimated template with a finite sample, they deduce that this expectation does not go to zero as the size of the sample goes to infinity. This work was done in a functional space, where functions only observed at a finite number of points of the functions were observed. In this case one can model these observable values on a grid. When the resolution of the grid goes to zero, one can show the consistency [Panaretos 2016] by using the Fréchet mean with the Wasserstein distance on the space of measures rather than in the space of functions. But in (medical) images the number of pixels or voxels is finite.

In [START_REF] Miolane | [END_REF]], the authors demonstrated the inconsistency in a finite dimensional manifold with Gaussian noise, when the noise level tends to zero.

In chapter 3, we focused our study on the inconsistency with Hilbert Space (including infinite dimensional case) as ambient space, for isometric action. Although we gave some bounds of the consistency bias when the noise level tends to infinity, we did not gave an asymptotic behaviour of the consistency bias.

Settings and Notation

In this paper, we suppose that observations belong to a Hilbert space pM, x¨, ¨yq, we denote by } ¨} the norm associated to the dot product x¨, ¨y. We also consider a group of transformation G which acts on M the space of observations. This means that1 g 1 ¨pg ¨xq " pg 1 gq ¨x and e G ¨x " x for all x P M , g, g 1 P G, where e G is the identity element of G.

The generative model is the following: we transform an unknown template t 0 P M with Φ a random and unknown element of the group G and we add some noise. Let σ be a positive noise level and ε a standardized noise: Epεq " 0, Ep}ε} 2 q " 1. Moreover we suppose that ε and Φ are independent random variables. Finally, the only observable random variable is:

Y " Φ ¨t0 `σε. (4.1)
This generative model is commonly used in Computational anatomy in diverse frameworks, for instance with currents [Durrleman 2014], but also in functional data analysis [Kurtek 2011b].

For instance: if we assume that the noise is independent and identically distributed on each pixel or voxel with a standard deviation w on each pixel/voxel, then σ " ? N w, where N is the number of pixels/voxels. But the noise which we consider can be more general: we do not require the fact that the noise is independent over each region of the space M .

Note that the inconsistency of template estimation can be also studied with an alternative generative model, called backward model where Y " Φ ¨pt 0 `σεq chapter 3. Some authors also use the term perturbation model see [Huckemann 2011, Rohlf 2003, Goodall 1991].

Quotient space: the random transformation of the template by the group leads us to project the observation Y into the quotient space. The quotient space is defined as the set containing all the orbit rxs " tg ¨x, g P Gu for x P M . The set which is constituted of all orbits is called the quotient space M by the group G and is noted by: Q " M {G " trxs, x P M u.

As we want to do statistics on this space, we aim to equip the quotient with a metric. One often requires that d M the distance in the ambient space is invariant under the group action G (see figure 4.1), this means that @m, n P M, @g P G d M pg ¨m, g ¨nq " d M pm, nq.

If d M is invariant and if the orbits are closed sets2 , then

d Q prxs, rysq " inf gPG d M px, g ¨yq,
is well defined, and d Q is a distance in the quotient space. The quotient distance d Q prxs, rysq is the distance between x and y 1 where y 1 is the registration of y with respect to x. We say in this case that y 1 is in optimal position with respect to x.

' 0 ' ' p " p0, 1q q " p´2, 0q d Q prps, rqsq " 1 Figure 4
.1: Due to the invariant action, the orbits are parallel. Here the orbits are circles centred at 0. This is the case when the group G is the group of rotations.

One particular distance in the ambient space M , which we use in all this chapter, is the distance given by the norm of the Hilbert space: d M pa, bq " }a ´b}. Moreover we say that G acts isometrically on M , if x Þ Ñ g ¨x is a linear map which leaves the norm unchanged. In this case d M the distance given by the norm of the Hilbert space is invariant under the group action. The quotient (pseudo)-distance is, in this case, d Q pras, rbsq " inf gPG }a ´g ¨b}. Thus, even if the quotient space is not a linear space, we have a "polarization identity" in the quotient space:

sup gPG xa, g ¨by " 1 2 `}a} 2 `}b} 2 ´d2 Q pras, rbs " 1 2 `d2 Q pras, r0sq `d2 Q prbs, r0sq ´d2 Q pras, rbsq ˘(4.2)
When the distance given by the norm is invariant under the group action, we define the variance of the random orbit rY s as the expectation of the (pseudo)distance between the random orbit rY s and the orbit of a point x in M :

F pxq " Epd 2 Q prxs, rY sqq " Epinf gPG }g ¨x ´Y } 2 q " Epinf gPG }x ´g ¨Y } 2 q.
Note that F pxq is well defined for all x P M because Ep}Y } 2 q is finite. Moreover, since F pg ¨xq " F pxq, for all x P M and g P G, the variance F is well defined in the quotient space: rxs Þ Ñ F pxq does have a sense. Besides, in presence of a sample of the observable variable Y noted Y 1 , . . . , Y n , one can define the empirical variance of a point x in M :

F n pxq " 1 n n ÿ k"1 ˆinf gPG }g ¨x ´Yi } 2 ˙" 1 n n ÿ k"1 ˆinf gPG }x ´g ¨Yi } 2 ˙.
Definition 4.1. The template estimation is performed by minimizing F n :

t0n " argmin xPM F n pxq.

In order to study this estimation method, one can look the limit of this estimator when the number of data n tends to `8, in this case, the estimation becomes:

t08 " argmin xPM F pxq.

If m ‹ P M minimizes F , then rm ‹ s is called a Fréchet mean of rY s.

Definition 4.2. We say that the estimation is consistent if t 0 minimizes F . Moreover the consistency bias, noted CB, is the (pseudo)-distance between the orbit of the template rt 0 s and rm ‹ s:

CB " d Q prt 0 s, rm ‹ sq.
If such a m ‹ does not exists, then the consistency bias is infinite.

Note that, if the action is not isometric and is not either invariant, a priori d Q is no longer a (pseudo)-distance in the quotient space (this point is discussed in section 4.3). However one can still define F and wonder if the minimization of F is a consistent estimator of t 0 . In this case we call F a pre-variance.

Questions and Contributions

This setting leads us to wonder about few things listed below: Questions:

• Is t 0 a minimum of the variance or the pre-variance?

• What is the behavior of the consistency bias with respect to the noise level?

• How to perform such a minimization of the variance? Indeed, in practice we have only a sample and not the whole distribution.

Contribution: In the case of an isometric action, we provide a Taylor expansion of the consistency bias when the noise level σ tends to infinity. As we do not have the whole distribution, we minimize the empirical variance given a sample. An element which minimizes this empirical variance is called an empirical Fréchet mean. We already know that the empirical Fréchet mean converges to the Fréchet mean when the sample size tends to infinity [Ziezold 1977]. Therefore our problem is reduced to finding an empirical Fréchet mean with a finite but sufficiently large sample. One algorithm called the "max-max" algorithm [START_REF] Stéphanie Allassonnière | Towards a coherent statistical framework for dense deformable template estimation[END_REF]] aims to compute such an empirical Fréchet mean. We establish some properties of the convergence of this algorithm. In particular, when the group is finite, the algorithm converges in a finite number of steps to an empirical Karcher mean (a local minimum of the empirical variance given a sample). This helps us to illustrate the inconsistency in this very simple framework.

We would like to insist on this point: the noise is created in the ambient space with our generative model and the computation of the Fréchet mean is done in the quotient space, this interaction induces an inconsistency. On the opposite, if one models the noise directly in the quotient space and compute the Fréchet mean in the quotient space, we have no reason to suspect any inconsistency.

Moreover it is also possible to define and use isometric actions on curves [Hitziger 2013, Kurtek 2011b] or on surfaces [Kurtek 2011a] where our work can be directly applied. The previous works related to the inconsistency of the template estimation [Bigot 2011[START_REF] Miolane | [END_REF]] and chapter 3 focused to isometric action, which is a restriction to real applications. That is why, we provide, in section 4.3, some insights of the non invariant case: the inconsistency also appears as soon as the noise level is large enough.

This chapter is organized as follows: Section 4.2 is dedicated for isometric action. More precisely, in section 4.2.1, we study the presence of the inconsistency and we establish the asymptotic behavior when the noise parameter σ tends to 8. In section 4.2.3 we detail the max-max algorithm and its properties. In section 4.2.4 we illustrate the inconsistency with synthetic data. In section 4.2.5, we see some examples of the registration score surface. This surface, seen in section 4.2.1 is a deformation of the unit sphere: for each unit vector, the amount of deformation of this vector is given by the quality of registration of data with respect to this vector.

Inconsistency of the template estimation with an isometric action when the noise level tends to infinity. 89

Finally in section 4.3, we prove the inconsistency for more general group action, when the noise level is large enough. We do it in two settings, firstly, the group contains a subgroup acting isometrically on M , secondly the group acts linearly on the space M .

4.2 Inconsistency of the template estimation with an isometric action when the noise level tends to infinity.

Inconsistency and quantification of the consistency bias

We start with theorem 4.1 which gives us an asymptotic behavior of the consistency bias when the noise level σ tends to infinity. One key notion in theorem 4.1 is the concept of fixed point under the action G: a point x P M is a fixed point if for all g P G, g ¨x " x. We require that the support of the noise ε is not included in the set of fixed points. But this condition is almost always fulfilled. For instance in R n the set of fixed points under a linear group action is a null set for the Lebesgue measure (unless the action is trivial: g ¨x " x for all g P G but this situation is irrelevant).

Theorem 4.1. Let us suppose that the support of the noise ε is not included in the set of fixed points under the group action. Let Y be the observable variable defined in Equation (4.1). If the Fréchet mean of rY s exists, then we have the following lower and upper bounds of the consistency bias noted CB:

σK ´2}t 0 } ď CB ď σK `2}t 0 }, (4.3)
where K " sup }v}"1 E ˜sup gPG xv, g ¨εy ¸P p0, 1s, K is a constant which depends only of the standardized noise and of the group action. In particular, K does not depends of the template. The consistency bias has the following asymptotic behavior when the noise level σ tends to infinity:

CB " σK `opσq as σ Ñ `8. (4.4)

In the following we note by S the unit sphere of M . For v P S, we call θpvq " E ˜sup gPG xv, g ¨εy ¸, so that K " sup vPS θpvq. The sketch of the proof is the following:

• K ą 0 because the support of ε is not included in the set of fixed points under the action of G.

• K ď 1 is the consequence of the Cauchy-Schwarz inequality.

• The proof of Inequalities (4.3) is based on the triangular inequalities:

}m ‹ } ´}t 0 } ď CB " inf gPG }t 0 ´g ¨m‹ } ď }t 0 } `}m ‹ },
where m ‹ minimizes F : having a piece of information about the norm of m ‹ is enough to deduce a piece of information about the consistency bias.

• The asymptotic Taylor expansion of the consistency bias (4.4) is the direct consequence of inequalities (4.3).

Proof of theorem 4.1. We note S the unit sphere in M . In order to prove that K ą 0, we take x in the support of ε such that x is not a fixed point under the action of G. There exists g 0 P G such that g 0 ¨x ‰ x. We note v 0 " g 0 ¨x }x} P S, we have xv 0 , g 0 ¨xy " }x} ą xv 0 , xy and by continuity of the dot product there exists r ą 0 such that: @y P Bpx, rq xv 0 , g 0 ¨yy ą xv 0 , yy as x is in the support of ε we have Ppε P Bpx, rqq ą 0, it follows: P ˜sup gPG xv 0 , g ¨εy ą xv 0 , εy ¸ą 0.

(4.5)

Thanks to Inequality (4.5) and the fact that sup gPG xv 0 , g ¨εy ě xv 0 , εy we have: θpv 0 q " E ˜sup gPG xv 0 , g ¨εy ¸ą Epxv 0 , εyq " xv 0 , Epεqy " xv 0 , 0y " 0.

Then we get K ě θpv 0 q ą 0. Moreover, if we use the Cauchy-Schwarz inequality:

K ď sup vPS Ep}v} ˆ}ε}q ď Ep}ε} 2 q 1 2 " 1.
In order to prove Inequalities (4.3), we use the "polar" coordinates of a point in M , every point in M can be represented by pr, vq where r ě 0 is the radius, and v belong to S the unit sphere in M , v represents the "angle". We compute F pmq as a function of pr, vq. In a first step, we minimize this expression as a function of r, in a second step we minimize this expression as a function of v. This makes appear the constant K (see figure 4.2).

As we said, let us take r ě 0 and v P S, we expand the variance at the point rv:

F prvq " E ˆinf gPG }rv ´g ¨Y } 2 ˙" r 2 ´2rE ˜sup gPG xv, g ¨Y y ¸`Ep}Y } 2 q. (4.6)
Indeed }g ¨Y } " }Y } thanks to the isometric action. We note x `" maxpx, 0q the positive part of x. Moreover we define the two following functions:

λpvq " Epsup gPG xv, g ¨Y yq " Epsup gPG xg ¨Y, vyq and λpvq " λpvq `for v P S, since that f : x P R `Þ Ñ x 2 ´2bx `c reaches its minimum at the point r " b ànd f pb `q " c ´pb `q2 , the r ‹ ě 0 which minimizes (4.6) is λpvq and the minimum value of the variance restricted to the half line R `v is:

F p λpvqvq " Ep}Y } 2 q ´λpvq 2 .

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when the noise level tends to infinity. 91

M ' t 0 ' 0 ' λpvqv v ' m ‹ ' m 1 ‹ ' m 2 ‹ ' m 3 ‹ Figure 4.2:
We minimize the variance on each half-line R `v where }v} " 1. The element which minimizes the variance on such a half-line is λpvqv, where λpvq ě 0. We get a surface in M by S P v Þ Ñ λpvqv (which is a curve in this figure since we draw it in dimension 2). The proof of theorem 4.1 states that if rm ‹ s is a Fréchet mean then m ‹ is an extreme point of this surface. On this picture there are four extreme points which are in the same orbit: we took here the simple example of the group of rotations of 0, 90, 180 and 270 degrees.

To find rm ‹ s the Fréchet mean of rY s, we need to maximize λpvq 2 with respect to v P S: m ‹ " λpv ‹ qv ‹ with v ‹ P argmax vPS λpvq.

Note that we remove the positive part and the square because argmax λ " argmax pλ `q2 indeed λ takes a non negative value. In order to prove it let us remark that: λpvq ě Epxv, Φ ¨t0 `εyq " xv, EpΦ ¨t0 qy `0, then there are two cases: if EpΦ ¨t0 q " 0 then for any v P S we have λpvq ě 0, if w " EpΦ ¨t0 q ‰ 0 then we take v " w }w} P S, and we get λpvq ě A w }w} , w E " }w} ě 0. As we said in the sketch of the proof we are interested in getting information about the norm of }m ‹ }:

}m ‹ } " λpv ‹ q " sup vPS λ.
Let v P S, we have: ´}t 0 } ď xv, gΦ ¨t0 y ď }t 0 } because the action is isometric. Now we decompose Y " Φ ¨t0 `σε and we get: Moreover we recall the triangular inequalities:

}m ‹ } ´}t 0 } ď CB " inf gPG }t 0 ´g ¨m‹ } ď }t 0 } `}m ‹ }, (4.11)
Thanks to (4.10) and to (4.11), Inequalities (4.3) are proved.

4.2.2

Remarks about theorem 4.1 and its proof

We can ensure the presence of inconsistency as soon as the signal to noise ratio satisfies }t 0 } σ ă K 2 . Moreover, if the signal to noise ratio verifies }t 0 } σ ă K 3 then the consistency bias is not smaller than }t 0 } i.e.: CB ě }t 0 }. In other words, the Fréchet mean in quotient space is too far from the template: the template estimation with the Fréchet mean in quotient space is useless in this case. In chapter 3 we also gave lower and upper bounds as a function of σ but these bounds were less informative than bounds given by theorem 4.1. These bounds did not give the asymptotic behaviour of the consistency bias. Moreover, in chapter 3 the lower bound goes to zero when the template becomes closed to fixed points. This may suggest that the consistency bias was small for this kind of template. We prove here that it is not the case.

Note that theorem 4.1 is not a contradiction with [Kurtek 2011b] where the authors proved the consistency of the template estimation with the Fréchet mean in quotient space for all σ ą 0. Indeed their noise was included in the set of constant functions which are the fixed points under their group action.

The constant K appearing in the asymptotic behaviour of the consistency bias (4.4) is a constant of interest. We can give several (but similar) interpretations of K:

• It follows from Equation (4.3) that K is the consistency bias with a null template t 0 " 0 and a standardized noise (σ " 1).

• From the proof of theorem 4.1 we know that 0 ă K ď Ep}ε}q ď 1. On the one hand, if G is the group of rotations then K " Ep}ε}q, because for all v s.t.

}v} " 1, sup gPG xv, gεy " }ε}, by aligning v and ε. On the other hand if G acts trivially (which means that g ¨x " x for all g P G, x P M ) then K " 0. The general case for K is between two extreme cases: the group where the orbits are minimal (one point) and the group for which the orbits are maximal (the whole sphere). We can state that the more the group action has the ability to align the elements, the larger the constant K is and the larger the consistency bias is.

Inconsistency of the template estimation with an isometric action

when the noise level tends to infinity. 93

• The squared quotient distance between two points is:

d Q pras, rbsq 2 " }a} 2 ´2 sup gPG xa, g ¨by `}b} 2 ,
thus the larger sup gPG xa, g ¨by, the smaller d Q pras, rbsq. We get:

K " 1 ´1 2 inf }v}"1 Epd 2 Q prvs, rεsqq
Therefore, the constant K encodes the level of contraction of the quotient distance (or folding). The larger K is, the more contracted the quotient space is.

• In chapter 3, we say that if the random variable was not included in the cone of the template, there was inconsistency. However in theorem 4.1 there is apparently no this notion of cone. In fact, this notion of cone appears, indeed we have:

K ą 0 ðñ Dv P M s.t. }v} " 1Supportpεq Ć Conepvq.
Besides, K ą 0 is necessary to see inconsistency in theorem 4.1.

One disadvantage of theorem 4.1 is that it ensures the presence of inconsistency for σ large enough but it says nothing when σ is small, in this case one can refer to [START_REF] Miolane | [END_REF]] or chapter 3. We can give a refinement of theorem 4.1.

Corollary 4.1. Under the hypothesis of theorem 4.1, we have:

σK ´2distpt 0 , FixpM qq ď CB ď σK `2distpt 0 , FixpM qq,
where FixpM q is the set of fixed points under the group action.

Proof. Indeed, in chapter 3 we have seen lemma 3.8 which states that we can translate the random variable by a fixed point, besides this translation is a congruent map. Therefore, it suffices to apply theorem 4.1 to Y ´m0 " Φpt 0 ´m0 q `σε where m 0 is the orthogonal projection of t 0 on FixpM q. We deduce immediately from corollary 4.1 a result which was already proved in chapter 3 (proposition 3.11): Corollary 4.2. Under the hypothesis of theorem 4.1, if t 0 is a fixed point, we have an expression of the consistency bias: CB " σK.

In the proof of theorem 4.1, we have seen that the minimum of the variance restricted to the half-line R `v for v P S, was We can also wonder if the converse of theorem 4.1 is true: if ε is a non biased noise always included in the set of fixed points, is rt 0 s a Fréchet mean of rΦ¨t 0 `σεs? Proposition 4.1. Let G be a group acting isometrically on M an Hilbert space. We consider a template t 0 and ε a standardized noise. We define the observable variable Y by Y " Φ ¨t0 `σε, where σ ą 0. If ε belongs almost surely in the set of fixed points, then t 0 is a Fréchet mean of rY s.

Ep}Y } 2 q ´˜ˆE ˆinf gPG xv,
Proof. A simple computation show that t 0 is a minimum of the variance:

F pmq " E ˆinf gPG }m ´g ¨pΦ ¨t0 `σεq} 2 " }m} 2 `Ep}Φ ¨t0 `σε} 2 q ´2Epsup gPG xm, gΦ ¨t0 y `xm, g ¨σεyq " }m} 2 `Ep}Φ ¨t0 `σε} 2 q ´2E ˜sup gPG xm, g ¨t0 y ¸´2 xm, Epσεqy " }m} 2 `Ep}Φ ¨t0 `σε} 2 q ´2E ˜sup gPG xm, g ¨t0 y ¸(4.12)
We see that the element m which minimizes (4.12) does not depend of σ, in particular we can assume σ " 0, and wonder which elements minimizes F pmq " Epinf gPG }m ´gΦ ¨t0 } 2 q " inf gPG }m ´g ¨t0 } 2 it becomes clear that only the points in the closure of the orbit of t 0 can minimize this variance.

Then when ε is included in the set of fixed points, the estimation is always consistent for all σ. This is an alternative proof of the Theorem of consistency done by Kurtek et al. [Kurtek 2011b].

Inconsistency of the template estimation with an isometric action when the noise level tends to infinity. 95

In the proof of theorem 4.1, we have seen that the direction of the Fréchet mean of rY s is given by the supremum of this quantity (4.7):

E ˜sup gPG xv, g ¨σεy `xv, gΦ ¨t0 y ¸.
This Equation is a good illustration of the difficulty to compute the Fréchet mean in quotient space. Indeed, we have on one side the contribution of the noise xv, g ¨σεy and on the other side the contribution of the template xv, gΦ ¨t0 y, and we take the supremum of the sum of these two contributions over g P G. Unfortunately the supremum of the sum of two terms is not equal to the sum of the supremum of each of these terms. Hence, it is difficult to separate these two contributions. However, we can intuit that when the noise is large, xv, g ¨σεy prevails over xv, gΦ ¨t0 y, and the use of the Cauchy-Schwarz inequality in Equations (4.8) and ( 4.9) proves it rigorously. We can conclude that, when the noise is large, the direction of the Fréchet mean in the quotient space depends more on the noise than on the template.

Template estimation with the Max-Max Algorithm

Max-Max Algorithm Converges to a Local Minima of the Empirical Variance

Section 4.2.1 can be understood as follows: if we want to estimate the template by minimizing the Fréchet mean in the quotient space, then there is a bias. This supposes that we are able to compute such a Fréchet mean. In practice, we cannot minimize the exact variance in quotient space, because we have only a finite sample and not the whole distribution. In this section we study the estimation of the empirical Fréchet mean with the max-max algorithm. We assume that the group is finite. In this case, the registration can always be found by an exhaustive search. Hence, the numeric experiments which we conduct in section 4.2.4 lead to an empirical Karcher mean in a finite number of steps. In a compact group acting continuously, the registration also exists but is not necessarily computable without approximation.

If we have a sample: Y 1 , . . . , Y I of independent and identically distributed copies of Y , then we define the empirical variance in the quotient space:

M Q x Þ Ñ F I pxq " 1 I I ÿ i"1 d 2 Q prxs, rY i sq (4.13) " 1 I I ÿ i"1 min g i PG }x ´gi ¨Yi } 2 " 1 I I ÿ i"1 min g i PG }g i ¨x ´Yi } 2 .
The empirical variance is an approximation of the variance. Indeed thanks to the law of large number we have lim IÑ8 F I pxq " F pxq for all x P M . One element which minimizes globally (respectively locally) F I is called an empirical Fréchet mean (respectively an empirical Karcher mean). For x P M and g P G I : g " pg 1 , . . . , g I q where g i P G for all i " 1..I we define J an auxiliary function by: Jpx, gq " 1

I I ÿ i"1 }x ´gi ¨Yi } 2 " 1 I I ÿ i"1 }g ´1 i ¨x ´Yi } 2 .
The max-max algorithm (algorithm 1) iteratively minimizes the function J in the variable x P M and in the variable g P G I (see also figure 4.3). This algorithm is nothing else than a gradient descent, it has also known as Procrustes Analysis [Gower 1975, Goodall 1991].

Algorithm 1 Max-Max algorithm

Require: A starting point m 0 P M , a sample Y 1 , . . . , Y I . n " 0. while Convergence is not reached do Minimizing g P G I Þ Ñ Jpm n , gq: we get g n i by registering Y i with respect to m n .

Minimizing x P M Þ Ñ Jpx, g n q: we get m n`1 " 1

I I ř i"1 g n i Y i . n " n `1. end while m " m n
First, we note that this algorithm is sensitive to the the starting point. However we remark that m 1 " 1 I ř I i"1 g i ¨Yi for some g i P G, thus without loss of generality, we can start from m 1 " 1 I ř I i"1 g i ¨Yi for some g i P G. The empirical variance does not increase at each step of the algorithm since: F I pm n q " Jpm n , g n q ě Jpm n`1 , g n q ě Jpm n`1 , g n`1 q " F I pm n`1 q Proposition 4.2. As the group is finite, the convergence is reached in a finite number of steps.

Inconsistency of the template estimation with an isometric action

when the noise level tends to infinity. 97

x g m 0 m 1 m 2 m n´1 mn g 0 g 1 g n´1 g n
Jpm 0 , g 0 q Jpm 1 , g 0 q Jpm 1 , g 1 q Jpm n´1 , g n´1 q Jpmn, g n´1 q Jpmn, g n q Figure 4.3: Iterative minimization of the function J on the two axes, the horizontal axis represents the variable in the space M , the vertical axis represents the set of all the possible registrations G I . Once the convergence is reached, the point pm n , g n q is the minimum of the function J on the two axis in green. Is this point the minimum of J on its whole domain? There are two pitfalls: firstly this point could be a saddle point, it can be avoided with proposition 4.3, secondly this point could be a local (but not global) minimum, this is discussed in section 4.2.4.3

Proof of proposition 4.2. The sequence pF I pm n qq nPN is non-increasing. Moreover the sequence pm n q nPN takes value in a finite set which is: t 1 I ř I i"1 g i ¨Yi , g i P Gu. Therefore, the sequence pF I pm n qq nPN is stationary. Let n P N such that F I pm n q " F I pm n`1 q. Hence the empirical variance did not decrease between step n and step n `1 and we have:

F I pm n q " Jpm n , g n q " Jpm n`1 , g n q " Jpm n`1 , g n`1 q " F I pm n`1 q, as m n`1 is the unique element which minimizes m Þ Ñ Jpm, g n q we conclude that m n`1 " m n .
This proposition gives us a shutoff parameter in the max-max algorithm: we stop the algorithm as soon as m n " m n`1 . Let us call m the final result of the max-max algorithm. It may seem logical that m is at least a local minimum of the empirical variance. However this intuition may be wrong: let us give a toy counterexample, suppose that we observe Y 1 , . . . , Y I , due to the transformation of the group it is possible that ř n i"1 Y i " 0. We can start from m 1 " 0 in the max-max algorithm, as Y i and 0 are already registered, the max-max algorithm does not transform Y i . At step two, we still have m 2 " 0, by induction the max-max algorithm stays at 0 even if 0 is not a Fréchet or Karcher mean of rY s. Because 0 is equally distant from all the points in the orbit of Y i , 0 is called a focal point of rY i s. The notion of focal point is important for the consistency of the Fréchet mean in manifold [Bhattacharya 2008]. Fortunately, the situation where m is not a Karcher mean is almost always avoided due to the following statement: Proposition 4.3. Let m be the result of the max-max algorithm. If the registration of Y i with respect to m is unique, in other words, if m is not a focal point of Y i for all i P 1..I then m is a local minimum of F I : r ms is an empirical Karcher mean of rY s.

Note that, if we call z the registration of y with respect to m, then the registration is unique if and only if xm, z ´g ¨zy ‰ 0 for all g P Gzte G u. Once the max-max algorithm has reached convergence, it suffices to test this condition for m obtained by the max-max algorithm and Y i for all i. This condition is in fact generic and is always obtained in practice.

Proof of proposition 4.3. We call g i the unique element in G which register Y i with respect to m, for all h P Gztg i u, } m ´gi ¨Yi } ă } m ´hi ¨Yi }. By continuity of the norm we have for a close enough to m: }a ´gi ¨Yi } ă }a ´hi ¨Yi } for all h i ‰ g i (note that this argument requires a finite group). The registrations of Y i with respect to m and to a are the same:

F I paq " 1 I I ÿ i"1 }a ´gi ¨Yi } 2 " Jpa, gq ě Jp m, gq " F I p mq,
because m Þ Ñ Jpm, gq has one unique local minimum m. This condition of the unique registration may be seem, odd, in fact this is a natural condition. Indeed, we can state a result which states that if rm n‹ s is a empirical Fréchet mean, then the registration of the data is unique: gpY i , m n‹ q ¨Yi , where gpY i , m ‹ q P argmin gPG }g ¨Yi ´mn‹ }.

Furthermore, gpY i , m ‹ q is unique up to an element of IsopY i q, this means that: gpY i , m n‹ q, gpY i , m n‹ q P argmin gPG }gY i ´mn‹ } ùñ gpY i , m n‹ q ¨Yi " gpY i , m n‹ q ¨Yi Proof. We want to find:

argmin g 1 ,...gn argmin mPM 1 n n ÿ i"1 }m ´gi ¨Yi } 2 .
Then, if rm n‹ s, we have m n‹ " 1 n n ř i"1 }m ´gi ¨Yi } 2 . Then we need to minimize:

argmin g 1 ,...gn 1 n n ÿ i"1 › › › › › ˜1 n n ÿ i"1 g i ¨Y ¸´g i ¨Yi › › › › › 2 .

Inconsistency of the template estimation with an isometric action when the noise level tends to infinity. 99

Therefore m n‹ " 1 n n ř i"1 gpY i , m n‹ q ¨Yi . For every j, let us consider: gpY j , m n‹ q ¨Yj , gpY j , m n‹ q ¨Y P argmin gPG }g ¨Yj ´mn‹ }.

Then we have:

m n‹ " 1 n n ÿ i"1 gpY i , m n‹ q ¨Yi " 1 n ˜n ÿ i"1,i‰j
pgpY i , m n‹ q ¨Yi q `gpY j , m n‹ q ¨Yj ¸.

By simplifying the sum, we get gpY j , m n‹ q ¨Yj " gpY j , m n‹ q ¨Yj . In chapter 5, proposition 5.2, we generalize proposition 4.4, at the Fréchet mean of rY s instead of the empirical Fréchet mean of Y 1 , . . . Y n .

Remark 4.2. We remark the max-max algorithm is in fact a gradient descent. The gradient descent is a general method to find the minimum of a differentiable function.

Here we are interested in the minimum of the variance F : let m 0 P M and we define by induction the gradient descent of the variance m n`1 " m n ´ρ∇F pm n q, where ρ ą 0 and F the variance in the quotient space. In chapter 3, the gradient of the variance in quotient space for finite group and for a regular point m was computed (m is regular as soon as g ¨m " m implies g " e), this leads to:

m n`1 " m n ´2ρ rm n ´EpgpY, m n q ¨Y qs ,
where gpY, m n q is the almost-surely unique element of the group which registers Y with respect to m n . Now if we have a set of data Y 1 , . . . , Y n we can approximated the expectation which leads to the following approximated gradient descent:

m n`1 " m n p1 ´2ρq `ρ 2 I I ÿ i"1
gpY i , m n q ¨Yi , now by taking ρ " 1 2 we get m n`1 " 1 I ř I i"1 gpY i , m n q ¨Yi . So the approximated gradient descent with ρ " 1 2 is exactly the max-max algorithm. But the max-max algorithm for finite group, is proved to be converging in a finite number of steps which is not the case for gradient descent in general.

Simulation on synthetic data

In this section, we consider data in an Euclidean space R N equipped with its canonical dot product x¨, ¨y, and G " Z{NZ acts on R N by horizontal translation:

Z{N Z ˆRN Ñ R N p k, px 1 , . . . , x N qq Þ Ñ px 1`k , x 2`k , . . . x N `kq ,
where indexes are taken modulo N . This space models the discretization of functions defined on r0, 1s with N points. This action is found in [START_REF] Stéphanie Allassonnière | Towards a coherent statistical framework for dense deformable template estimation[END_REF]] and used for neuroelectric signals in [Hitziger 2013]. The registration between two vectors can be made by an exhaustive research but it is faster with the fast Fourier transform [Cooley 1965]. Empirical variance at the template in blue and at the estimated template in red (b) Variation of FI pt0q (in blue) and of FI p mq (in red) as a function of I the size of the sample. Since convergence is already reached, F p mq, which is the limit of red curve, is below F pt0q: F pt0q is the limit of the blue curve. Due to the inconsistency, m is an example of point such that F p mq ă F pt0q. We display an example of a template and the template estimation with the max-max algorithm on figure 4.4a. This experiment was already conducted in [START_REF] Stéphanie Allassonnière | Towards a coherent statistical framework for dense deformable template estimation[END_REF]], but no explanation of the appearance of the bias was provided. We know from section 4.2.3 that the max-max output is an empirical Karcher mean, and that this result can be obtained in a finite number of steps. Taking σ " 10 may seem extremely high, however the standard deviation of the noise at each point is not 10 but σ ? N " 1.25 which is reasonable. The sample size is 10 5 , the algorithm stopped after 247 steps, and m the estimated template (in red on the figure 4.4a) is not a focal points of the orbits rY i s, then proposition 4.3 applies. We call empirical bias (noted EB) the quotient distance between the true template and the point m given by the max-max result. On this experiment we have EB σ » 0.11. Of course, one could think that we estimate the template with an empirical bias due to a too small sample size which induces fluctuation. To reply to this objection, we keep in memory m obtained with the max-max algorithm. If there was no inconsistency then we would have F pt 0 q ď F p mq. We do not know the value of the variance F at these points, but thanks to the law of large number, we know that:

F pt 0 q " lim IÑ8 F I pt 0 q and F p mq " lim IÑ8 F I p mq,

Inconsistency of the template estimation with an isometric action when the noise level tends to infinity. 101

Given a sample, we compute F I pt 0 q and F I p mq thanks to the definition of the empirical variance F I (4.13). We display the result on figure 4.4b, this tends to confirm that F pt 0 q ą F p mq. In other word, the variance at the template is larger that the variance at the point given by the max-max algorithm. Figure 4.4a shows that the main source of the inconsistency was the discontinuity of the template. One may think that a continuous template would lead to a better behaviour. But it is not the case as presented on figure 4.5. Even with a large number of observations created from a continuous template we do not observe a convergence to the template. In the example of figure 4.5, the empirical bias satisfies EB σ " 0.23. In green we also display the mean of data knowing transformations. This means that if the data are on the form Y i " Φ i ¨t0 `σε i , and if we know Φ i then we can estimate the template by

1 n n ÿ i"1 Φ ´1 i ¨Yi " t 0 `σ n n ÿ i"1 Φ ´1 i ε i ,
this produces a much better result, since that in this case we have EB σ " 0.04. But in practice, we do not know Φ i . A sample of size 10 3 may seem a too small. Thus we can do the same thing with a sample of size 10 6 on figure 4.6. We do not observe a much better estimation. 4.1: Empirical variances at the template and at 5 different outputs of the max-max algorithm coming from the same sample of size 10 5 p mi q 1ďiď5 , but with different starting points. We use σ " 10 and the action of horizontal translation in R 64 . Conclusion: on these five points, only m1 is an eventual global minima.

We observe that the variances at these points are very close. We also display these local minima on figure 4.7

Moreover we can compute all the quotient distances given two points in the set tt 0 , mi u (table 4.2). We observe that all the Karcher means are concentrated compared to the distance to the real template. One can intuit that the whole set of Karcher means (including the Fréchet means) are concentrated in this neighborhood.

Of course all these results are sensitive to the size of the sample. In the following We zoomed in order to see better the small differences between the different outputs of the max-max algorithm given five different starting points.

we launch 100 times the max-max algorithm with 100 different starting points. We did it for a sample of size 10 5 and with a sample of size 10 4 . In the figure 4.8 we plot an histogram of the distance between the template and the outputs of the max-max algorithm. We observe that this distance depends less on the starting point with a large sample. On this example, we can intuit that with an infinite number of observations the distance between the template and the Fréchet mean is unique: whatever the starting point we would have the same bias.

Example of registration score surface

In the proof of theorem 4.1, we have seen that the direction of the Fréchet mean was given by maximizing the λ function on the sphere:

S Q v Þ Ñ ˜E ˜sup gPG xv, g ¨Y y ¸¸`.
Through this process, we obtain a hyper-surface in the Hilbert space M which is a deformation of the unit sphere. The current section aims to study such hypersurfaces on an example. In order to obtain such a hyper-surface, we approximate the expectation in the computation of λ by the empirical expectation: we have an random hyper-surface depending on the sample of Y we simulate. Then we can draw S Q v Þ Ñ λpvqv in dimension 2 or 3. In dimension 2 for the action of rotation of angle Example 4.1. For the action of horizontal translation on R 3 . We can change this noise level and observe the resulting movie on http://loic.devilliers.free.fr/rosace3animation.gif.

For σ " 0 we observe that the hyper-surface is three pieces of three spheres. This can be shown and generalized in any dimension (including infinite). Indeed for σ " 0 we have:

λpvq " E ˜sup gPG xgΦ ¨t0 , vy ¸`.
Since λ is invariant under the group action: λpvq " λpg¨vq, without loss of generality we can assume that v belongs to the cone of the template. In this case we have λv " pxv, t 0 yq `. Then the point on the hyper-surface is xv, t 0 y v or 0 for v belonging to Conept 0 q, and xv, t 0 y ě 0. Moreover:

t0u ď txv, t 0 y v, v P Su Ă # p P R 3 › › › › p ´}t 0 } 2 › › › › 2 " }t 0 } 2 4 + .
In other words, these points are on the sphere of center t 0 2 and radius }t 0 } 2 . In particular the segment r0, t 0 s is a diameter of this circle. We can conclude that, when σ " 0 this hyper-surface is a union of part of the spheres. There are as many spheres as points in the orbits of the template.

When σ ą 0, note that it is possible to have λpvq " 0. For instance assume that the template is t 0 P pR `q3 , if the noise is sufficiently small, then Y belongs almost surely in pR `q3 . Let v P S be an unit vector in pR ´q3 then sup gPG xv, g ¨Y y ă 0 almost surely: the angle between v and g ¨Y is obtuse for every g P G. Then λv " 0. This makes the hyper-surface singular to zero, since some vectors of the sphere collapse to zero.

When σ is not equal to zero, it is harder to interpret the resulting surface. However, we believe that there is a link between the geometrical property of the surface and the statistical property. We advocate, that the discovery of such a links matters for a better understanding of the consistency bias.

Inconsistency in the case of non invariant distance under the group action

Notation and hypothesis

In this section, data still come from a Hilbert space M . But we take a group of deformation G which acts in a non invariant way on M . Starting from a template t 0 we consider a random deformation in the group G namely a random variable Φ which takes value in G and ε an standardized noise in M independent of Φ. We suppose that our observable random variable is:

Y " Φ ¨t0 `σε with σ ą 0, Epεq " 0, Ep}ε} 2 q " 1, where σ is the noise level. We suppose that Ep}Y } 2 q ă `8, and we define the pre-variance of Y in M {G as the map defined by:

F pmq " E ˆinf gPG }g ¨m ´Y } 2 ˙.
In this part we still study the inconsistency of the template estimation by minimizing F . We present two frameworks where we can ensure the presence of inconsistency: in section 4.3.3 we suppose that the group G contains a non trivial group H which acts isometrically on M . But some groups do not satisfy this hypothesis, that is why, in section 4.3.4 we do not suppose that G contains a subgroup acting isometrically but we require that G acts linearly on M . In both sections we prove inconsistency as soon as the variance σ 2 is large enough.

These hypothesis are not unacceptable as for example, deformations that are considered in computational anatomy may include rotations which form a subgroup

Inconsistency in the case of non invariant distance under the group action 107

H of the diffeomorphic deformations which acts isometrically. Concerning the second case, an important example is:

Example 4.2. Let G be a subgroup of the group of C 8 diffeomorphisms on R n G acts linearly on L 2 pR n q with the map:

@ϕ P G @f P L 2 pR n q ϕ ¨f " f ˝ϕ´1 .
Note that this action is not isometric: indeed, f ˝ϕ´1 has generally a different L 2 -norm than f , because a Jacobian determinant appears in the computation of the integral.

Where did we need an isometric action previously?

Let M be a Hilbert space, and G a group acting on M . Can we define a distance in the quotient space Q " M {G defined as the set which contains all the orbits? When the action is invariant, the orbits are parallel in the sense where d M pm, nq " d M pg ¨m, g ¨nq for all m, n P M and for all g P G. This implies that:

d Q prms, rnsq " inf gPG }m ´g ¨n},
is a distance on Q. But it is not necessarily the case when the action is no longer invariant. Let us take the following example:

Example 4.3. We call C 8 diff pR 2 q the set of the C 8 diffeomorphisms of M " R 2 . We equip R 2 with its canonical Euclidean structure. We take p " p´1, ´1q, q " p1, 1q and r " p2, 0q and we define a group G by:

G " ! f P C 8 diff pR 2 q | f pqq " pqq, f ppq " ppq, @x P R f px, 0q P Rr ) , G acts on R 2 by f ¨px, yq " f px, yq. In this case d Q (define by d Q pa, bq " inf gPG d M pa, gb q) is not a distance.
Indeed, first let us notice that q and p are fixed points under this group action and the orbit of r is the horizontal line tpx, 0q, x P Ru. On this example: inf gPG }q ´g ¨r} " }q ´p1, 0q} " 1 however inf gPG }r ´g ¨q} " }r ´q} " ? 2, then the function d Q is not symmetric. One could think define a distance by: dQ prms, rnsq " inf h,gPG }h ¨m ´g ¨n}.

Unfortunately, in this case we have:

dQ prps, rqsq " }p ´q} " 2 ? 2 and dQ prps, rrsq " 1 " dQ prrs, rqsq, then we do not have dQ prps, rqsq ď dQ prps, rrsq `d Q prrs, rqsq. In other words we do not have the triangular inequality (see figure 4.10). Therefore when the action is no longer invariant, a priori one cannot define a distance in the quotient anymore. If Y is a random variable in M , F pmq " Epinf gPG }g ¨m ´Y } 2 q cannot be interpreted as the variance of rY s.

However inf gPG }g ¨a ´b} is positive and is equal to zero if a " b, then inf gPG }g ä ´b} is a pre-distance in M . Then inf gPG }g ¨m ´Y } measures the discrepancy between the random point Y and the current point m. Even if the discrepancy measure is not symmetric or does not satisfy the triangular inequality, we can still define F pxq " Epinf gPG }g ¨x ´Y } 2 q and call it the pre-variance of the projection of Y into M {G, if Ep}Y } 2 q ă `8. The elements which minimize this function are the elements which orbit are the closest of the random point Y . Hence, we wonder if the template can be estimated by minimizing this pre-variance. Note that, once again F pxq " F pg ¨xq for all x P M and g P G. Then the pre-variance is well defined in the quotient space by rxs Þ Ñ F pxq.

It is not surprising to use a discrepancy measure which is not a distance, for instance the Kullback-Leibler divergence [Kullback 1951] is not symmetric although it is commonly used.

In the proof of inconsistency of theorem 4.1, we used that the action was isometric in order to simplify the expansion of the variance in Equation (4.6):

F pmq " E ˆinf gPG }m ´g ¨Y } 2 " E ˆinf gPG " }m} 2 ´xm, g ¨Y y `}g ¨Y } 2 ‰ ˙,
with }g ¨Y } 2 " }Y } 2 there was only one term which depends on g: xg ¨m, Y y and the two other terms could be pulled out of the infimum. When the action is no longer isometric we cannot do this trick anymore. To remedy this situation, in this chapter we require that the orbit of the template is a bounded set.

Non invariant group action, with a subgroup acting isometrically

In this Subsection, G acts on M a Hilbert space. We assume that there exists a subgroup H Ă G such that H acts isometrically on M . As H is included in 4.3. Inconsistency in the case of non invariant distance under the group action 109

G, we deduce a useful link between the variance of Y projected in M {H and the pre-variance of Y projected in M {G:

F pmq " E ˆinf gPG }g ¨m ´Y } 2 ˙ď E ˆinf hPH }h ¨m ´Y } 2 ˙" F H pmq.
The orbit of a point m under the group action G is rms " tg ¨m, g P Gu, whereas the orbit of the point m under the group action H is rms H " th ¨m, h P Hu. Moreover, we call F H the variance of rY s H in the quotient space M {H, and F the pre-variance of rY s in the quotient space M {G.

Inconsistency when the template is a fixed point

We begin by assuming that the template t 0 is a fixed point under the action of G:

Proposition 4.5. Suppose that t 0 is a fixed point under the group action G. Let ε be a standardized noise which support is not included in the fixed points under the group action of H, and Y " Φ ¨t0 `σε " t 0 `σε. Then t 0 is not a minimum of the pre-variance F Proof. We have:

1. Thanks to corollary 4.2 of section 4.2.1 we know that rt 0 s H " rEpY qs H is not the Fréchet mean of rY s H the projection of Y into M {H: we can find m P M such that:

F H pmq ă F H pt 0 q. (4.14)
Note that in order to apply corollary 4.2, we do not need that Φ is included in H, because t 0 is a fixed point.

2. Because we take the infimum over more elements we have:

F pmq ď F H pmq. (4.15)
3. As t 0 is a fixed point under the action of G and under the action of H:

F H pt 0 q " F pt 0 q " Ep}t 0 ´Y } 2 q. (4.16)
With Equations (4.14), (4.15) and (4.16), we conclude that t 0 does not minimize F .

Inconsistency in the general case for the template

The following proposition 4.6 tells us that when σ is large enough then there is an inconsistency.

Proposition 4.6. We suppose that the template is not a fixed point and that its orbit under the group G is bounded. We consider A ě sup gPG }g¨t 0 } }t 0 } and a ď inf gPG }g¨t 0 } }t 0 } , note that a ď 1 ď A and we have:

@g P G a}t 0 } ď }g ¨t0 } ď A}t 0 }.
We note:

θpt 0 q " 1 }t 0 } Epsup gPG xg ¨t0 , εyq and θ H " 1 }t 0 } E ˆsup hPH xh ¨t0 , εy ˙.
We suppose that θ H ą 0. If σ is bigger than a critical noise level noted σ c defined as:

σ c " }t 0 } θ H » -ˆθpt 0 q θ H `A˙`dˆθ pt 0 q θ H `A˙2 `A2 ´a2 fi fl . (4.17)
Then we have inconsistency.

Note that in section 4.2.1 we have proved inconsistency in the isometric case as soon as σ ą 2}t 0 } K , where K ě θ H , then we find in this theorem an analogical sufficient condition on σ where

« ´θpt 0 q θ H `A¯`c´θ pt 0 q θ H `A¯2 `A2 ´a2
ff is a corrective term due to the non invariant action.

We have shown in chapter 3 that if the orbit of the template rt 0 s H is a manifold, then θ H ą 0 as soon as the support of ε is not included in T t 0 rt 0 s K (the normal space of the orbit of the template t 0 at the point t 0 ). If rt 0 s is not a manifold, we have also seen in chapter 3 that θ H ą 0 as soon as t 0 is a limit point of rt 0 s H and the support of ε contains a ball Bp0, ηq for η ą 0. Hence, θ H ą 0 is a rather generic condition. Condition (4.17) can be reformulated as follows: as soon as the signal to noise ratio }t 0 } σ is sufficiently small:

}t 0 } σ ă θ H ´θpt 0 q θ H `A¯`c´θ pt 0 q θ H `A¯2 `A2 ´a2
, then there is inconsistency. We remark the presence of the constants θpt 0 q and θ H in proposition 4.6. This kind of constants were already here in the isometric case under the form θp t 0 }t 0 } q " 1 }t 0 } Epsup gPG xt 0 , g ¨εyq, due to the polarization identity (4.2), we can state that it measures how much the template looks like to the noise after registration, but only in the isometric case. However we can intuit that this constant plays a analogical role in the non isometric case.

Example 4.4. Let G acting on M , we suppose that G contains H " OpM q the orthogonal group of M . Assume that G can modify the norm of the template by 4.3. Inconsistency in the case of non invariant distance under the group action 111 multiplying its norm by at most 2. Then we can set up A " 2 and a " 0. By aligning ε and }t 0 } we have θ H " Ep}ε}q ą 0, and θpt 0 q " AEp}ε}q then when the signal to noise ratio }t 0 } σ is smaller that Ep}ε}q 4`?20 then there is inconsistency. By Cauchy-Schwarz inequality we have Ep}ε}q ď Ep}ε} 2 q " 1, thus the signal to noise ratio has to be rather small in order to fulfill this condition.

Proof of proposition 4.6

We define the following values:

λ H " 1 }t 0 } 2 E ˆsup hPH xh ¨t0 , Y y ˙and λpt 0 q " 1 }t 0 } 2 E ˜sup gPG xg ¨t0 , Y y ¸.
Note that λ H and λpt 0 q are registration scores which definitions are the same than the registration score used in the proof of theorem 4.1 in section 4.2 (only the normalization by }t 0 } is different). The proof of proposition 4.6 is based on the following Lemma:

Lemma 4.1. If: λ H ě 0, (4.18)
a 2 ´2λpt 0 q `λ2 H ą 0, (4.19)
then t 0 is not a minimizer of the pre-variance of rY s in M {G.

How condition (4.19) can be understood? In order to answer to that question, let us imagine that G " H acts isometrically, then a can be set up to 1, and λpt 0 q " λ H the condition (4.19) becomes λ 2 H ´2λ H `1 " pλ H ´1q 2 ą 0 and the condition (3.26) of theorem 3.4 aimed to ensure that λ H ą 1. Now let us return to the non invariant case: if H is strictly included in G such that a is closed enough to 1 and λpt 0 q closed enough to λ H , then on can think that condition (4.19) still holds. But the closed enough seems hard to be quantified.

Proof of lemma 4.1. The proof is based on the following points:

1. F pλ H t 0 q ď F H pλ H t 0 q, 2. F H pλ H t 0 q ă F pt 0 q.
With items 1 and 2 we get that F pλ H t 0 q ă F pt 0 q. Item 1 is just based on the fact that in the map F , we take the infimum on a larger set than on F H . We now prove item 2, in order to do that we expand the two quantities, firstly:

F H pλ H t 0 q " E ˆinf hPH }h ¨λH t 0 } 2 `}Y } 2 ´2 xh ¨λH t 0 , Y y ˙(4.20) " λ 2 H }t 0 } 2 `Ep}Y } 2 q ´2λ H E ˆsup hPH xh ¨t0 , Y y ˙(4.21) " Ep}Y } 2 q ´λ2 H }t 0 } 2 ,
We use the fact that H acts isometrically between Equations (4.20) and ( 4.21) and the fact that λ H ě 0 because inf aPA ´λa " ´λ sup aPA a is true for any A subset of R if λ ě 0. Secondly:

F pt 0 q " E ˆinf gPG }g ¨t0 } 2 `}Y } 2 ´2 xg ¨t0 , Y y ě a 2 }t 0 } 2 `Ep}Y } 2 q ´2E ˜sup gPG xg ¨t0 , Y y ȩ a 2 }t 0 } 2 `Ep}Y } 2 q ´2λpt 0 q}t 0 } 2 Then: F pt 0 q ´FH pλ H t 0 q ě }t 0 } 2 " a 2 ´2λpt 0 q `λ2 H ‰ ą 0,
thanks to hypothesis (4.19). Proof of proposition 4.6. In order to prove proposition 4.6, all we have to do is proving λ H ě 0 and proving that Condition (4.19) is fulfilled when σ ą σ c . Firstly, thanks to Cauchy-Schwarz inequality, we have:

λ H " 1 }t 0 } 2 E ˆsup hPH xh ¨t0 , Φ ¨t0 `σεy ě 1 }t 0 } 2 " ´A}t 0 } 2 `Epsup hPH xh ¨t0 , σεyq  ě ´A `σ θ H }t 0 }
Note that as σ ą σ c ě A }t 0 } θ H we get λ H ě 0, this proves (4.18). We also have:

λpt 0 q " 1 }t 0 } 2 E ˜sup gPG xg ¨t0 , Φ ¨t0 `σεy ḑ 1 }t 0 } 2 « A 2 }t 0 } 2 `σE ˜sup gPG xg ¨t0 , εy ¸ff ď A 2 `σ θpt 0 q }t 0 } ,
Then we can find a lower bound of a 2 ´2λpt 0 q `λ2 H :

a 2 ´2λpt 0 q `λ2 H ě a ´2 ˆA2 `σ θpt 0 q }t 0 } ˙`ˆσ θ H }t 0 } ´A˙2 ě a 2 ´A2 ´2 σθ H }t 0 } ˆθpt 0 q θ H `A˙`ˆσ θ H }t 0 } ˙2 :" P pσq
For σ ą σ c where σ c is the biggest solution of the quadratic Equation P pσq " 0, we get a 2 ´2λpt 0 q `λ2 H ą 0 and the template estimation is inconsistent thanks to lemma 4.1. The critical σ c is exactly the one given by proposition 4.6.

4.3.4

Linear action

The result of the previous part has a drawback, it requires that the group of deformations contains a non trivial subgroup which acts isometrically. Now, we remove this hypothesis, but we require that the group acts linearly on data. This means that for all g P G, m Þ Ñ g ¨m is a linear map. In this Subsection we suppose that the group G acts linearly on M . Once again, we can give a criteria on the noise level which leads to inconsistency:

Proposition 4.7. We suppose that the orbit of the template is bounded, therefore we can consider the following two constants:

Da ě 0, A ą 0 such that @g P G a}t 0 } ď }g ¨t0 } ď A}t 0 }.

We suppose that A ă ? 2. In other words, the deformation of the template can multiply the norm of the template by less than ? 2. We also suppose that:

θpt 0 q " 1 }t 0 } E ˜sup gPG xg ¨t0 , εy ¸ą 0. (4.22)
There is inconsistency as soon as

σ ě σ c " }t 0 } θpt 0 q « A 2 `1 `a1 ´a2 p2 ´A2 q 2 ´A2
ff .

Example 4.5. For instance if A ď 1.2, then there is inconsistency if σ ě 7 }t 0 } θpt 0 q . Once again we find a condition which is similar to the isometric case, but due to the non invariant action we have here a corrective term which depends on A and a. In chapter 3, we have seen lemma 3.7 which states that θpt 0 q ą 0. However G does not act isometrically, therefore we can no longer apply lemma 3.7 in order to fulfill Condition (4.22). However it is easy to fulfill this Condition thanks to the following Proposition: Proposition 4.8. If t 0 is not a fixed point, and if the support of ε contains a ball Bp0, ηq for η ą 0 then

θpt 0 q " 1 }t 0 } E ˜sup gPG xg ¨t0 , εy ¸ą 0.
Remark 4.3. It is possible to remove the condition A ă ? 2 in proposition 4.7. Indeed Let be h P G such that:

sup gPG }g ¨t0 } }h ¨t0 } ă ? 2.
The template t 0 can be replaced by h ¨t0 since Φ ¨t0 `σε is equal to Φh ´1 ¨ht 0 and applying proposition 4.7 to the new template h ¨t0 . We get that h ¨t0 does not minimize the variance F with A ď ? 2 (because the new template is h ¨t0 q. Since h ¨t0 does not minimize F , the original template t 0 does not minimize the pre-variance F neither, since F pt 0 q " F ph ¨t0 q.

This changes the critical σ c since we apply proposition 4.7 to h ¨t0 instead of t 0 itself.

Proofs of proposition 4.7 and proposition 4.8

As in section 4.3.3 we first prove a Lemma: Lemma 4.2. We define:

λpt 0 q " 1 }t 0 } 2 E ˜sup gPG xg ¨t0 , Y y ¸.
Suppose that λpt 0 q ě 0 and that: a 2 ´2λpt 0 q `λpt 0 q 2 p2 ´A2 q ą 0.

(4.23)

Then t 0 is not a minimum of F . Proof of lemma 4.2. Since @g P G a}t 0 } ď }g ¨t0 } ď A}t 0 }, (4.24)
then by linearity of the action we get:

@g P G, µ P R a}µt 0 } ď }g ¨µt 0 } ď A}µt 0 }. (4.25)
We remind that:

F pmq " E ˆinf gPG }g ¨m} 2 ´2 xg ¨m, Y y `}Y } 2 ˙.
By using Equations (4.24) and (4.25) we get:

F pt 0 q ě a 2 }t 0 } 2 ´2λpt 0 q}t 0 } 2 `Ep}Y } 2 q,
We get:

F pλpt 0 qt 0 q ď E ˆA2 }λpt 0 qt 0 } 2 `}Y } 2 `inf gPG p´2λpt 0 q xg ¨t0 , Y yq ˙(4.26) ď A 2 λpt 0 q 2 }t 0 } 2 `Ep}Y } 2 q ´2λpt 0 q 2 }t 0 } 2 .
Note that we use the fact that the action is linear in Equation (4.26). We obtain that t 0 is not the minimum of the F :

F pt 0 q ´F pλpt 0 qt 0 q ě }t 0 } 2 " a 2 ´2λpt 0 q `λpt 0 q 2 p2 ´A2 q ‰ ą 0.

Proof of proposition 4.7. By solving the following quadratic inequality we remark that: a 2 ´2λpt 0 q `p2 ´A2 qλpt 0 q 2 ą 0 if λpt 0 q ą 1 `a1 ´a2 p2 ´A2 q 2 ´A2 ,

Inconsistency in the case of non invariant distance under the group action 115

Besides, as in section 4.3.3.2 we can take a lower bound of λpt 0 q by decomposing Y " Φ ¨t0 `σε and applying Cauchy-Schwarz inequality xΦ ¨t0 , g ¨t0 y ě ´A2 }t 0 } 2 , we get:

λpt 0 q ě ´A2 `σ }t 0 } θpt 0 q. (4.27)
Thanks to Condition (4.27) and the fact that σ ą σ c we get:

λpt 0 q ě ´A2 `σ }t 0 } θpt 0 q ą 1 `a1 ´a2 p2 ´A2 q 2 ´A2
Then λpt 0 q ě 0 and Condition (4.23) is fulfilled. Thus, there is inconsistency, according to lemma 4.2. Proof of proposition 4.8. First we notice that:

}t 0 }θpt 0 q " E ˜sup gPG xg ¨t0 , εy ¸ě Epxt 0 , εyq " xt 0 , Epεqy " 0. (4.28)
In order to have θpt 0 q ą 0, first we show that there exists x P Bp0, ηq and g 0 P G such that sup gPG xg ¨t0 , xy ě xg 0 ¨t0 , xy ą xt 0 , xy .

Let g 0 P G such that g 0 ¨t0 ‰ t 0 . There are three cases to be distinguished (see figure 4.11: ' 0

' 0 ' g 0 t 0 ' t 0 'x Bp0, ηq
' g 0 t 0 't 0 'x Bp0, ηq (b) Case 2: g ¨t0 is pro- portional to t0 with a factor ą 1. ' 0 ' g 0 t 0 't 0 'x Bp0, ηq (c)
Case 3: g ¨t0 is proportional to t0 with a factor ă 1.

Figure 4.11: Representation of the three cases, on each we can find an x in the support of the noise such as xx, g 0 ¨t0 y ą xx, t 0 y and by continuity of the dot product xε, g 0 ¨t0 y ą xε, t 0 y with is an event with a non zero probability, (for instance the ball in gray). This is enough in order to show that θpt 0 q ą 0.

1. The vectors g 0 ¨t0 and t 0 are linearly independent. In this case t K 0 Ć pg 0 ¨t0 q K , then we can find x P t K 0 and x R pg ¨t0 q K . Then xt 0 , xy " 0 and xg ¨t0 , xy ‰ 0, without loss of generality we can assume that xg ¨t0 , xy ą 0 (replacing x by ´x if necessary). We also can assume that x P Bp0, ηq (replacing x by xη 2}x} if necessary). Then we have x P Bp0, ηq and:

xg 0 ¨t0 , xy ą 0 " xt 0 , xy .
2. If g 0 ¨t0 " wt 0 with w ą 1, we take x " η 2}t 0 } t 0 P Bp0, ηq and we have:

xg ¨t0 , xy " w η 2 }t 0 } ą η 2 }t 0 } " xt 0 , xy .
3. If g 0 ¨t0 " wt 0 with w ă 1 we take x " ´η 2}t 0 } t 0 P Bp0, ηq and we have:

xg 0 ¨t0 , xy " ´w η 2 }t 0 } ą ´η 2 }t 0 } " xt 0 , xy .
In all these cases, we can find x such that xg 0 ¨t0 , xy ą xt 0 , xy By continuity there exists r ą 0 such that for all y on this ball we have xg ¨t0 , yy ą xt 0 , yy. Then the event tsup gPG xg ¨t0 , εy ą xt 0 , εyu has non zero probability, since x is in the support of ε we have Ppε P Bpx, rqq ą 0. Thus Inequality in (4.28) will be strict. This proves that θpt 0 q ą 0.

4.3.5

Example of a template estimation which is consistent

In order to underline the importance of the hypotheses, we give an example where the method is consistent:

Example 4.6 (affine action). Let M be a Hilbert space and V a closed sub-linear space of M (see figure 4.12). Then G " V acts on M by:

pv, mq P G ˆM Þ Ñ m `v. This action is not isometric, indeed m Þ Ñ m `v is not linear (except if v " 0).
However the distance is invariant under this group action (}v ¨m ´v ¨n} " }m ´v}), let us consider V K the orthogonal space of V . The variance in the quotient space is:

F pmq " E ˆinf vPV }m `v ´Y } 2 ˙" Ep}ppmq ´ppY q} 2 q " Ep}ppmq ´ppt 0 q `ε} 2 q,
where p : M Ñ V K the orthogonal projection on V K . Then it is clear that t 0 minimizes F .

The map rxs Þ Ñ ppxq will be called an congruent section in section 5.4.1. Hence, is there a contradiction with proposition 4.6 or proposition 4.7 which prove inconsistency as soon as the noise level is large enough? In proposition 4.6, we require that there is a subgroup acting isometrically, in this example the only element which acts linearly is the identity element m Þ Ñ m `0, then H " t0u is the only possibility, however the support of the noise should not be included in the set of fixed point under the group action of H. Here, all points are fixed under H, hence it is not possible to fulfill this condition. Example 4.6 is not a contradiction with proposition 4.6, it is also not a contradiction with proposition 4.7 since it does not act linearly on data.

4.3. Inconsistency in the case of non invariant distance under the group action 117

V K V ' x ' y ' ppxq ' ppyq d Q prxs, rysq rxs rys Figure 4
.12: In the case of vertical translation by vectors of V , the orbits are affine subspaces parallel to V . The distance between two orbits rxs and rys is given by the distance between the orthogonal projection of x and y in V K . This is an example where the template estimation is consistent.

Inconsistency with non invariant action and regularization

In practice people add a regularization term in the function they minimize in LD-DMM [Beg 2005, Durrleman 2014], or in Demons [START_REF] Lombaert | Spectral Log-Demons: Diffeomorphic Image Registration with Very Large Deformations[END_REF]] etc. Because, if one considers two points, one does not want necessarily to fit one with the other. Indeed, even if one deformation matches exactly these two points, it may be an unrealistic deformation. So far, we did not study the use of such a term in the inconsistency.

Case of deformations close to the identity element of G

If we suppose that the deformations Φ of the template is closed to identity, it is useless to take the infimum over G because G contains big deformations. Perhaps one of these big deformations can reache the infimum in F , but this element is not the one which deforme the template in the generative model. Then such big deformations should not be taken into account. That is why, if we suppose that G can be equipped with a distance d G , then we can assume that there exists r ą 0 such that the deformation Φ belongs almost surely to B " Bpe, rq " tg P G, d G pe, gq ă ru.

Instead of defining F pmq as Epinf gPG }g ¨m ´Y } 2 q, one can define F pmq "

Epinf gPB }g ¨m ´Y } 2 q, and the previous proofs will still be true, when replacing for instance λpt 0 q by λpt 0 q " 1 }t 0 } 2 Epsup gPB xg ¨t0 , Y yq etc. Likewise we need to replace the hypothesis "the support of ε is not included in the set of fixed points" by "the support of ε in not included is the set of fixed points under the action restricted to B.

Note that restraining ourselves to B is equivalent to add a following regularization on the function F :

F pmq " E ˆinf gPG }g ¨m ´Y } 2 `Regpgq ˙with Regpgq " " 0 if g P B `8 if g R B .
Moreover considering only the elements in B will automatically satisfy the condition A ă ? 2 in proposition 4.7 as long as the group G acts continuously on the template, if r is small enough. 4.3.6.2 Inconsistency in the case of a group acting linearly with a bounded regularization

In this section we suppose that the group G acts linearly. We also suppose that A ă ? 2. The regularization term is a bounded map Reg : G Ñ r0, Ωs. With this framework, we still able to prove that there is inconsistency as soon as the noise level is large enough: Proposition 4.9. Let G be a group acting linearly on M . We suppose that the orbit of the template t 0 is bounded with A " sup

gPG }g¨t 0 } }t 0 } ă ?
2, the generative model is still

Y " Φ ¨t0 `σε. We define the pre-variance as:

F pmq " E ˆinf gPG `}Y ´g ¨m} 2 `Regpgq ˘˙.
Then as soon as the noise level is large enough, i.e.:

σ ą σ c " }t 0 } θpt 0 q » -A 2 `1 `b1 ´pa 2 `Ω }t 0 } 2 qp2 ´A2 q 2 ´A2 fi fl .
Then t 0 is not a minimizser of F .

The proof is exactly the same as the Proof of proposition 4.7, we take 0 as a lower bound of the the regularization term in the lower bound of F pt 0 q, and we take Ω as a upper bound of the regularization term in the upper bound of F pλpt 0 qt 0 q. We solve a similar quadratic equation in order to find the critical σ.

Conclusion and discussion

We provided an asymptotic behavior of the consistency bias when the noise level σ tends to infinity in the case of isometric action. As a consequence, the inconsistency can not be neglected when σ is large. When the action is no longer isometric, inconsistency has been also shown when the noise level is large. However we have not answered this question: can the inconsistency be neglected? When the noise level is small enough, then the consistency bias is small [START_REF] Miolane | [END_REF]] or chapter 3, hence it can be neglected. Note that the quotient space is not a manifold, this prevents us to use a priori the Central Limit theorem for manifold proved in [Bhattacharya 2008]. But if the Central Limit theorem could be applied to quotient space, the fluctuations induces an error which would be approximately equal to σ ? I and if K ! 1 ? I , then the inconsistency could be neglected because it is small compared to fluctuation. One way to avoid the inconsistency is to use another framework, for a instance a Bayesian paradigm [Cheng 2016].

In the numerical experiments we presented, we have seen that the estimated template is more crispy that the true template. The intuition is that the estimated template in computational anatomy with a group of diffeomorphisms is also more detailed. But the true template is almost always unknown. It is then possible that one think that the computation of the template succeeded to capture small details of the template while it is just an artifact due to the inconsistency. Moreover in order to tackle this question, one needs to have a good modelisation of the noise, for instance in [Kurtek 2011b], the observations are curves, what is a relevant noise in the space of curves?

In this chapter, we have considered actions which do not let the distance invariant. Although we have only shown the inconsistency as soon as the noise level is large enough, the inequality used where not optimal at all, surely future works could improve this work and prove that inconsistency appears for small noise level. Moreover a quantification of the inconsistency should be established.

Introduction

Let us make a brief overview of what we did so far. In chapter 3, we established inconsistency for isometric action in Hilbert space. In chapter 4, we provided an asymptotic behaviour of the consistency bias when the noise level goes to infinity also for isometric action in Hilbert space. Besides, we extended the study of the consistency for non isometric action in Hilbert space. As a result, we saw that the inconsistency also appears when the noise level was large enough. Therefore, our previous results have two flaws. The first one is that those results say nothing in the case of small noise level for non isometric action. The second one is that our results are restricted to ambient spaces which are assumed to be Hilbert spaces. But ambient spaces are not always Hilbert spaces. This chapter gathers some results which complete our previous study, to solve the two flaws mentioned below. In this chapter, we consider only backward estimation.

In section 5.2, we concentrate on compact and continuous group action. We find an implicit expression of an element which minimizes the variance/pre-variance in the quotient space which is used this implicit expression in order to prove inconsistency in the Hilbert space in the case of a invariant distance under the group action. Moreover this work can be generalized to Riemannian manifolds contrary to most of our previous works. Section 5.3 is a conjecture to generalize the presence of inconsistency when data belong to ambient space which can be metric spaces (including infinite dimensional manifold) restricted to compact group action.

In section 5.4, we discuss the existence of a section of the quotient which satisfies geometrical properties. The first property is that the section is a congruent map between the quotient and the ambient space. When such a section exists, the quotient space can be embedded into the ambient space. The section is said to be congruent, when the distance is preserved though the section. Moreover we can find an explicit equation of the Fréchet mean in the quotient space. However, we advocate that the congruent section does not exists often. We give some examples and we provide an algorithm which is able to dismiss the existence of a congruent section. The second property is that the section is a measurable map. The existence of a measurable map which registers elements with respect to a point is a cornerstone in order to complete the proofs of Theorems seen in section 5.2.

In section 5.5 we prove that the template estimation can be inconsistent even for small σ for non isometric action which are a perturbation of an isometric action.

Implicit equation of an element which minimizes the

variance/pre-variance and proofs of inconsistency

In this section, we consider a compact group. We also assume that the group acts continuously on our ambient space. This implies that the registration between two points has always a solution (non necessarily unique). We exhibit an implicit equation of an element which minimizes the variance/pre-variance in the quotient space. We use this expression to show inconsistency for isometric action in Hilbert space, but also in more general spaces as complete Riemannian manifolds.

One difficulty in all this thesis is that we do not have an explicit equation of the Fréchet mean in the quotient space. Therefore, proving the consistency or the inconsistency cannot be reduced to show that the template is a solution of this possible explicit equation. In order to overcome this difficulty, we have worked with inequalities in chapter 4 and we have studied the variance restricted to an halfline in chapter 3 (theorem 3.4). But these two strategies were based on avoiding the difficulty. On the contrary in theorem 3.1, we have found the gradient of the variance in the quotient space. Solving this gradient equal to zero gives an implicit equation of the Fréchet mean in the quotient space. In this section, we aim to do the same: we want to have an implicit (or even better an explicit) equation of the elements which minimize the variance in the quotient space.

In the following, we will use the theoretical advantages of the backward estimation method: minimizing F pmq " E ˆinf gPG }m ´g ¨Y } 2 q ˙. This estimation differs from the forward estimation method namely minimizing F pmq " E ˆinf gPG }g ¨m ´Y } 2 ˙, used in section 4.3. However, these two estimations are the same in the case of a invariant distance under the group action. Note that both estimation methods are used in practice, see [Glaunes 2006, Joshi 2004[START_REF] Du | [END_REF][START_REF] Glasbey | A penalized likelihood approach to image warping[END_REF]] for instance.

Implicit equation of an element which minimizes the variance/pre-variance in quotient spaces

Let M be our ambient space; we just assume that M is a metric space, with d M its metric. Let G be a group acting on M and Y be a variable in M . We define the pre-variance of Y in the quotient space as:

F pmq " E ˆinf gPG d M pm, g ¨Y q 2
˙.

If d M is invariant under the group action, then we say that F is the variance in the quotient space For the moment, we make no assumption of the random variable Y , except that Epd M pm, Y q 2 q ă `8 for at least one m in M . Therefore F is well defined, thanks to the triangular inequality. We assume the existence of a minimum of the pre-variance (otherwise the template estimation is necessarily inconsistent). We denote by m ‹ one minimizer of 5.2. Implicit equation of an element which minimizes the variance/pre-variance and proofs of inconsistency 123 the pre-variance F . We assume that the group G is compact and that the action is continuous (namely g Þ Ñ g ¨x is a continuous map for all x P M ). Therefore the registration of Y with respect to m ‹ does exist. Then it exists gpY, m ‹ q P G such that: F pm ‹ q " Epd 2 M pm ‹ , gpY, m ‹ q ¨Y qq. We define by E the variance of gpY, m ‹ q ¨Y : Epmq " Epd 2 M pm, gpY, m ‹ q ¨Y qq.

For Z a random variable in M , we define FMpZq as the set of all Fréchet means of Z:

FMpZq " argmin mPM Epd 2 M pm, Zqq.

Proposition 5.1. Let pM, d M q be a metric space, G a compact group acting continuously on M . We assume that the variance/pre-variance, defined by

F pmq " E ˆinf gPG d M pm, g ¨Y q 2 ˙for m P M,
reaches its minimum a point m ‹ . We assume that there exits y Þ Ñ gpy, m ‹ q ¨y a deterministic measurable function such that:

@y P M gpy, m ‹ q P argmin gPG d M pm ‹ , g ¨yq.

Then, the set FMpgpY, m ‹ q ¨Y q is non empty, besides we have:

m ‹ P FMpgpY, m ‹ q ¨Y q (5.1)

Remark 5.1. Note that we do not need the distance to be invariant under the group action. However it requires a backward estimation. It requires also a compact group contrarily to previous results as theorem 3.4 for instance.

When the ambient space M is a Hilbert space and Z a squarred integrable variable, FMpZq exists, and is unique and equals EpZq (proposition 1.4), in this case we can conclude that: m ‹ " EpgpY, m ‹ q ¨Y q.

(5.2)

Note that equation (5.2) was already obtained in the case of the finite and isometric group action (see equation (3.8)). However, this was done by differentiating the variance, which is not possible everywhere as we have already see it (section 3.4.7).

In the case of a finite sample Y 1 , . . . , Y n , we define the empirical pre-variance by:

F n pmq " 1 n n ÿ i"1 inf gPG }m ´g ¨Yi } 2
if m ‹ is a minimizer of the empirical pre-variance F n , we can also deduce that:

m ‹ " 1 n n ÿ i"1 gpY, m ‹ q ¨Y
Proof of proposition 5.1. We have two things to prove: FMpgpY, m ‹ q ¨Y q is not empty as a first step and m ‹ P FMpgpY, m ‹ q ¨Y q as a second step.

• First, let us prove that the set FMpgpY, m ‹ q ¨Y q is not empty. If FMpgpY, m ‹ q ¨Y q was empty, then in particular, m ‹ would not minimize E the variance of gpY, m ‹ q¨Y . Therefore, we can find z P M such that Epzq ă Epm ‹ q in this case we have:

F pm ‹ q " Epm ‹ q ą Epzq ą E ˆinf gPG d 2 M pz, g ¨Y q ą F pzq,
which is a contradiction, since m ‹ minimizes F .

• Secondly, let us take z P FMpgpY, m ‹ q ¨Y q, we get:

F pm ‹ q " E ˆinf gPG d 2 M pg ¨m, Y q " Epm ‹ q ě Epzq " Epd 2 M pz, gpY, m ‹ q ¨Y qq ě E ˆinf gPG d 2 M pz, g ¨Y q ě F pzq ě F pm ‹ q,
Then we conclude that Epm ‹ q " Epzq " min E, then:

m ‹ P FMpgpY, m ‹ q ¨Y q.
It may seem odd, that we get an implicit equation of m ‹ which depends on the choice we made on the element gpY, m ‹ q of G which registers Y with respect to m ‹ . Indeed, there is a priori no uniqueness of the element which register Y to m ‹ . What if there are such two elements gpY, m ‹ q and gpY, m ‹ q? The following proposition prove that gpY, m ‹ q ¨Y " gpY, m ‹ q ¨Y almost surely. In other words, if there is no uniqueness in the registration of Y to m, then the choice of the element gpY, m ‹ q, has no consequence: 5.2. Implicit equation of an element which minimizes the variance/pre-variance and proofs of inconsistency 125 Proposition 5.2. Let G be a compact group acting continuously on M a Hilbert space, let Y be a random variable, and m ‹ an element which minimizes the prevariance (defined in section 5.2.1). We assume that there are two random variables gpY, m ‹ q and gpY, m ‹ q which minimize d M pm ‹ , g ¨Y q. In this case we have almost surely: gpY, m ‹ q ¨Y " gpY, m ‹ q ¨Y

Proof. Applying equation ( 5.2) to gpY, m ‹ q and gpY, m ‹ q leads to:

m ‹ " EpgpY, m ‹ q ¨Y q " EpgpY, m ‹ q ¨Y q.

Now we mix g and g: let us choose A a measurable set, and define the following random variable:

Z " gpY, m ‹ q ¨Y 1 Y PA `gpY, m ‹ q ¨Y 1 Y RA .
For every Y , Z is an element which reach the minimum in d M pm ‹ , yq for y P rY s.

Once again, we can apply equation ( 5.2) to Z, we get m ‹ " EpZq, we can deduce that:

m ‹ " E pgpY, m ‹ q ¨Y 1 Y PA `gpY, m ‹ q ¨Y 1 Y RA q " E pgpY, m ‹ q ¨Y q .
By splitting the second expectation into two parts and simplifying, we have:

EpgpY, m ‹ q ¨Y 1 Y PA q " EpgpY, m ‹ q ¨Y 1 Y PA q,
and we do it for any measurable set A, which yields gpY, m ‹ q ¨Y " gpY, m ‹ q ¨Y, Y -almost surely.

We can generalize the uniqueness of the registration variable gpY, m ‹ q ¨Y in the case of complete Riemannian manifolds: Proposition 5.3. Let G be a compact group acting continuously on M a complete Riemannian manifold, let Y be a random variable and m ‹ an element which minimizes the pre-variance. We assume that gpY, m ‹ q P argmin gPG d M pm ‹ , g ¨Y q and gpY, m ‹ q P argmin gPG d M pm ‹ , g ¨Y q, for gpY, m ‹ q and gpY, m ‹ q two random variables in G. We assume that PpgpY, m ‹ q ¨Y P Cpm ‹ qq " PpgpY, m ‹ q ¨Y P Cpm ‹ qq " 0.

Where Cpm ‹ q is the cut locus of m ‹ . In this case we have: gpY, m ‹ q ¨Y " gpY, m ‹ q ¨Y almost surely.

Proof. Applying equation ( 5.1) to gpY, m ‹ q and gpY, m ‹ q leads to: m ‹ " FMpgpY, m ‹ q ¨Y q " FMpgpY, m ‹ q ¨Y q. Now we can mix g and g: let us choose A a measurable set, and define:

Z " gpY, m ‹ q ¨Y 1 Y PA `gpY, m ‹ q ¨Y 1 Y RA ,
The random variable Z reaches the infimum which is in the definition of F pm ‹ q. Then from equation ( 5.1), we get m ‹ " FMpZq. By differentiation of the variance at m ‹ (because of the probability to fall in the cut locus is zero), we have:

EpLog m‹ gpY, m ‹ q ¨Y 1 Y PA q " EpLog m‹ gpY, m ‹ q ¨Y 1 Y PA q,
and we do it for any measurable sets which yields Log m‹ gpY, m ‹ q ¨Y " Log m‹ gpY, m ‹ q ¨Y, Y -almost surely.

This proves that gpY, m ‹ q ¨Y " gpY, m ‹ q ¨Y , Y -almost surely.

5.2.2

Interpretation of equation ( 5.2) in Hilbert spaces

' rm ‹ s ' rY s 'Qlog m‹ pY q Q M Figure 5
.1: Representation of the quotient as if it was a manifold, with M the tangent plane. If rm ‹ s is a Fréchet mean of rY s then EpQlog m‹ pY qq " 0, where Qlog is a quotient logarithm function.

In the previous subsection, we have found an implicit equation of the elements which minimize the pre-variance in the quotient spaces. Can we explain this equation? First, we suppose that the ambient space is an Hilbert space. We can rewrite equation (5.2) as follows:

EpgpY, m ‹ q ¨Y ´m‹ q " 0.

Then, if we define a quotient logarithm function as: Qlog m‹ pyq " gpy, m ‹ q ¨y ´m‹ .

Implicit equation of an element which minimizes the variance/pre-variance and proofs of inconsistency 127

We have the equation EpQlog m‹ pY qq " 0 in M . Then even if Q is not a manifold and M is not the tangent plane of Q at the point m ‹ , equation ( 5.2) can be reinterpreted as the equation of a exponential barycenter: EpQlog m‹ pY qq " 0 when m ‹ is one element which minimizes the pre-variance (see figure 5.1). This equation looks like the one find for the Fréchet mean in a manifold, when Fréchet mean are, in particular, exponential barycenter (see definition 1.5).

Interpretation of equation (5.1) in complete Riemannian manifolds

Now, we want to find also an explanation of equation ( 5.1) when the ambient space is a complete Riemannian manifold. Contrarily to Hilbert space, there is no easy expression of FMpZq for Z a random variable in M when M is a manifold. This may complicate our analysis of equation ( 5.1). However we can adapt our previous interpretation. We call Qlog of Y the element gpY, m ‹ q ¨Y , then we have m ‹ " FMpQlog m‹ pY q. Therefore m ‹ is a critical point of the variance of Qlog m‹ pY q. By differentiating the variance of Qlog m‹ pY q (defined as m Þ Ñ Epd 2 pm, Qlog m‹ pY qqq (see lemma 3.9) we have that:

E `Log m‹ pQlog m‹ pY qq ˘" 0.

In other words, m ‹ is a double exponential barycenter, since it is the expectation of the log of the log which is equal to zero (see figure 5.2).

' rm ‹ s ' rY s ' gpY, m ‹ q ¨Y 'Log m‹ pgpY, m ‹ q ¨Y q Q M T m‹ M Figure 5
.2: Representation of the quotient, M acts as an intermediary between the quotient space and the tangent plan at m ‹ . If m ‹ is a minimizer of the pre-variance in the quotient space, then EpLog m‹ pQlog m‹ qY q " 0. Where Qlog m‹ pY q is the registration of Y to m ‹ and Log m‹ is the Riemannian logarithm map.

Note that this interpretation has been done without differentiating the prevariance in the quotient space. Because, we have already seen that it may not be possible. Moreover gpY, m ‹ q may be not unique, this is an analogy with the cut locus issue in manifolds. In manifold when points are in the cut locus, the differentiability fails because, for instance, of the non uniqueness of the geodesics. However, the non uniqueness in this case, is not really an issue, indeed if gpY, m ‹ q ¨Y and gpY, m ‹ q ¨Y are two choices in order to register Y with respect to m ‹ , then EpgpY, m ‹ q ¨Y q " EpgpY, m ‹ q ¨Y q " m ‹ for any measurable choice of gpY, m ‹ q. Therefore, we have avoided the difficulty of differentiating the pre-variance in the quotient space. Without differentiating the pre-variance, we have found that if m ‹ minimizes the pre-variance, then m ‹ satisfies a certain implicit equation, which is an analogy with the implicit equation ∇F pm ‹ q " 0 found in section 3.3 for finite group.

Remark 5.2. Let f be the function defined by: f pmq P FMpgpY, mq ¨Y q (5.3)

then starting from a point m 0 , we define the sequence m n`1 " f pm n q. If this sequence is well defined and converge to a point m, and if f is continuous at m, this point will satisfied: m P FMpgpY, mq ¨Y q, in other words the limit of this sequence is a good candidate to be a Fréchet mean of rY s. This sequence is nothing else than the max-max algorithm. Indeed, we have seen in chapter 4, that the max-max algorithm is the repetition of two steps, the first one registers data to a current point, the second one takes the mean of the registered data in order to update the current point. This is exactly what the sequence pm n q n defined by equation (5.3) does. The only difference is that the max-max algorithm seen in chapter 4 was restricted to a finite sample. Equation (5.3) deals with the whole distribution.

5.2.4

Inconsistency in Hilbert space thanks to equation (5.2) In this subsubsection we give an alternative proof of theorem 3.4 based on equation (5.2).

The first advantage of this current proof is that the point which variance is strictly smaller than the variance at the template is not proportional to the template. The second advantage is that this proof works with forward generative model contrarily to theorem 3.4 which was stated and proved for backward generative model only.

Theorem 5.1. Let M be a Hilbert space, and G a compact group acting isometrically and continuously. We assume a forward model: Y " Φ ¨t0 `σε, where ε is a standardized noise pEpεq " 0 and Ep}ε} 2 q " 1) and σ ą 0. We assume that there exits y Þ Ñ gpy, t 0 q ¨y a deterministic measurable function such that: @y P M gpy, t 0 q P argmin gPG d M pt 0 , g ¨yq.

We assume that PpY R ConepΦ ¨t0 qq ą 0. Then rt 0 s is not a Fréchet mean of rY s in the quotient space.

Implicit equation of an element which minimizes the variance/pre-variance and proofs of inconsistency 129

Proof. Thanks to equation (5.2), in order to show inconsistency, all we have to do is proving that the following equation is not true: t 0 " EpgpY, t 0 q ¨Y q.

(5.4)

By a reductio ad absurdum, if we assume that this equation holds. By taking the dot product of (5.4) with t 0 , one get:

}t 0 } 2 " Epsup gPG xY, g ¨t0 yq
Indeed, because the action is isometric we have that maximizing the dot product is the same thing than minimizing the distance: xgpY, t 0 q ¨Y, t 0 y " max gPG xg ¨Y, t 0 y. By replacing g by Φ, and thanks to PpY R ConepΦ ¨t0 qq ą 0, we get:

}t 0 } 2 ą EpxY, Φ ¨t0 y " EpxΦ ¨t0 `σε, Φ ¨t0 yq " }t 0 } 2 `0.
Thus we have a contradiction, EpgpY, t 0 q ¨Y q is different of t 0 , then t 0 does not satisfy equation ( 5.2). We can conclude that t 0 does not minimize the variance in the quotient space.

This proves the inconsistency of the template estimation by computing the Fréchet mean in quotient space when the observable variable verifies a forward model.

Moreover,

we know that the variance F at the point EpgpY, t 0 q ¨Y q is strictly smaller than the variance at the template. In the proof of theorem 3.4 (for Hilbert spaces) we also found λpt 0 qt 0 , a point which variance was smaller than the variance at the template. By reading the proof of theorem 3.4, one could argue that: "If the only points which variance is smaller than the variance at the template are proportional to the template, inconsistency is not really an issue. Because the estimated template would be the real template up to a scaling.". This results exhibits an example of point with a smaller variance than at the template, which is not necessairly proportional to t 0 .

Inconsistency: variation of the isotropy group

We use equation (5.2) in order to prove inconsistency, by studying the isotropy group of the template, of the observations Y and of m ‹ . Indeed, we want to prove that the template t 0 is not equal to m ‹ the Fréchet mean of rY s. One way to prove that two quantities are not equal, is to prove that these two quantities do not share the same properties. Here we prove that t 0 and m ‹ do not share the same isotropy group. We remind that the isotropy group of a point m is the group of element g P G such that g ¨m:

Isopmq " tg P G g ¨m " mu.

Let us take a simple example. Let M " C 0 pr0, 1s, Rq be the set of continuous real fonctions defined on r0, 1s. Then the group G defined by G " tϕ : r0, 1s Ñ r0, 1s such as ϕis an increasing homeomorphismu, acts on M by: pϕ, f q Þ Ñ f ˝ϕ.

Starting from a template t 0 , there can be bijective maps which leave this template invariant: any bijection which lets every isolevel invariant, in this case Isopt 0 q ‰ 0. An isolevel associated to a fonction f defined on Ω and a real y is defined as: isolevelpf, yq " tx P Ω, such as f pxq " yu Now by adding some noise, it is likely that the observations Y with IsopY q " te G u almost surely. Then it is no longer possible to find a bijective map in G which lets every isolevel invariant except for the identity map. We give this explanation for 1D functions, but we could also explain it in 2D ou 3D. For an image, we can find many diffeomorphisms which leaves the isolevel invariant, when such isolevel are smooth enough: it is sufficient to follow the tangent at each point of the isolevel.

If we prove that m ‹ have the same property Isopm ‹ q " te G u then it will become obvious that t 0 ‰ m ‹ , in other words t 0 does not minimize the variance. Theorem 5.2. Let G be a compact group acting continuously on M . We assume that M is a Hilbert space or a complete Riemannian manifold. We suppose that the distance d M (the Hilbert norm or the Riemannian distance) is invariant under the group action G.

Let Y be a random variable in M , we assume that it exists m 0 P M such that Epd 2 M pm 0 , Y qq ă `8. Suppose that PpIsopY q " te G uq ą 0 (5.5)

Let us note m ‹ a minimizer of the variance F pmq " E ˆinf gPG d M pm, g ¨Y q 2 ˙. We assume that it exists a deterministic and measurable function y Þ Ñ gpy, m ‹ q such that:

5.2. Implicit equation of an element which minimizes the variance/pre-variance and proofs of inconsistency 131 gpy, m ‹ q P argmin gPG d M pm ‹ , g ¨yq.

In the case of a complete Riemannian manifold M , we make the extra assumption that PpgpY, m ‹ q ¨Y P Cpm ‹ qq " 0, where Cpm ‹ q is the cut locus of m ‹ .

Then m ‹ satisfies Isopm ‹ q " te G u.

Corollary 5.1. Under the same hypotheses as theorem 5.2, we take a template t 0 such that Isopt 0 q ‰ te G u. Then we create an observable variable Y from this template, (for instance in a Hilbert space Y " Φpt 0 `σεq `σ1 ε 1 : a mixture of backward/forward model). Then the template t 0 is not a minimizer of the variance in the quotient space (since Isopt 0 q ‰ te G u " Isopm ‹ q for any m ‹ minimizer of the variance.

We can understand corollary 5.1 as follows: m ‹ , obtained by minimization of the variance, looses all symmetry due to the noise. In a sense we can state that m ‹ looks like Y and do not look like t 0 ; m ‹ is noisy.

Note that Huckemann [START_REF] Huckemann | On the meaning of mean shape: manifold stability, locus and the two sample test[END_REF]] has already proved that the Fréchet mean never lie on a singular orbit, in finite dimensional Riemannian manifolds for Lie group proper action (stability theorem). However, the proof we give here, works also in infinite dimensional Hilbert spaces. It is our understanding that the proof of Huckemann relies, for instance, on the fact that the set of singular points in the quotient space is a null set. Such results are not that obvious to generalize in infinite dimensional spaces. However, we should point out that Huckemann proves the measurability of the congruent section. This last result may be generalizable in our current setting.

Proof of theorem 5.2. The detail of the proof is based on these two steps:

• If Isopmq ‰ te G u then it exist h P Gzte G u such that h ¨m " m. We have:

F pm ‹ q " Epd M pm ‹ , gpY, m ‹ q ¨Y qq 2 q.
As m ‹ " h ¨m‹ , we also have m ‹ " h ´1 ¨m‹ . Thanks to the invariant distance:

d M pm ‹ , gpY, m ‹ q ¨Y q " d M ph ´1 ¨m‹ , gpY, m ‹ q ¨Y q " d M pm ‹ , hgpY, m ‹ q ¨Y q.
Therefore, hgpY, m ‹ q also reaches the infimum in the computation of F pm ‹ q, which proves that we have two different ways for reaching this minimum.

• Thanks to propositions 5.2 and 5.3, we get a relation between the two registration variables which reach the infimum:

gpY, m ‹ q ¨Y " hgpY, m ‹ q ¨Y Y ´almost surely gpY, m ‹ q ´1hgpY, m ‹ q ¨Y " Y Y ´almost surely gpY, m ‹ q ´1hgpY, m ‹ q P IsopY q Y ´almost surely gpY, m ‹ q ´1hgpY, m ‹ q " e G with a non zero probability (5.5) h " e G with a non zero probability This is a contradiction, because we supposed h ‰ e G .

Conclusion: Isopm ‹ q " te G u. Thus t 0 ‰ m ‹ , the estimation of the template is inconsistent. Remark 5.3. We also need to prove that gpY, m ‹ q ¨Y is a measurable variable, if not m Þ Ñ Epd 2 M pm, gpY, m ‹ q ¨Y qq could have no sense. At this moment, we do not have a proof that gpY, m ‹ q ¨Y is a measurable variable. This technical point will be discussed in section 5.4.3.

Remark 5.4. When M is a Hilbert space, and when G " V is a closed linear subspace of M acting by vertical translation, we have seen that the template estimation is consistent. However, d M is invariant under the group action. Therefore, is there a contradiction with corollary 5.1? First G is not a compact set, but as previously said, this is not a problem, indeed the compacity was used in order to have an element in the group which reaches the infimum in }m ´g ¨n}. In the affine action, we have the existence of such g P G " V : this is ppmq ´ppnq, where p is the orthogonal projection into V .

For this action, we have Isopmq " te G u for every m P M . Therefore on this example, it is not possible to choose a template t 0 with Isopt 0 q ‰ te G u, this example is not a counter example of corollary 5.1.

Towards an extension to other spaces

We previously assumed that the ambient space was a Hilbert or a complete Riemannian manifold only to have this result: if Z and Z 1 are two random variables in M such that the Fréchet mean of Z1 A `Z1 1 A c does not depend of the measurable set A, then Z " Z 1 almost surely. Here we noted A c the complementary of the measurable set A. In other words, if all possible mixture of Z and Z 1 have the same Fréchet mean, then Z " Z 1 . This leads us to the following definition.

Definition 5.1. Let pM, d M q a metric space, we say that M is a space with good mixtures if we have the following property: where L 2 pΩ, M q is the set of all random variable which takes value in M and such that Epd 2 M pm, Xqq ă `8 for one m P M (and thus for all m P M by triangular inequality).

@Z, Z 1 P L 2 pΩ, M q " @A measurable FMpZ1 A `Z1 1 A c q " FMpZq ‰ ùñ Z " Z 1 a.s., 5
Theorem 5.2 can be immediately generalized in any spaces with good mixture. Can we exhibit other spaces with good mixtures different from Hilbert spaces, for instance metric spaces?

5.3 Conjecture of inconsistency for metric space with non invariant distance under the group action.

This section is an attempt to generalize the proof of inconsistency in spaces which are not Hilbert spaces anymore but just metric spaces. We do not suppose that the distance in the ambient space is invariant under the group action. Although we do not give a proof, we provide an intuition that it will not be consistent either. We state two conjectures: the first one is dedicated to metric space when no regularization is added. On the contrary, the second one is a generalization when a regularization term is added. Working in a metric space with a regularization term is the real framework of applications.

Conjecture 1: in metric space without regularization

Let pM, d M q be a metric space, G a compact group acting continuously. We do not suppose that the distance d M is invariant under the group action. In particular, inf gPG d M px, g ¨yq will not define a distance in the quotient space.

In this section, we consider the backward model: Y " Φ ¨X where Φ is a random variable in G, X is a random variable defined as a noisy version of the template: the template is t 0 , the unique Fréchet mean of X:

t 0 " argmin mPM Epd 2 M pm, Xqq.
We also assume that Φ and X are independent random variables. We are interested in the template estimation given the observation Y by the minimization of the pre-variance:

F pmq " E ˆinf gPG d 2 M pm, g ¨Y q ˙.
Because X " Φ ¨Y we have also:

F pmq " E ˆinf gPG d 2 M pm, g ¨Xq ˙.
We use the term pre-variance, because it is not a variance (because we do not have a quotient distance). As the group is compact and the action is continuous inf gPG d M pm, g ¨Xq is reached for some g P G; we note gpX, t 0 q one element which reaches the infimum. Therefore, F pt 0 q " Epd 2 M pt 0 , gpX, t 0 q ¨Xqq. We define the variance of the random variable gpX, t 0 q ¨X: m Þ Ñ Epmq " Epd 2 M pm, gpX, t 0 q ¨Xqq. We define the Voronoï cell associated to the point x defined as the set of elements closer to x than to all the point in the orbit of x:

VCpxq " tt P M s.t. @g P G d M px, tq ď d M pg ¨x, tqu.

Conjecture 5.1. Let pM, d M q be a metric space, let G be a compact group acting continuously on M . We assume that a point t 0 is the unique Fréchet mean of X. Let us suppose that the template t 0 does not always belong to VCpXq:

Ppt 0 R VCpXqq ą 0.
In this case, we think that t 0 is not a minimizer of the pre-variance F .

Unfortunately, this conjecture is false, indeed, there is a counter example: the action of vertical translation pv, mq Þ Ñ v `m is a counter example for m P M (M a Hilbert space), and v P V (V is a closed linear sub-space. It may be possible to fix this conjecture by adding an hypothesis which excludes this counter-example.

This hypothesis is very similar to the one in theorems 3.1 and 3.4 (the theorems of inconsistency in chapter 3). Indeed in these theorems, the condition was PpX R Conept 0 qq ą 0, where Conept 0 q " VCpt 0 q was the Voronoï cell associated to t 0 , (this cell was a cone, due to the isometric action). In theorems 3.1 and 3.4, the action was isometric, and this condition could be also written Ppt 0 R VCpXqq ą 0. Therefore in conjecture 5.1, we have the same condition, however, as the space is no longer a linear space, the voronoï cell associated to X is no longer a cone.

Let us detail why we think that this conjecture may be true:

Proposition 5.4. There are two cases:

• If gpX, t 0 q ¨X does not have a Fréchet mean in M , then t 0 does not minimize F : the template estimation is inconsistent.

• If z is one Fréchet mean of gpX, t 0 q ¨X, then F pzq ď F pt 0 q. Therefore z is a good candidate to show that t 0 does not minimize F .

Proof of proposition 5.4.

• Let us prove the first point: if gpX, t 0 q ¨X does not have a Fréchet mean in M , this means in particularly that t 0 does not minimize E. In this case, it exists z 1 P M such that Epz 1 q ă Ept 0 q. Then once again:

F pt 0 q " Ept 0 q ą Epz 1 q ą Epd 2 M pz 1 , gpX, t 0 q ¨Xqq ą E ˆinf gPG d 2 M pz 1 , g ¨Xq ą F pz 1 q, 5.
3. Conjecture of inconsistency for metric space with non invariant distance under the group action. 135

• For the second point, if z is one Fréchet mean of gpX, t 0 q ¨X, then:

F pt 0 q " E ˆinf gPG d 2 M pt 0 , g ¨Xq " Ept 0 q ě Epzq ě Epd 2 M pz, gpX, t 0 q ¨Xqq ě E ˆinf gPG d 2 M pz, g ¨Xq ě F pzq,

Is

it possible that t 0 P FMpgpX, t 0 q ¨Xq ? In this case t 0 would minimize two functions:

m Þ Ñ Epd 2 M pm, Xqq and m Þ Ñ E 2 pd 2 M pm, gpX, t 0 q ¨Xqq.
In the case where gpX, t 0 q " e G X-a.s. It is obvious that t 0 minimizes these two functions. This would mean that t 0 belongs almost surely in the Voronoï Cone of X. Therefore, as we have assumed that it is not the case, we can conclude that gpX, t 0 q ‰ e G with a non zero probability. Therefore, the only thing remaining to prove that t 0 does not minimize the pre-variance, is proving that z ‰ t 0 , for every z P FMpgpX, t 0 q ¨Xq. Maybe, this can be done by adding extra assumptions. The difficulty, in order to finish the proof of this conjecture, is that we are not able to compute the minimum of the variance E.

Note that we require a compact group acting continuously so that the registration problem with respect to the template has at least a solution. Therefore an immediate extension for non compact group is when the registration problem has at least a solution.

Remark 5.5. As in section 5.2.5, we also need to prove that gpX, t 0 q ¨X is a measurable variable, if not m Þ Ñ Epd 2 M pm, gpX, t 0 q ¨Xqq could have no sense. At this point, we do not have a proof that gpX, t 0 q¨X is a measurable variable. This technical point will be discussed in section 5.4.3.

Conjecture 2: metric space with regularization

Let pM, d M q be a metric space, G a compact group acting continuously. We take Y " Φ ¨X where X is a noisy template: the template is t 0 the unique Fréchet mean of X:

t 0 " argmin mPM Epd 2 M pm, Xqq.
This means that the template t 0 belongs almost surely to VC reg pX, Φq, which is exactly the hypothesis we excluded. The only thing we need to prove is that t 0 is not a Fréchet mean of gpY, t 0 q ¨Y . Because in this case, for z P FMpgpY, t 0 q ¨Y q, we would have F pzq ă F pt 0 q proving the inconsistency. The difficulty in order to prove that t 0 does not minimize FMpgpY, t 0 q ¨Y q is that we do not know gpY, t 0 q therefore it hard to know if t 0 minimize FMpgpY, t 0 q ¨Y q.

Congruent and measurable sections

In this section, we discuss the existence of a section of the quotient space satisfying a certain property (being congruent or being measurable). First, a section of the quotient space is a map which associate at each orbit an element of this orbit. Note that a section always exists: it suffices, for each orbit, to chose one element in the orbit (with the axiom of choice).

In sections 5.4.1 and 5.4.2, we discuss the existence of congruent section, this means that this section leaves the distance invariant. We see that if such a section exists, then computing the Fréchet mean in the quotient space, and the consistency bias is straightforward. For instance, for the group of rotations acting on R n a congruent section exists, and this way, we are able to have an explicit formula of the consistency bias. Unfortunately, we show that a congruent section does not always exists.

In section 5.4.3, we discuss the existence of a measurable section such that the image of this section is included in a Voronoï cell. If a measurable section exists, then we solve the missing technical details of sections 5.2 and 5.3: we needed to prove that gpY, t 0 q ¨Y and gpY, m ‹ q ¨Y are measurable variables, Y is the observable variable, t 0 the template, and m ‹ the element which minimizes the pre-variance, and gpa, bq is one element which registers a with respect to b.

Congruent section and Computation of Fréchet Mean in Quotient Space

Let M be an Hilbert space. Given points m P M and y P M , there is a priori no closed-form expression in order to compute the quotient distance inf gPG }g ¨m ´y}.

Therefore computing and minimizing the variance in the quotient does not seem straightforward. There is one case where it may be possible: the existence of a congruent section. We say that s : Q Ñ M is a section if π ˝s " Id, where π : M Ñ Q is the canonical projection into the quotient space. Moreover we say that the section s is congruent if:

@o, o 1 P Q }spoq ´spo 1 q} " d Q po, o 1 q.
Then the image of the quotient by the section S " spQq is a part of M which has an interesting property: @p, q P S, }p ´q} " d Q prps, rqsq.

In other words, the section gives us a part of M containing a point of each orbit such that all points in S are already registered. Moreover, if s is a section, s1 : rms Þ Ñ g ¨sprmsq is also a section, without loss of generality we can assume that t 0 " sprt 0 sq. In this case, the variance is equal to:

F pmq " Ep}sprmsq ´sprY sq}2 q, where we recognize the variance of the random variable sprY sq. As we know that the element which minimizes the variance in a linear space is given by the expected value, we have that:

F pmq ě F pEpsprY sqqq.
Moreover this inequality is strict if and only if m and EpsprY sqq are not in the same orbit. Therefore, we have a method to determine if the estimation is consistent or not: computing EpsprY sqq and verifying if t 0 and EpsprY sqq are in the same orbit, and the consistency bias is given by d Q prt 0 s, rEpsprY sqqsq. Moreover if we take m P S, we have F pmq " Ep}m ´sprY sq} 2 q and it is now straightforward that the restriction of F to S, noted F | S , is differentiable 1 on S, and that ∇F | S pmq " m ´EpsprY sq. In particular }∇F | S pt 0 q} " }t 0 ´EpsprY sqq} gives us the value of the bias.

Note that in this thesis, we have already seen two examples of action with a congruent section see section 3.3.6 and example 4.6. Each time, we were able to compute the bias, as in the following example of rotations (studied in [START_REF] Miolane | [END_REF]]): This can be done for every η ą 0, therefore, we can also conclude that a local congruent section around f 1 does not exist either: because lim

ηÑ0 f 2 " lim ηÑ0 f 3 " f 1 .
Ziezold [Ziezold 1977] had already noticed that being registered (he uses the expression "optimal position" instead of registered) was not always a transitive relation in other examples.

The existence of a congruent section indicates us that the quotient space is not so complicated. Indeed when a congruent section exists, the quotient space is embedded in the ambient space with respect to the distances in the quotient space and in the ambient space. The computation of the Fréchet mean would be easier, it suffices to project data on S and to take the mean. When such a congruent section does not exist, computing the Fréchet mean in quotient space is not so easy. However, we can established proofs of inconsistency which are less tight.

Remark 5.6 (Link between congruent section and equation (5.2)). When M is a Hilbert space, equation (5.2) is an implicit equation which gives the expression of m ‹ , an element which minimizes the pre-variance in the quotient space:

m ‹ " EpgpY, m ‹ q ¨Y q.
If we knew m ‹ , we could compute gpY, m ‹ q. Therefore we could compute EpgpY, m ‹ qq and therefore we could compute m ‹ . The serpents eats it own tail. However, if there is a congruent section s, things will become much easier. Indeed let us assume that a congruent section s exists, if m ‹ minimizes F . Without loss of generality we can assume that sprm ‹ sq " m ‹ , then gpY, m ‹ q ¨Y " sprY sq does not depend on m ‹ , and equation (5.2) becomes m ‹ " EpsprY sqq: finding m ‹ is just the computation of an expected value.

' y 1 ' ´y1 ' y 2 ' ´y2 ' y 3 ' ´y3 ' y 4
(a) The four orbits in R 2 , the segments are the lines which realize the quotient distance. Therefore, if a congruent section exists, it should be possible to find 4 points in R 2 such that any two different points among these 4 points would have a distance equal to 1. Of course, this is not possible, in order to prove it, we compute the matrix A as explained above, we find:

A " 1 2 ¨2 1 1 1 2 1 1 1 2 ',
then rankpAq " 3 ą dim M . In order to build a regular tetrahedron, we need to be at least in R 3 .

Let us return to a general isometric action on a general Euclidean space. By choosing the y 1 i s randomly, if we get an example where rankpAq ą dim M , then we reject the existence the x i such that }x i ´xj } 2 " d Q pry i s, ry j sq 2 for every pi, jq P 1, n 2 . Therefore, we reject the existence of a congruent section for the chosen group action.

Note that the rank of A is very sensitive to error in the computation of the squared quotient distance. That is why it is preferable to chose some vectors y i with coordinates which are integer.

We implement this algorithm, with the action of horizontal translation in R 64 . For instance, if we take 100 random points we find that the rank of the matrix A is 99. In other words, in order to embed the quotient space R 64 {pZ{64Zq with respect to the quotient and Euclidean distances, we need at least an Euclidean space of dimension 99. But if we take more points for instance 500 we find that we need a at least an Euclidean space of dimension 499. The more we take points the larger the dimension must be. In other words it is probably not possible to embed the quotient space in an Euclidean space with respect to the quotient distance. In particular, the original ambient space of dimension 64 is not sufficient: there is no congruent section.

Remark 5.7. Here, we can notice that the Nash embedding theorem [Nash 1956] states that any Riemannian manifold can be included in R d with d P N sufficiently large. Furthermore, this embedding is cconsistent with the Riemannian metric. Then, if the embedded Riemannian manifold is totally geodesic in R d . Then the quotient distance is given by the Euclidean norm in R d . Here the situation is more complex, indeed the quotient space Q is not a manifold, and even if we remove some points in Q to obtain a Riemannian manifold Q ‹ , we do not want to embed Q ‹ into R d with d sufficiently large, we want to embed Q ‹ into M with an application s such that sprxsq P rxs.

Measurable section

We now assume that pM, d M q is a metric set, and G a compact group acting continuously on M . We have seen in sections 5.2 and 5.3 that we need to prove the existence of the measurable variables gpY, t 0 q ¨Y and gpY, m ‹ q ¨Y , where gpa, bq is one element which registers a to b (see remarks 5.3 and 5.5). In fact, in these previous sections, we have admitted the existence of such measurable variables. This technical detail matters because we need it in order to make the proof of theorem 5.2 rigorous. In this section we make the link between this problem and the existence of an measurable section. Definition 5.2. Let µ be a point in M (µ could be t 0 or m ‹ depending on the context). We say that s is a µ-measurable-section if:

• s is a section namely, s is a map; s : M {G Ñ M with π ˝s " Id (where π is the canonical projection into the quotient space).

• s is a measurable map.

• S " spM {Gq is included in the Voronoï cell of µ:

s : M {G Ñ VCpµq rxs Þ Ñ y where y P rxs .

Once again, it is easy to show that a section s : M {G Ñ VCpµq exists: for every orbit, it suffices to choose one element among the element of the orbit which are the closest to µ. As the group is compact and acts continuously, the existence of this element is ensured. Therefore, by the axiom of choice, s exits. However, nothing in this argument ensure the measurability of the section. On the contrary, using the axiom of choice is the best thing to do in order to build non measurable functions or non measurable sets.

When such a µ-measurable-section exists, then gpY, µq¨Y " spY q is a measurable variable because Y and s are measurable. Therefore in order to solve the technical problem seen in sections 5.2 and 5.3, all we have to do is reduced to prove the existence of µ-measurable-section.

If we have the existence of a congruent section, the result is straightforward: let s be a congruent section, without loss of generally we can assume that sprµsq " µ, then for every point x P S " spQq we have d Q prxs, rµsq " d M px, µq, then x P VCpµq therefore S Ă VCpµq, then s : Q Ñ VCpµq is measurable since continuous since congruent. Unfortunately, we have already seen that a congruent section does not always exists.

Fortunately, being measurable is much weaker assumption that being congruent. Then, it should be possible to show that a µ-measurable-section exists even if there is no congruent section.

Inconsistency for non isometric action by perturbation of an isometric group action

In this Section, we exhibit a sufficient condition in order to prove inconsistency even for non isometric action. However, knowing if this condition is satisfied is difficult, except if the action is isometric. Therefore, we propose to prove that this condition is only verified for action which are small perturbation of an isometric action.

Notation, hypothesis and theorem of inconsistency

Let M be a Hilbert space, G acting on M , Y a random variable such that Ep}Y } 2 q ă `8. The observable variable Y is built with a template t 0 added to unbiased noise ε and a random transformation Φ in the group G. The random deformation Φ and the noise ε are independent variables. This leads to two different generative models:

Y " Φ ¨t0 `ε, which is called forward model. Or Y " Φ ¨pt 0 `εq,
which is called a backward model.

In this Section we exhibit results for this two models, however in the case of a forward model we will need an extra hypothesis: the action is linear which means that x Þ Ñ g ¨x is linear for all g P G.

In this Section, we suppose that there exists some non negative constants A, a, B, b such that: 5.8) This can be seen as a relaxation of the isometric action (previously studied) where we had: @x P M @g P G }x} 2 " }g ¨x} 2 .

@x P M @g P G a}x} 2 `b ď }g ¨x} 2 ď A}x} 2 `B, (

Inconsistency for non isometric action by perturbation of an isometric group action 145

Note that it is always possible to take a " b " 0, but we will see that it is possible to have a more precise result if a and b are greater than 0. • For an isometric action we have A " a " 1 and B " b " 0.

• More generally, if one has a compact group of linear operators for the euclidean norm } }, we can take:

A " sup gPG ~g~2 a " 1 A B " b " 0,
where ~g~is the subordinate norm of the linear application x Þ Ñ g ¨x.

• In Example (4.2), this condition is fulfilled as soon as the diffeomorphims in the group G has a Jacobian determinant uniformly bounded since:

}f ˝φ´1 } 2 2 " ż R n f pxq 2 |Jac x |dxφ ď A}f } 2 2 .
We can define the function m Þ Ñ F pmq " Epinf gPG }m ´g ¨Y } 2 q and provide frameworks in which t 0 does not always minimize F . As we do not suppose an isometric action, F is a called pre-variance instead of being called variance. We define λpt 0 q " E ˜sup gPG xt 0 , g ¨Y y }t 0 } 2 . Proposition 5.5. Let G a group acting on M a Hilbert space. We assume that Y satisfies a backward or forward generative model from a template t 0 (with the extra assumption that G acts linearly in the case of a forward model). We suppose that: r}t 0 }pλpt 0 q ´1qs 2 ą pA ´aqEp}Y } 2 q `B ´b.

(5.9)

We also assume that λpt 0 q ą 1. Then t 0 does not minimize F .

Condition (5.9) can be understood as follows: the non isometric action leads to some constants A, a, B, b such that A and a are closed enough and so are B and b. In the case of an isometric action B " b " 0 and A " a " 1 and the right member is equal to 0, besides we have seen that λpt 0 q ą 1 as soon as the quotient distance is a contraction with respect to the ambient distance (see chapter 3). Therefore, condition (5.9) is a generalization of the sufficient condition for inconsistency in the case of isometric action.

Proof. We define for λ ě 1, f pλq " F pλt 0 q. Then we have:

f pλq " λ 2 }t 0 } 2 `E ˆinf gPG }g ¨Y } 2 ´2λ xt 0 , g ¨Y y ˙. λ P pλq F pλt 0 q ppλq 1 P p1q F pt 0 q pp1q λ ‹ P pλ ‹ q F pλ ‹ t 0 q ppλ ‹ q Figure 5
.6: Sketch of the proof: thanks to condition (5.8) we find two parabola P and p such that ppλq ď F pλt 0 q ď P pλq for all λ ě 0. P and p reach their minimum at λpt 0 q ą 1. Condition (5.9) ensures that P pλpt 0 qq ă pp1q. Then F pλpt 0 qt 0 q ă F pt 0 q as a conclusion t 0 is not the minimum of F . In chapter 3, the action was isometric and P pλq " ppλq " F pλt 0 q for all λ ą 0 leading to a simpler proof.

By using condition (5.8), we get:

f pλq ď λ 2 }t 0 } 2 ´2λEpsup gPG xt 0 , g ¨Y y `AEp}Y } 2 q `B " P pλq.
Similarly,

f pλq ě λ 2 }t 0 } 2 ´2λEpsup gPG xt 0 , g ¨Y y `aEp}Y } 2 q `b " ppλq.
This allows us to determine the unique λ P R `which minimizes P and p: λpt 0 q " argmin P " argmin p "

Epsup gPG xt 0 , g ¨Y yq }t 0 } 2 .
Now, we know that: " F pt 0 q " f p1q ě pp1q F pλpt 0 qt 0 q " f pλpt 0 qq ď P pλpt 0 qq .

thus if pp1q ą P pλpt 0 qq, we have F pt 0 q ą F pλpt 0 qt 0 q. So the only thing we have to prove is pp1q ą P pλpt 0 qq: P pλpt 0 qq " λpt 0 q 2 }t 0 } 2 ´2λpt 0 q}t 0 } 2 λpt 0 q `AEp}Y } 2 q `B, (5.10)

pp1q " }t 0 } 2 ´2}t 0 } 2 λpt 0 q `aEp}Y } 2 q `b.

(5.11)

Inconsistency for non isometric action by perturbation of an isometric group action 147

Now we can compute pp1q ´P pλpt 0 qq, thanks to equations (5.10) and (5.11):

pp1q ´P pλpt 0 qq " r}t 0 }pλpt 0 q ´1qs 2 ´"pA ´aqEp}Y } 2 q `B ´b‰ ą 0, by using Condition (5.9). This concludes that t 0 does not minimize F .

5.5.2

Is it possible to fulfill condition (5.9)?

As we have seen, we need condition (5.9) in order to prove that pp1q ą P pλpt 0 qq which suffices to prove that t 0 do not minimizes F . But is it possible to have this condition? First, we can notice that when the action is isometric we can always take A " a " 1 and B " b " 0 then as soon as t 0 ‰ 0 and λpt 0 q ‰ 1 (and we have seen that λpt 0 q ą 1 ) we have the condition (5.9), since the left member of condition (5.9) is positive whereas its right member is equal to zero. So in fact, in the case of isometric action we just prove again what we proved in chapter 3. However let us suppose that we have a isometric which satisfies (5.9) then the left hand side is strictly greater than the right hand side. Now if it was possible to deform our group action on a continuous way, such that the left and right member are continuous with respect to the group action, then starting from an isometric group action we can deform a little this group action to have a non isometric group action which still satisfies (5.9).

However, we need to specify what is meant by deforming our group action in a continuous way. Definition 5.3. We suppose that for all τ P r0, 1s we have G τ a group acting on M . Moreover we suppose that G 0 acts isometrically on M . We say that the group action is continuous with respect to τ if for all τ 0 P r0, 1s we have: (5.12) This means that for τ sufficiently close to τ 0 , g τ ¨x will behave like a g τ 0 ¨x for some g τ 0 P G τ 0 . Note that a priori we do not know how to find this g τ 0 .

In particular, if G τ acts linearly for all τ , condition (5.12) is equivalent to: @τ 0 P r0, 1s lim (5.13) where ~~is the subordinate norm associated to the euclidean norm } }. Let us give an example:

Example 5.6. We take M " R n , let be G a compact group of the linear group GL n pRq which linearly. We assume additionally that G do not acts isometrically.

In other words G is not included in O n pRq the orthogonal group. A classical result [START_REF] Bourbaki | Algebre: Chapitre 8[END_REF]] on maximal compact subgroup states that it exists S a symmetric definite positive matrix such that:

S ´1GS Ă O n pRq.
Now, the space of symmetric and definite positive matrices is path connected. Therefore we can find τ P r0, 1s Þ Ñ S τ such that S 0 " S and S 1 " I n . We define then G τ " S ´1 τ GS τ . In particular, G 0 " S ´1GS Ă O n pRq and G 1 " G. For every τ P r0, 1s G τ acts linearly on M . It is possible that, for some τ , G τ is included in O n pRq but we can remove these τ in order that G 0 is the only group which acts isometrically: We define τ ‹ P r0, 1p as: τ ‹ " sup tt P r0, 1s such that G t Ă O n pRqu.

Then by the fact that O n pRq is a closed set we have that G τ‹ Ă O n pRq. Then we can consider G τ only for rτ ‹ , 1s and re-scale the τ parameter so that τ ‹ " 0 then G τ acts linearly but not isometrically as soon as τ ą 0. Now let us prove that equation (5.13) is satisfied:

Proof. Let g τ " S ´1 τ gS τ P G τ with g P G. We have S τ ´1gS τ P G τ 0 then we have: because it exists N such that for all g P G ~g~ď N (the group is compact). Now, if one have a continuous path of group action pG τ , ¨qτPr0,1s such that G 0 acts isometrically on M then (5.9) is a inequality between two terms which are continuous maps with respect of τ . When (5.9) is fulfilled for τ " 0 (in chapter 3 we have seen that this condition is fulfilled as soon as the quotient distance to the template is contracted compared to the ambient distance), then this condition (5.9) will be still fulfilled for τ P r0, cs for some c ą 0 by continuity. Besides, if the condition λpt 0 q ą 1 is true for τ " 0 (the isometric action), λpt 0 q ą 1 for a small interval r0, c 1 s.

Conclusion: if is possible to connect a non isometric action to an isometric one with a continuous path and if the template estimation is inconsistent for the isometric group, then it will also be inconsistent for a few non isometric groups among the path connecting the two groups. The inconsistency appears even for small σ and for non isometric groups. However, we do not know how to estimate the c P p0, 1s such that the action of G τ leads to inconsistency for τ ă c.

Conclusion

In section 5.2, we have found an implicit equation of an element which minimizes the variance/pre-variance. This implicit equation was used in order to prove that the element which minimize the variance in Hilbert space have less symmetry than the original template in theorem 5.2. In section 5.3, we have provided a conjecture of inconsistency. This conjecture is based on the Voronoï cell, this concept is similar to the notion of the cone of the template seen in chapter 3. It is important to notice that, we do neither suppose an invariant distance under the group action nor that the ambient space is a Hilbert space in this conjecture,. Besides, in this conjecture, we could add a regularization term in the template estimation. Finally, in section 5.5, we do not assume a isometric action, and we give a condition which leaves to inconsistency. Unfortunately, this condition is not easy to verify. However, we can state that when the action is sufficiently closed to an isometric one, then inconsistency holds.

Chapter 6 In this thesis, we have considered the estimation of the template in computational anatomy with the Fréchet mean in quotient space in the case where data in the ambient space were deformed and noisy. We have mathematically proved that the presence of the noise and the presence of the deformation on data leads to an inconsistent estimator of the template. In chapter 3 we proved the inconsistency mostly for isometric action in Hilbert space. Chapter 4 generalized the inconsistency in more general action in Hilbert space, but we restrict ourselves on the case where the noise level was high. We extended our results in chapter 5 for nonlinear spaces, where we proved in theorem 5.2 that the Fréchet mean in quotient space can be more noisy than the original template. Table 6.1, summaries of the main assumptions of the different results leading to inconsistency.

Conclusion and Perspectives

We have highlighted the origin of the inconsistency: it is the contraction of the quotient distance with respect to the distance in the ambient space (see theorems 3.1, 3.4 and 4.1 for instance). We have also provided a quantification of the consistency bias (theorem 4.1 for instance). As a result, the consistency bias is asymptotically linear with respect to the noise level σ, when σ Ñ `8 in the case of an isometric action in Hilbert spaces.

Besides, we have done a little bit more than providing quantification of the consistency bias. We have proved that the estimated element is less symmetric than the original template (theorem 5.2). The examples of the diffeomorphism on landmarks or images were described in section 1.3.3, the vertical translation in example 4.6, the horizontal translation on two points in section 3.3.6, the horizontal translation on more than two points in examples 3.1 and 5.2, the continuous horizontal translation in examples 3.3 and 5.3, rotations in section 3.4.6.3 and example 5.1 of the significance of these constants and how they are related to each other:

• In theorem 3.5, we have found a lower bound of the consistency bias which depended on λpt 0 q. We recall, that λpmq " 1 }m} 2 Epsup gPG xm, g ¨Y yq. This quantity indicates how much the observable variable Y looks like the template t 0 after registration. Later, in proposition 3.7, we take a Taylor expansion of the lower bound found in theorem 3.5 when σ tends to infinity. This Taylor expansion was linear with respect to σ. And the linearity constants was depending on θpt 0 q defined as θpt 0 q " 1 }t 0 } Epsup gPG xε, g ¨t0 yq. This quantity indicates how much the template t 0 looks like the standardized noise ε after registration. In proposition 3.8, we found an upper bound of the consistency bias, this upper bound depends on θpm ‹ q " 1 }m‹} Epsup gPG xε, g ¨m‹ yq, where m ‹ is a minimizer of the variance.

• Therefore, in chapter 3, we have seen that we could have lower and upper bounds of the consistency bias. However the Taylor expansion of these two bounds were different (since they depends on two different constants θpt 0 q and θpm ‹ q), this prevent us to find a Taylor expansion of the consistency bias. This flaw is fixed in chapter 4, as we found in the proof of theorem 4.1 new lower and upper bounds. These bounds are also asymptotically linear with respect to σ, but with the same constant K " sup vPS θpvq, where S is the unit sphere, and θpvq " Epsup gPG xv, g ¨εyq. Therefore K is bigger than θpt 0 q and θpm ‹ q. Therefore we were able to establish a Taylor expansion of the consistency bias for isometric action with respect to σ when σ tends to infinity: CB " Kσ `opσq. This result is complementary with the one of [START_REF] Miolane | [END_REF]] which provides a Taylor expansion of CB when σ Ñ 0.

• In propositions 4.6 and 4.7, we found that the threshold for the noise level, depends on θpt 0 q, besides the proofs make λpt 0 q intervenes.

• Finally, in proposition 5.5, we have also give a sufficient condition for inconsistency where λpt 0 q also appeared.

Questions to be investigated

In order to conclude this thesis, we give some questions which need to be solved. We can classify the questions to be investigated into two parts: on the one hand, there are statistically issues directly related to the template estimation with the Fréchet mean in quotient space. On the other hand, there are also geometrical questions, which can be solved independently from the template estimation. But whose solutions would bring new answers to the template estimation issue.

Statistical questions on the template estimation

• What is the behaviour of the consistency bias for σ Ñ 0 in Hilbert space for isometric action or in infinite dimensional manifolds?

• How can we prove that the inconsistency appears for all σ for non isometric action? Solving conjectures 5.1 and 5.2, for instance?

• What is the behaviour of the consistency bias with respect to σ when a regularization term is added? Solving this issue could tell which regularization minimizes the inconsistency.

• In all this thesis, the consistency bias was studied under the condition that the Fréchet mean in quotient space exists. If not, the consistency is, by convention, infinite. Can we given some results ensuring such existence? Indeed, the theorems on the existence or the uniqueness of the Fréchet means [Kendall 1989, Karcher 1977] are all, (to the best of our knowledge), restricted 6.2. Questions to be investigated 155 to random variable with a support included in a ball which is small enough. Therefore these interesting results applies only in the case of small noise level.

Geometrical questions

• Given a metric space and a group action on this metric space, is it possible to obtain a criteria which indicates the existence (or the non existence) of a congruent section? Indeed, we have seen in chapter 5, than when such a congruent section exists, computing the bias is just a computation of an expectation.

• Is it possible to establish that there exists a measurable section of the quotient space which takes value in the Voronoï cell of a certain point? This would make the proof of theorem 5.2 perfectly rigorous.

• Can we find spaces with good mixtures (defined in definition 5.1) different from Hilbert space? This question may matter since we have seen that theorem 5.2 can be directly extended to this kind of space. Dans cette thèse, on a divisé les preuves d'inconsistances en plusieurs résultats avec différentes hypothèses sur l'action ou sur le template. Sur la figure 7.1, on a dessiné un diagramme de Venn de certaines propriétés de l'action avec quelques exemples vus dans cette thèse.

Questions ouvertes

Pour conclure cette thèse, on peut donner quelques questions ouvertes concernant les problèmes statistiques et géométriques.

Questions statistiques reliées à l'estimation de template

• Quel est le comportement du biais de consistance lorsque σ Ñ 0 dans des espaces de Hilbert pour des actions isométriques ou dans des variétés de dimension infinie ?

• Comment prouver l'inconsistance pour tout σ ą 0 et pour toute action isométrique. En résolvant les conjectures 5.1 et 5.2, par exemple ?

• Quel est le comportement du biais quand un terme de régularisation est ajouté ? Ainsi on pourrait dire quel type de régularisation il est préférable de considérer.

• Dans toute cette thèse, le biais de consistance a été étudié à la condition que la moyenne de Fréchet dans l'espace quotient existait. Sinon le biais de consistance est par convention infinie. Peut-on assurer cette existence ? En effet les théorèmes d'existence et d'unicité de la moyenne de Fréchet [Kendall 1989, Karcher 1977] sont tous restreints (nous semble-t'il ) à des variables aléatoires dont le support est contenu dans une boule suffisamment petite. Donc, ces résultats ne s'appliquent que dans le cas des niveaux de faible bruit.

Questions géométriques

• Pour un certain espace métrique et une certaine action de groupe, est-ce possible de trouver un critère indiquant l'existence (ou la non-existence) d'une section congruente ? En effet, au chapitre 5, nous avons vu que lorsqu'une section congruente existe, le calcul du biais est réduit au calcul d'une espérance.

• Est-il possible de montrer qu'il existe une section mesurable de l'espace quotient dont l'image est incluse dans la cellule de Voronoï d'un certain point ? Cela simplifierait le théorème 5.2.

• Peut-on trouver des espaces à bons mélanges (définition 5.1) différents des espaces de Hilbert ? En effet, nous avons vu que le théorème 5.2 peut être directement étendu à ce type d'espaces.

Cadre de la thèse

Le but de cette thèse est de démontrer l'inconsistence de l'estimation de template en calculant la moyenne de Fréchet dans l'espace quotient. Pour cela on se place dans un espace de départ noté M . On suppose souvent que M est un espace de Hilbert. On considère G un groupe agissant sur M . On définit alors l'orbite d'un élément m P M par : rms " tg ¨m, g P Gu De plus on définit le quotient de M par G noté M {G comme l'ensemble des orbites. Si on suppose que la distance d M de M est invariante sous l'action de G, c'està-dire que : @g P G @m P M @n P M d M pg ¨m, g ¨nq " d M pm, nq Alors on peut munir d Q d'une (pseudo)-distance définie par d Q prms, rnsq " inf gPG d M pa, g ¨bq.

Soit t 0 un élément de M , que l'on nomme template, partant de ce template on crée des données en déformant ce template par l'action de groupe et en ajoutant du bruit. Suivant que l'on ajoute le bruit avant ou après la déformation on obtient deux modèles génératifs :

Y " Φ ¨t0 `σε ou Y " Φ ¨pt 0 `σεq Où on a noté t 0 le template, Φ une variable aléatoire dans G, ε une variable aléatoire centrée réduite (Epεq " 0 et Ep}ε} 2 q " 1, on suppose également que Φ et ε sont deux variables aléatoires indépendantes. Le premier modèle est appelé «forward » et le second «backward ». Une fois que l'on a généré Y , on s'intéresse au problème inverse : on cherche à estimer le template, ou plus précisément on cherche à estimer son orbite. Une méthode souvent utilisée est le calcul de la moyenne de Fréchet dans l'espace quotient, c'est-à-dire que l'on minimise : m Þ Ñ F pmq " Epd Q prms, rY sq 2 q " Epinf gPG }m ´g ¨Y } 2 q On obtient donc un estimateur du template. Dans cette thèse, on étudie donc les propriétés statistiques de cet estimateur.

Dans cette thèse, on va montrer que généralement cet estimateur est inconsistant. De plus on cherche à estimer le biais de consistance, c'est-à-dire la distance entre le template et la moyenne de Fréchet dans le quotient.

Simulation sur des données synthétiques

Avant d'étudier de manière théorique l'estimation de template, on peut essayer de faire des expériences numériques pour se donner une intuition de la (in)consistance

Preuve d'inconsistance pour des actions isométriques

On appelle action isométrique sur un espace de Hilbert M , toute action d'un groupe G tel que pour tout g P G, m Þ Ñ g ¨m soit linéaire et que }g ¨m} " }m} pour tout m P M .

Pour étudier la consistance on peut commencer par se restreindre aux groupes finis. On a le résultat suivant : Théorème 8.1. Soit G un groupe fini agissant sur M " R n effectivement et isométriquement. Supposons que la variable aléatoire X est absolument continue par rapport à la mesure de Lebesgue. On suppose aussi que Ep}X} 2 q ă `8 et que t 0 " EpXq est un point régulier (son groupe d'isotropie est réduit à t0u).

On définit Conept 0 q comme l'ensemble des points plus proches de t 0 que des autres points gt 0 pour g P G (voir figure 8.2). Conept 0 q est en fait définit comme l'ensemble des points déjà recalés avec t 0 . Si: De plus, on peut prouver que la condition 8.2 est souvent vérifiée. Par exemple, lorsque l'orbite du template est une sous variété de M , il suffit que le support de X ne soit pas inclus dans l'orthogonal de l'espace tangent de rt 0 s au point t 0 . Dans cette thèse, on prouve également le théorème suivant qui fournit un équivalent du biais lorsque le niveau de bruit diverge vers `8 : Théorème 8.3. Soit M un espace de Hilbert, et G agissant isométriquement sur M . Soit Y " Φ ¨t0 `σε, t 0 est le template, Φ une variable aléatoire dans G, ε une variable aléatoire centrée réduite (Epεq " 0 et Ep}ε} 2 q " 1. Si le support du bruit ε n'est pas inclus dans l'ensemble des points fixes, alors on a l'encadrement du biais de consistance suivant :

P pX R Conept 0 qq ą 0, ( 8
σK ´2}t 0 } ď CB ď σK `2}t 0 }, (8.4

Figure 1 . 1 :

 11 Figure 1.1: Comparison between two fishes.

Figure 1

 1 Figure 1.2: Example of the unit sphere in R 2 , a Riemannian manifold.

  gPG d M pa, g ¨bq " inf gPG d M ph ¨a, g ¨pj ¨bqq.

  rcsq `dQ prcs, rbsq. Proposition 1.3. When the orbits are closed set in M for the topology defined by the distance d M , d Q is a distance in Q.

2. 1

 1 Anatomie computationnelleAu début du 20ème siècle, D'Arcy Thompson effectua un travail visionnaire en étudiant les formes des animaux. Une des idées principales était de créer une classification entre deux espèces. Pour ce faire, il introduisit une grille sur l'image du premier animal et déforma cette grille pour faire coïncider l'image du premier animal sur le second. Plus la grille doit être déformée, plus les espèces sont différents.

Figure 2 . 1 :

 21 Figure 2.1: Comparison entre deux poissons.

  gPG d M pa, g ¨bq Remarque 2.1. Pour tout a, b P M , d Q pras, rbsq ď d M pa, bq. De plus, si a et b sont en position générique, on a: 1 d Q pras, rbsq ă d M pa, bq.

  (a) If d ą 0 then w Þ Ñ ρpd, wq has an asymptotic linear expansion: ρpd, wq " If d ą 0, then ρpd, wq " opw k q as w Ñ 0, for all k P N.

  .46) Secondly, we have for all m P M , (in particular for m ‹ ): F pmq " Epinf gPG p}m ´g ¨t0 } 2 `σ2 }ε} 2 ´2 xg ¨σε, m ´g ¨t0 yqq ě d Q prms, rt 0 sq 2 `σ2 ´2Epsup gPG xσε, g ¨myq. (3.47) With Inequalities (3.46) and (3.47) one gets:

Figure 3 . 8 :

 38 Figure3.8: Different signals and their θpt 0 q approximated with a sample of size 10 3 in R Z{100Z . ε is here a Gaussian noise in R Z{100Z , such that Epεq " 0 and Ep}ε} 2 q " 1. For instance the blue signal is a signal defined randomly, and when we approximate the θpt 0 q which corresponds to that t 0 we find » 0.25.

Remark 4 . 1 .

 41 When G acts isometrically on M a Hilbert space, by expansion of the squared norm we have:d Q pras, rbsq 2 " }a} 2 ´2sup gPG xa, g ¨by `}b} 2

  λpvq " E ˜sup gPG xv, g ¨Y y ¸" E ˜sup gPG pxv, g ¨σεy `xv, gΦ ¨t0 yq ¸(4.7) λpvq ď E ˜sup gPG pxv, g ¨σεy `}t 0 }q ¸" σE ˜sup gPG xv, g ¨εy ¸`}t 0 } (4.8) λpvq ě E ˜sup gPG pxv, g ¨σεyq ´}t 0 } ¸" σE ˜sup gPG xv, g ¨εy ¸´}t 0 }. (4.9) By taking the largest value in these inequalities with respect to v P S, we get by definition of K: ´}t 0 } `σK ď }m ‹ } " sup vPS λpvq ď }t 0 } `σK. (4.10)

Proposition 4 . 4 .

 44 Let G a group acting isometrically on M a Hilbert space, let Y 1 , . . . , Y n being a sample of a random variable Y , let rm n‹ s being a empirical Fréchet mean of Y . Then m n‹ "

Figure 4 . 4 :

 44 Figure 4.4: Template t 0 and template estimation m on figure 4.4a. Empirical variance at the template and the template estimation with the max-max algorithm as a function of the size of the sample on figure 4.4b.

Figure 4 . 5 :

 45 Figure 4.5: Example of an other template (here a discretization of a continuous function) and the template estimation with a sample size 10 3 in R 64 , ε is Gaussian noise and σ " 10. Even with a continuous function the inconsistency appears.

Figure 4 . 6 :

 46 Figure 4.6: Example of the continuous template (here a discretization of a continuous function) and the template estimation with a sample size 10 6 in R 64 , ε is Gaussian noise and σ " 10.

Figure 4 . 7 :

 47 Figure 4.7: Several local minima computed from the same sample of size 10 5 , we took the same step function as template. We zoomed in order to see better the small differences between the different outputs of the max-max algorithm given five different starting points.

Figure 4 . 8 :

 48 Figure 4.8: Histogram of the quotient distance between the template and the output of the max-max algorithm, in red we have the distribution of the distance of 100 max-max algorithm with the same sample of size 10 5 , in blue the distribution with a sample of size 10 4 . The larger the size of the sample, the more concentrated the distance is.

Chapter 4 .Figure 4 . 9 :

 449 Figure 4.9: Rose in the case of horizontal translation in R 3 with a noise level σ " 5.5, we use a parametrization of a discretized of the sphere with 2500 points.

Figure 4 .

 4 Figure 4.10: Example of three orbits, when dQ does not satisfy the inequality triangular.

  (a) Case 1: t0 and g ¨t0 are linearly independent.

  The original template has a large isotropy group: any bijective map which is equal to identity on p0, 0.2q Y p0.8, 1q. Due to the noise, there is no more bijective map which leaves invariant this noisy template.

Figure 5 . 3 :

 53 Figure 5.3: Example of signals, which are 1D images.

Figure 5 . 4 :

 54 Figure 5.4: Representation of the three functions f 1 , f 2 and f 3 with η " 0.05. the functions f 2 and f 3 are registered with respect to f 1 . However f 2 and f 3 are not registered with each other, since it is more profitable to shift f 2 in order to align the highest parts of f 2 and f 3 .

  Representation of the four orbits in the quotient space, the quotient space can not be embedded into R 2 in a congruent way.

Figure 5 . 5 :

 55 Figure 5.5: Representation of the four points in R 2 and representation of the four orbits

Example 5 . 5 .

 55 Let us give three examples where (5.8) is verified:

inf gτ 0

 0 PGτ 0 ~gτ ´gτ 0 ~ď ~gτ ´Sτ 0 ´1gS τ 0 ď ~Sτ ´1gpS τ ´Sτ 0 q~`~pS τ ´1 ´Sτ 0 ´1qgS τ 0 sup gτ PGτ inf gτ 0 PGτ 0 ~gτ ´gτ 0 ~ď ~Sτ ´1~N ~Sτ ´Sτ 0 ~`~S τ ´1 ´Sτ 0 ´1~N ~Sτ 0 ~Ñ τ Ñτ 0 0.

Contents 6 . 1

 61 Synthesis of contributions . . . . . . . . . . . . . . . . . . . . 151 6.1.1 Hypotheses leading to inconsistency . . . . . . . . . . . . . . 151 6.1.2 The role played by the constants . . . . . . . . . . . . . . . . 152 6.2 Questions to be investigated . . . . . . . . . . . . . . . . . . . 154 6.2.1 Statistical questions on the template estimation . . . . . . . . 154 6.2.2 Geometrical questions . . . . . . . . . . . . . . . . . . . . .

'

  Figure 6.1: Different kind of actions according to their property, the actions in bullet are examples of each cases. The definition and the existence of such congruent section (CS) was discussed in section 5.4.1, isometric actions were studied in chapters 3 and 4, linear actions in section 4.3.4. The examples of the diffeomorphism on landmarks or images were described in section 1.3.3, the vertical translation in example 4.6, the horizontal translation on two points in section 3.3.6, the horizontal translation on more than two points in examples 3.1 and 5.2, the continuous horizontal translation in examples 3.3 and 5.3, rotations in section 3.4.6.3 and example 5.1

'

  Figure 7.1: Différents types d'actions selon leurs propriétés. Les actions représentées par des points sont des exemples de chaque case. La définition et l'existence des sections congruentes (SC) a été discuté dans la sous-section 5.4.1, les actions isométriques ont été étudiées aux chapitres 3 et 4, les actions linéaires en soussection 4.3.4.

8. 3 .d

 3 Preuve d'inconsistance pour des actions isométriques 163 de l'estimation du template. Pour ce faire, on prend par exemple des signaux vus comme des fonctions discrétisées de r0, 1s dans R avec N points. On définit une action de groupe par translation :τ P Z{N Z, px 1 , . . . , x n q P R n Þ Ñ px 1`τ , . . . , x n`τ qPartant d'un template t 0 fixé, on crée un échantillon Y 1 , . . . Y I en agissant sur ce template et en y ajoutant du bruit; Y k " Φ k ¨t0 `εk . Ainsi avec cet échantillon, on peut définir la variance empirique :m Þ Ñ F I pmq " Q prms, rY i sqPuis en minimisant cette variance (via un algorithme de minimisation alternée appelé «max-max »), on peut comparer la moyenne de Fréchet empirique au vrai template :

Figure 8 . 1 :

 81 Figure 8.1: Exemple d'un template (une fonction escalier) et du template estimé avec un échantillon de taille 10 5 dans R 64 , ε est un bruit gaussien et σ " 10.

8. 4 .

 4 Quantification du biais pour les actions isométriques 165 8.4 Quantification du biais pour les actions isométriques

  Computational anatomy. . . . . . . . . . . . . . . . . . . . . . 16 1.2 Statistics in computational anatomy . . . . . . . . . . . . . . 17 1.3 Geometrical framework of template estimation . . . . . . . 18

	Chapter 1
	Introduction
	Contents
	1.1

1.3.1 A brief overview of manifolds and Riemannian manifolds . . . 18 1.3.2 Deformations and quotient space . . . . . . . . . . . . . . . . 20 1.3.3 What are the data? . . . . . . . . . . . . . . . . . . . . . . . 20 1.3.4 What are the actions? . . . . . . . . . . . . . . . . . . . . . . 21 1.3.5 Distance in quotient space . . . . . . . . . . . . . . . . . . . . 21 1.4 Statistical estimation method . . . . . . . . . . . . . . . . . . 24 1.4.1 Fréchet mean in Hilbert space . . . . . . . . . . . . . . . . . . 24 1.4.2 Fréchet mean in metric spaces . . . . . . . . . . . . . . . . . . 25 1.4.3 Geometrical and statistical sufficient conditions for inconsistency 27 1.4.4 Noise and probability in Hilbert Spaces . . . . . . . . . . . . 28 1.4.5 Backward/forward model, backward/forward estimation . . . 29 1.5 Template estimation in this thesis and in related works . . 30 1.6 Heuristic of inconsistency . . . . . . . . . . . . . . . . . . . . 31 1.7 Manuscript overview . . . . . . . . . . . . . . . . . . . . . . . 31 16 Chapter 1. Introduction 1.1 Computational anatomy.

  Contents 2.1 Anatomie computationnelle . . . . . . . . . . . . . . . . . . . 34 2.2 Statistique en anatomie computationnelle . . . . . . . . . . . 35 2.3 Cadre géométrique pour l'estimation de template . . . . . . 36

2.3.1 Distance dans le quotient . . . . . . . . . . . . . . . . . . . . 36 2.3.2 Moyenne de Fréchet dans les espaces de Hilbert . . . . . . . . 36 2.3.3 Moyenne de Fréchet dans les espaces métriques . . . . . . . . 37 2.4 Estimation de template dans cette thèse et dans d'autres travaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.5 Organisation du manuscrit . . . . . . . . . . . . . . . . . . . . 39 34 Chapter 2. Introduction (français)

  An orbit rms P Q which minimizes this map is called a Fréchet mean of rXs.If we have an i.i.d sample of observations Y 1 , . . . , Y n we can write the empirical quotient variance:

  XPConept 0

	3.4. Inconsistency for finite and infinite group when the template is
	not a fixed point	63
	get:
	E ˜sup	xg ¨X, t 0 y ¸"E ˜sup
	gPG

.29) For all x P M , we have sup gPG xg ¨x, t 0 y ě xx, t 0 y and thanks to condition (3.27) we gPG xg ¨X, t 0 y 1

  .s..

	Now let us choose m 1 " ´m we have:
	sup	xm, g ¨Y y `sup	´xm, g ¨Y y " 0 Y ´almost surely.
	gPG	gPG
	If we reformulate we have:
		sup

gPG xm, g ¨Y y " inf gPG xm, g ¨Y y Y ´almost surely.

  zq 2 is 3.5. Fréchet means ambient and quotient spaces are not consistent when the template is a fixed point 79 differentiable at any m P M zCpzq (i.e. z R Cpmq) with derivative ´2Log m pzq. Now since: |d M px, zq 2 ´dM py, zq 2 | " |d M px, zq ´dM py, zq}d M px, zq `dM py, zq| ď d M px, yqp2d M px, zq `dM py, xqq, we get in a local chart φ : U Ñ V Ă R n at t " φpmq we have locally around t that: h Þ Ñ d M pφ ´1ptq, φ ´1pt `hqq, is smooth and |d M pφ ´1ptq, φ ´1pt `hqq| ď C|h| for a C ą 0. Hence for sufficiently small h, |d M pφ ´1ptq, zq 2 ´dM pφ ´1pt `hq, zq 2 | ď C|h|p2d M pm, zq `1q. We get the result from dominated convergence Lebesgue theorem with Epd M pm, Xqq ď Epd M pm, Xq 2 `1q ă `8.

Table 3 .

 3 1: Behaviour of the consistency bias with respect to σ 2 the variability of X " t 0 `σε. The constants K i 's depend on the kind of noise, on the template t 0 and on the group action.Epsup gPG xv, g ¨εyq (proposition 3.11)

	Consistency bias noted CB	G is any group			Supplementary properties
							for G a finite group
	Upper bound of CB	CB ď σ	`2? σ 2 `K1 σ	CB ď K 2 σ (theorem 3.3)
		(proposition 3.8)
	Lower bound of CB for σ Ñ 8 when the template is not a		CB ě L " σÑ8	K 3 σ (proposition 3.7)
	fixed point Behavior of CB for σ Ñ 0 when the template is not a	CB ď U " σÑ0	K 4	?	σ	CB " 0 section 3.3.6, can we extend opσ k q, @k P N in the
	fixed point						this result for finite group?
	CB when the template is a	CB " σ sup
	fixed point		}v}"1

  a good idea to search the Fréchet mean of rY s in the direction pointed by v: the more λpvq is large, the more v is a good choice. On the contrary when this value is equal to zero, it is useless to search the Fréchet mean in this direction.Likewise, for v P S, θpvq " Epsup gPG xg ¨v, εyq is a registration score with respect to the noise, the larger θpvq, the more the unit vector v looks like to the noise ε after registration. If rm ‹ s is a Fréchet mean of rY s we have seen that its norm verifies:Then if there are two different Fréchet means of rY s noted rm ‹ s and rn ‹ s, we can deduce that }m ‹ } " }n ‹ }. Even if there is no uniqueness of the Fréchet mean in the quotient space, we can state that the representants of the different Fréchet means have all the same norm.

		ˆE ˆinf	˙˙`i
	therefore λpvq "	gPG	xv, g ¨Y y	s a registration score: λpvq tells you how
	much it is }m ‹ } " sup	Epsup	xv, g ¨Y yq.
			}v}"1	gPG
					˙˙`¸2
					g ¨Y y	.

  Chapter 4. Inconsistency when the noise level tends to infinity 4.2.4.1 Max-max algorithm with a step function as template

			template and max max output
	1.5					97.2
					template
					max max output
						97
	1
						96.8
						96.6
												F_I(t_0)
	0.5
												F_I(m)
						96.4
						96.2
	0
						96
	-0.5					95.8
	0	0.2	0.4	0.6	0.8	1	0	2e+06	4e+06	6e+06	8e+06	1e+07
									I: size of the sample

(a) Example of a template (a step function) and the estimated template m with a sample size 10 5 in R 64 , ε is Gaussian noise and σ " 10. At the discontinuity points of the template, we observe a Gibbs-like phenomena.

  4.2. Inconsistency of the template estimation with an isometric action when the noise level tends to infinity. 103

	1.4
	1.2
	1
	0.4	0.45	0.5	0.55	0.6

Table 4 .

 4 2: Quotient distance between two pairs of points among the template and the five Karcher means, we remark that the different Karcher means are closed to each other compared to the distance between the template, in other words the set of Karcher means seems localized in a small neighborhood far from the template roughly a rose with n petals. Moreover this curve depends on the noise level.

	2π n we observe

  .3. Conjecture of inconsistency for metric space with non invariant distance under the group action. 133

Table 7 .

 7 Contents 7.1 Synthèse des contributions . . . . . . . . . . . . . . . . . . . . 157 7.2 Questions ouvertes . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.2.1 Questions statistiques reliées à l'estimation de template . . . 159 7.2.2 Questions géométriques . . . . . . . . . . . . . . . . . . . . . 160 1: Récapitulatif des hypothèses principales dans les différents résultats de cette thèse qui conduisent à l'inconsistance. Avec les abréviations suivantes, B : backward, F : forward, EH : espace de Hilbert, VRC : variété riemannienne complète.

	Chapter 7

  .1) alors rt 0 s n'est pas une moyenne de Fréchet de rXs. Figure 8.2: Conept 0 q en gris.Théorème 8.2. Soit G un groupe agissant isométriquement sur un espace de Hilbert M . Soit X une variable aléatoire dans M telle que Ep}X} 2 q ă `8. On suppose que t 0 " EpXq ‰ 0. Si:P pd Q prt 0 s, rXsq ă }t 0 ´X}q ą 0 (8.2)Alors rt 0 s n'est pas une moyenne de Fréchet de rXs dans Q " M {G.

	g ¨t0	Conept 0 q
	0	t 0
	g 1 ¨t0
	ou de manière équivalente
	P ˜sup	xg ¨X, t 0 y ą xX, t 0 y ¸ą 0	(8.3)
	gPG

  La constante K dépend seulement du bruit standardisé et de l'action de groupe mais pas du template. On obtient alors l'équivalent du biais suivant quand σ tend vers plus l'infini :CB " σK `opσq as σ Ñ `8.(8.5)8.5 Actions non isométriquesLorsque les actions sont non isométriques, les preuves des théorèmes vues précédemment ne s'appliquent pas. Cependant, si on suppose que l'orbite du template est bornée, on peut montrer l'inconsistance dès que le niveau de bruit est suffisamment grand. On prouve, en particulier, la proposition suivante :Proposition 8.1. Soit G un groupe agissant sur M un espace de Hilbert. On suppose que le template t 0 n'est pas un point fixe, et que son orbite sous l'action du groupe G est bornée. On suppose que G contient un sous-groupe H agissant isométriquement. On considère A ě sup } . Notons que a ď 1 ď A et on a : @g P G a}t 0 } ď }g ¨t0 } ď A}t 0 }.On suppose que θ H ą 0. Si σ est plus grand qu'un certain niveau de bruit critique noté σ c défini par :

									)
	où K " sup	E ˜sup
	}v}"1
	gPG }t 0 On pose : }g¨t 0 } }g¨t 0 } }t 0 } et a ď inf gPG
	θpt 0 q "	1 }t 0 }	Epsup gPG	xg ¨t0 , εyq et θ H "	1 }t 0 }	hPH E ˆsup	xh ¨t0 , εy ˙.
				»					fi
	σ c "	}t 0 } θ H	-ˆθpt 0 q θ H	`A˙`dˆθ pt 0 q θ H	`A˙2	`A2 ´a2	fl .	(8.6)

gPG xv, g ¨εy ¸P p0, 1s.

Alors l'estimation est inconsistante.

This childish remark contains all this thesis, since it is the cause of the presence of the inconsistency.

Cette remarque peut sembler innocente, mais en fait c'est ce fait là qui conduit à l'inconsistance.

2.3. Cadre géométrique pour l'estimation de template

3.5.2.2 Proof of theorem 3.7 . . . . . . . . . . . . . . . . . .

3.6 Conclusion and discussion . . . . . . . . . . . . . . . . . . . . 80

The term max-max algorithm is used for instance in[Allassonnière

2007], and we prefer to keep the same name, even if it is a minimization.

Note that in this chapter, g ¨x is the result of the action of g on x, and ¨should not to be confused with the multiplication of real numbers noted ˆ.

If the orbits are not closed sets, it is possible to have dQpras, rbsq " 0 even if ras ‰ rbs, in this case we call dQ a pseudo-distance. Nevertheless, this has no consequence in this chapter if dQ is only a pseudo-distance.

We say that F |S is differentiable on S, even if S is not open, because m Þ Ñ Ep}m ´sprY sq}

q is defined and differentiable on M , and is equal to F |S .

q, y 4 " p0, 0q (see figure5.5). Then we have d Q pry i s, ry j sq " 1 for every pi, jq P 1, 4 2 (except if i " j; d Q pry i s, ry i sq " 0).

tions, Random Processes and of the 1974 European Meeting of Statisticians, pages591-602. Springer, 1977. (Cited on pages 27, 38, 71, 88 and 140.)

Remerciements

Chapter 5

Study of consistency with a backward estimation

Chapter 5. Study of consistency with a backward estimation

We are interested in the template estimation given the observation Y with the minimization of the regularized pre-variance defined by:

where Reg is a regularization over the group. Here we suppose that

is a continuous map, and that Regpe G q " 0. In most non linear registration algorithm, for instance, in LDDMM framework [Beg 2005], one makes a trades off between an inexact matching thanks to the term d 2 M pm, g ¨Y q, and a realistic selection of the chosen deformation via the regularization term Regpgq. In LDDMM, the regularization is the squared norm of the vector which defined the flow equation satisfying by the chosen diffeomorphim. For a point x P M and an element h P G, we define the regularized Voronoï cell of x as the element closer from x than the other element of the orbit of x. However, here the "closer" needs to be understood as the distance d M regularized by Reg, the regularization term:

The definition is similar to the definition of the Voronoï cell, but here, the regularization acts as a deformation of the metric d M . As Regpe G q " 0, and Regpgq ě 0, it is easy to see that: VCpxq Ă VC reg px, e G q, in other words, the regularized Voronoï cell is bigger than the original Voronoï cell.

Conjecture 5.2. Let X be a random variable, we assume that t 0 is the unique Fréchet mean of X. Y " Φ ¨X, where Φ is a random variable in G, X and Φ are assumed to be independant. We suppose that t 0 does not belong almost surely to the random regularized Voronoï cell VC r pX, Φq. Then we think that t 0 is not a minimizer of F .

Note that a priori for x and h, VC reg px, hq is a strictly subset of M . Therefore, it is possible to fulfill the condition t 0 does not belong almost surely in VC reg pX, Φq.

Once again, let us explain why we think that this conjecture may be true: As the group G is compact and acts continuously, we have the existence of an element which reaches the infimum in the regularized pre-variance:

Example 5.1. The action of rotations: G " SOpnq acts isometrically on M " R n . We notice that the quotient distance is d Q prxs, rysq " |}x} ´}y}|. We can check that sprxsq " }x}v is a section for an unitary vector v . Therefore the computation of the bias is given by d Q prt 0 s, rEpsprY sqsq " |Ep}Y }q ´}t 0 }q|.

Unfortunately, the congruent section generally does not exist. Let us give an example:

Example 5.2. Taking N P N with N ě 3, we consider the action of G " Z{N Z on M " R Z{N Z " R G by horizontal translation: for τ P Z{N Z, and px 1 , x 2 , . . . , x N q "

Let us take three points p 1 , p 2 and p 3 in M defined by: p 1 " p0, 5, 0, . . . , 0q, p 2 " p0, 3, 2, 0, . . . , 0q and p 3 " p2, 3, 0, . . . , 0q.

By hand we can check that there is no x P rp 1 s, y P rp 2 s and z P rp 3 s such that }x ´y} " d Q prp 1 s, rp 2 sq, }x ´z} " d Q prp 1 s, rp 3 sq, and }y ´z} " d Q prp 2 s, rp 3 sq. Thus, a congruent section in Q " M {G does not exists.

We can generalize this simple example by taking a non finite group:

Example 5.3. Let us take M " L 2 pR{Zq the set of 1-periodic functions such that ş 1 0 f 2 ptqdt ă `8 (this example was already introduced in 3.3). G " R{Z acts on L 2 pR{Zq by horizontal translation:

Then a congruent section in Q " M {G does not exists.

Proof. Let us take

4 `2ηs for some η P p0, 1 4 q (see figure 5.4). Let us suppose that a congruent section s exists. Without loss of generality we can assume that sprf 1 sq " f 1 , we should have

In other words, sprf 2 sq should be registered with respect to f 1 . For τ P R{Z we can verify that }f 1 ´τ ¨f2 } ě }f 1 ´f2 } and that this inequality is strict as soon as τ ‰ 0. Then f 2 is the only element of rf 2 s registered with f 1 then sprf 2 sq " f 2 . Likewise for sprf 3 sq " f 3 , then we should have:

However it is easy to verify that

This is a contradiction. Therefore, a congruent section does not exist. ˝

Congruent section in Euclidean space

We have seen that the existence of a congruent section is useful to compute an element which minimize the variance. In section 5.4.1, we were able to prove by an example that for horizontal translation action in R d a discrete space or L 2 pr0, 1sq a continuous space, we do not have the existence of a congruent section. But, for a given action, what if we are not able to find such an example to reject the existence of a congruent section by ourselves? In this section we propose an algorithm which is able to reject the existence of such a congruent section.

Let G be a group acting on M (a Hilbert space). First, we can notice that, if d Q pras, rbsq " inf gPG }a ´g ¨b} is not a distance in a quotient space, then a section can not exist. Indeed if we have d M a distance in the ambient space and s a congruent section, every property of the distance d M would be verified by d Q through the congruent section. Therefore in the following, we restrict ourselves to isometric action. Now, we propose a simple algorithm which can reject the existence of a congruent section in Euclidean space. Let M " R d be an Euclidean space. Taking random points in the ambient space py 1 , . . . , y n q. If there is a congruent section, in particular we have:

(5.6)

By expanding the square norms, we get:

(5.7)

Even if we do not know the family of vectors px i ´xn q 1ďiďn´1 , thanks to the chosen vectors py i q i , and thanks to equations (5.6) and (5.7), we can compute A P M n´1 pRq, the Gram matrix associated to the family of vectors px i ´xn q 1ďiďn´1 (if these vectors px i q exist):

@pi, jq P 1, n 2 A ij " xx i ´xn , x j ´xn y " 1 2 `dQ pry i s, ry n sq 2 `dQ pry j s, ry n sq 2 ´dQ pry i s, ry j sq 2 ˘.

If we denote by B the matrix such that the i-th column of B are the coordinate of x i ´xn , we have A " pBq T B, moreover the rank of B is equal to the rank of A. Then we should have rankpAq " rankpBq ď dim M (since the vectors x i ´xn belongs to M ).

Example 5.4. Let us chose G " tI 2 , ´I2 u acting isometrically on M " R 2 , an orbit of a point m is rms " tm, ´mu, the quotient distance is:

We chose y 1 " p1, 0q, y 2 " p 1 2 , ? 3 2 q, y 3 " p´1 In this thesis, we have divided the proofs of inconsistency in several results with all different hypothesis on the action or on the template. On figure 6.1 we draw a Venn diagram of some properties of the action with a few examples that we have seen in this thesis. One important property was the fact that the distance was invariant under the group action, a particular case was the isometric action in Hilbert spaces. The most advanced results was for isometric action in Hilbert spaces. Even if we were able to prove inconsistency for non isometric action, but for a large enough noise level (propositions 4.6 and 4.7).

The role played by the constants

One important constant in the consistency bias is the noise level σ. Indeed, in theorem 4.1 (for isometric action in Hilbert space), we have seen that the consistency bias was asymptotically linear with respect to σ. Besides for non isometric action, we were able to show inconsistency as soon as the noise level was above a certain threshold (propositions 4.6 and 4.7).

But σ is not the only constant which appears in the study of the consistency: there were also the constants K, λpt 0 q, θpm ‹ q and θpt 0 q. Let us make a brief overview

Variation du groupe d'isotropie

Un autre résultat de cette thèse, est la preuve que sous certaines conditions le groupe d'isotropie de la moyenne de Fréchet dans le quotient est plus petit que le template. Ce théorème est prouvé dans les espaces de Hilbert ainsi que les variétés riemanniennes complètes lorsque la distance est invariante sous l'action de groupe, ce résultat est similaire à celui de [START_REF] Huckemann | On the meaning of mean shape: manifold stability, locus and the two sample test[END_REF]].

Cela prouve non seulement une nouvelle fois l'inconsistance mais aussi que le template et la moyenne de Fréchet dans le quotient n'ont pas les mêmes propriétés géométriques.