Habib Ammari

Alexandre D'

Bénédicte, Isabelle Albane

Joëlle Lara

Michelle, Valérie Lise-Marie Laurence

Zaïna

Grégoire, Mathieu Vincent Édouard

aux Carmine

Guy, Ivan, Matthew, Sira Gilles

Xiu-Yuan

Michel Mia

Naoufal Paul

Zhuoran Au Dma

Merci À Cécile

Charles

Automatically understanding the content of a natural signal, like a sound or an image, is in general a difficult task. In their naive representation, signals are indeed complicated objects, belonging to high-dimensional spaces. With a different representation, they can however be easier to interpret.

This thesis considers a representation commonly used in these cases, in particular for the analysis of audio signals: the modulus of the wavelet transform. To better understand the behaviour of this operator, we study, from a theoretical as well as algorithmic point of view, the corresponding inverse problem: the reconstruction of a signal from the modulus of its wavelet transform.

This problem belongs to a wider class of inverse problems: phase retrieval problems. In a first chapter, we describe a new algorithm, PhaseCut, which numerically solves a generic phase retrieval problem. Like the similar algorithm PhaseLift, PhaseCut relies on a convex relaxation of the phase retrieval problem, which happens to be of the same form as relaxations of the widely studied problem MaxCut. We compare the performances of PhaseCut and PhaseLift, in terms of precision and complexity.

In the next two chapters, we study the specific case of phase retrieval for the wavelet transform. We show that any function with no negative frequencies is uniquely determined (up to a global phase) by the modulus of its wavelet transform, but that the reconstruction from the modulus is not stable to noise, for a strong notion of stability. However, we prove a local stability property. We also present a new non-convex phase retrieval algorithm, which is specific to the case of the wavelet transform, and we numerically study its performances.

Finally, in the last two chapters, we study a more sophisticated representation, built from the modulus of the wavelet transform: the scattering transform. Our goal is to understand which properties of a signal are characterized by its scattering transform. We first prove that the energy of scattering coefficients of a signal, at a given order, is upper bounded by the energy of the signal itself, convolved with a high-pass filter that depends on the order. We then study a generalization of the scattering transform, for stationary processes. We show that, in finite dimension, this generalized transform preserves the norm. In dimension one, we also show that the generalized scattering coefficients of a process characterize the tail of its distribution. 1

Résumé

Les tâches qui consistent à comprendre automatiquement le contenu d'un signal naturel, comme une image ou un son, sont en général difficiles. En effet, dans leur représentation naïve, les signaux sont des objets compliqués, appartenant à des espaces de grande dimension. Représentés différemment, ils peuvent en revanche être plus faciles à interpréter.

Cette thèse s'intéresse à une représentation fréquemment utilisée dans ce genre de situations, notamment pour analyser des signaux audio : le module de la transformée en ondelettes. Pour mieux comprendre son comportement, nous considérons, d'un point de vue théorique et algorithmique, le problème inverse correspondant : la reconstruction d'un signal à partir du module de sa transformée en ondelettes.

Ce problème appartient à une classe plus générale de problèmes inverses : les problèmes de reconstruction de phase. Dans un premier chapitre, nous décrivons un nouvel algorithme, Phase-Cut, qui résoud numériquement un problème de reconstruction de phase générique. Comme l'algorithme similaire PhaseLift, PhaseCut utilise une relaxation convexe, qui se trouve en l'occurence être de la même forme que les relaxations du problème abondamment étudié Max-Cut. Nous comparons les performances de PhaseCut et PhaseLift, en termes de précision et de rapidité.

Pendant les quatre années qu'a duré ma thèse, Stéphane Mallat m'a fait découvrir avec passion un domaine mathématique dont, initialement, je soupçonnais à peine l'existence. J'ai pu bénéficier de sa vision claire et profonde de tous les sujets que nous avons abordés, de ses idées foisonnantes et de sa grande culture scientifique. Je lui suis aussi reconnaissante de sa patience et de sa disponibilité.

Ce sont Albert Cohen et Francis Bach qui, lorsque j'étais en master, m'ont incitée à le contacter ; merci à eux pour ce conseil fructueux. J'ai beaucoup apprécié les collaborations aussi conviviales qu'instructives que nous avons menées avec d'autres chercheurs de l'ENS. Merci donc à Alexandre d'Aspremont et Fajwel Fogel, ainsi qu'à Habib Ammari et Han Wang. Merci également à Alexandre pour son aide précieuse dans ma recherche de post-doc.

En janvier 2013, j'ai eu la chance de passer un mois à l'université de Yale. Je suis très reconnaissante à Ronald Coifman de m'avoir invitée et de m'avoir consacré autant

Dans les deux chapitres suivants, nous étudions le cas particulier de la reconstruction de phase pour la transformée en ondelettes. Nous montrons que toute fonction sans fréquence négative est uniquement déterminée (à une phase globale près) par le module de sa transformée en ondelettes, mais que la reconstruction à partir du module n'est pas stable au bruit, pour une définition forte de la stabilité. On démontre en revanche une propriété de stabilité locale. Nous présentons également un nouvel algorithme de reconstruction de phase, non-convexe, qui est spécifique à la transformée en ondelettes, et étudions numériquement ses performances.

Enfin, dans les deux derniers chapitres, nous étudions une représentation plus sophistiquée, construite à partir du module de transformée en ondelettes : la transformée de scattering. Notre but est de comprendre quelles propriétés d'un signal sont caractérisées par sa transformée de scattering. On commence par démontrer un théorème majorant l'énergie des coefficients de scattering d'un signal, à un ordre donné, en fonction de l'énergie du signal initial, convolé par un filtre passe-haut qui dépend de l'ordre. On étudie ensuite une généralisation de la transformée de scattering, qui s'applique à des processus stationnaires. On montre qu'en dimension finie, cette transformée généralisée préserve la norme. En dimension un, on montre également que les coefficients de scattering généralisés d'un processus caractérisent la queue de distribution du processus.

Chapter 1 Introduction

The goal of data analysis is to develop methods to automatically understand the content of natural signals. Examples of possible tasks are finding which objects are pictured in an image, or transcribing an audio recording of a human voice.

These problems are in general difficult, because, under their raw form, signals tend to be complicated objects, living in high-dimensional spaces. An audio signal of 2 seconds, sampled at 44.1 kHz, is for example an element of R 88200 . To overcome this difficulty, a common method is to find a better representation of signals, which makes them easier to interpret.

Such representations must satisfy two essential properties. They must be discriminative: signals representing different things must have different representations. On the other hand, signals representing the same thing (two audio recordings of the same voice pronouncing the same words, for example) must have identical representations, even if they are not equal.

This thesis studies the wavelet transform modulus, which is an example of such a representation, used in particular in audio processing.

To understand to what extent it satisfies the previous two properties, we consider the corresponding inverse problem: the reconstruction of a function from its wavelet transform modulus. It is an example of a well-known class of inverse problems: phase retrieval problems. Chapter 2 of this thesis describes an algorithm to solve generic phase retrieval problems. Chapters 3 and 4 are devoted to the specific case of the wavelet transform.

In Chapters 5 and 6, we consider more sophisticated representations that are built from the wavelet transform modulus: scattering transforms.

Section 1.1 of this introduction is about the class of phase retrieval problems. In Section 1.2, we consider the specific case of phase retrieval for the wavelet transform. Section 1.3 is devoted to the scattering transform. In each section, an introduction explains the main definitions and known results, while the latter parts present the contributions of this thesis.

Phase retrieval problems

Phase retrieval problems are inverse problems of the form: Reconstruct x from {|L i (x)|} i∈I where: x is an unknown element of a complex vector space E I is an arbitrary set of indexes for any i ∈ I, L i : E → C is a known linear form |.| denotes the usual complex modulus Multiplying x by a unitary complex number has no influence on {|L i (x)|} i∈I :

if |λ| = 1, then ∀i ∈ I, |L i (x)| = |L i (λx)|
so we can not hope to reconstruct exactly x from the set of modulus {|L i (x)|} i∈I . We only try to recover x up to a global phase, that is up to multiplication by a unitary complex number.

Phase retrieval problems have been studied from the fifties because of their numerous physical applications, for example X-ray imaging [START_REF] Miao | Extending x-ray crystallography to allow the imaging of noncrystalline materials, cells, and single protein complexes[END_REF], diffractive imaging [START_REF] Bunk | Diffractive imaging for periodic samples: retrieving one-dimensional concentration profiles across microfluidic channels[END_REF] or astronomy [START_REF] Dainty | Phase retrieval and image reconstruction for astronomy[END_REF]. In this section, we describe the theoretical and algorithmic issues they raise. We then come to the first contribution of this thesis (chapter 2): a general algorithm to solve phase retrieval problems.

Theoretical issues

Given a specific phase retrieval problem, the main theoretical question that it raises is to know whether the problem is well-posed, in terms of uniqueness and stability: is the reconstruction unique, up to a global phase? If it is unique, is it stable under measurement noise?

We call uniqueness in a phase retrieval problem the fact that all vectors x ∈ E are uniquely determined by {|L i (x)|} i∈I , up to a global phase:

∀x, y ∈ E such that ∀i ∈ I, |L i (x)| = |L i (y)|, ∃λ ∈ C, |λ| = 1 such that x = λy (Uniqueness)
The definition of stability is less canonical. Broadly speaking, we wish that, when the modulus |L i (x)| are not exactly known, but only up to a small error, it is still possible to reconstruct an approximation of the unknown x:

∀x, y ∈ E such that ∀i ∈ I, |L i (x)| ≈ |L i (y)|, ∃λ ∈ C, |λ| = 1 such that x ≈ λy (Stability)
Various meanings are possible for the symbol "≈". When the spaces E and (R + ) I are endowed with metrics, which we respectively denote by d E and d I , we can for example decide that the reconstruction is stable if there exists C > 0 such that, for all > 0: ∀x, y ∈ E such that d I ({|L i (x)|} i∈I , {|L i (y)|} i∈I ) < , ∃λ ∈ C, |λ| = 1 such that d E (x, λy) < C (Strong stability)

This notion of stability can however be too strong. For the wavelet transform, we will have to introduce a weaker notion of local stability.

Uniqueness and stability results for phase retrieval problems can be grouped in two main families:

• when the measurements L i are randomly chosen: at least for particular choices of probability laws, the phase retrieval problem is then well-posed with high probability; (Uniqueness) and (Strong stability) both hold. The drawback of this case is that, in many applications, the measurements are fixed: they can not be chosen at random.

• when the measurements L i are imposed by concrete applications: although useful, this setting is in general more difficult to theoretically study. The main cases that can be handled are when the L i have a particular form, which allows to use harmonic analysis properties. Even in these cases, the results are often negative: there is no uniqueness or no stability.

For random measurements, when the ambient space E has finite dimension, there is uniqueness with probability 1 if the probability measure is uniformly continuous with respect to Lebesgue measure [START_REF] Balan | On signal reconstruction without noisy phase[END_REF][START_REF] Conca | Algebraic characterization of injectivity in phase retrieval[END_REF], and the number of measurements is large enough:

Card I ≥ 4 dim E -4

When E = C n and the measurements are independently chosen according to a normal random law, there is uniqueness and (strong) stability with high probability [START_REF] Candès | Phaselift: exact and stable signal recovery from magnitude measurements via convex programming[END_REF][START_REF] Candès | Solving quadratic equations via phaselift when there are about as many equations as unknowns[END_REF], provided that, for a known constant C:

Card I ≥ C dim E
Other probability laws can be considered, as in [START_REF] Alexeev | Phase retrieval with polarization[END_REF]Candès et al., 2015;Gross et al., 2015a].

For deterministic measurements, the most well-known case, and also the most important for applications, is the reconstruction of a compactly-supported function from the modulus of its Fourier transform:

Reconstruct a compactly-supported f ∈ L 2 (R) from | f |
Here, the recovery is only considered up to trivial ambiguities (multiplication by a global phase, translation and complex conjugation). Even up to these ambiguities, there is no uniqueness [START_REF] Akutowicz | On the determination of the phase of a Fourier integral[END_REF][START_REF] Walther | The question of phase retrieval in optics[END_REF]. For uniqueness to hold, it is necessary that the signals f verify additional hypotheses, such as sparsity [START_REF] Ranieri | Phase retrieval for sparse signals: uniqueness conditions[END_REF].

In higher dimensions (f ∈ L 2 (R d ) with d ≥ 2), there is uniqueness for "generic" functions f [ [START_REF] Barakat | Necessary conditions for a unique solution to two-dimensional phase recovery[END_REF] but no stability.

The study of the phase retrieval problem for the Fourier transform relies on tools from harmonic analysis. These tools can be adapted to handle other measurements than the Fourier transform, notably fractional Fourier transforms. In this latter case, uniqueness holds, under simple conditions on the parameters of the fractional Fourier transforms [START_REF] Jaming | Uniqueness results in an extension of pauli's phase retrieval problem[END_REF]. However, there is no strong stability [START_REF] Andreys | Zak transform and non-uniqueness in an extension of pauli's phase retrieval problem[END_REF].

An intermediate way between random measurements and fixed measurements imposed by a concrete situation is to design deterministic measurements so that they satisfy uniqueness or stability properties, as in [START_REF] Bodmann | Stable phase retrieval with low-redundancy frames[END_REF]. This method is interesting from a theoretical point of view, but it suffers from the same drawback as random measurements: it has in general no physical application.

Algorithms

Even when it is well-posed, numerically solving a phase retrieval problem is also difficult. Two main families of algorithms have been designed for generic phase retrieval problems, each with its advantages and drawbacks:

• iterative methods, which are simple to implement and relatively fast, but tend to return erroneous reconstructions

• methods by convexification, which often achieve exact reconstruction, but have high complexity

Iterative methods are the older family. They consist in building a sequence of approximate solutions, refined step by step until it converges, and returning the limit of this sequence. They notably include the alternate projection algorithm, introduced by Gerchberg and Saxton [1972], and improved by [START_REF] Fienup | Phase retrieval algorithms: a comparison[END_REF].

Unfortunately, there is in general no guarantee that the sequence of approximate solutions converges towards the true solution. Phase retrieval problems are indeed highly non-linear, and do not admit a convex formulation. The sequence of approximations can therefore converge only towards a local optimum.

Depending on the considered application, various heuristics have been developed to overcome convergence problems [START_REF] Millane | Phase retrieval in cristallography and optics[END_REF]. For particular choices of random measurements, iterative methods have been proven to converge towards the true solutions when correctly initialized [START_REF] Netrapalli | Phase retrieval using alternating minimization[END_REF]Candès et al., 2015]. However, in general, their convergence properties are not well understood. For phase retrieval problems which arise in audio processing (and to which we will come back), they are often disappointing: reconstructed signals tend to present audio artifacts [START_REF] Sturmel | Signal reconstruction from STFT magnitude: a state of the art[END_REF].

Methods by convexification have been introduced by [START_REF] Chai | Array imaging using intensity-only measurements[END_REF] and by [START_REF] Candès | Phaselift: exact and stable signal recovery from magnitude measurements via convex programming[END_REF], whose algorithm PhaseLift we briefly describe. The principle is to reformulate the phase retrieval problem under a matricial form. This reformulation is still a non-convex problem, but has a convex approximation. At least for precise choices of random measurements, the solution to the approximated problem can be proven to be the same as the solution to the initial problem, with high probability.

To describe the reformulation of the problem into a matricial form, we assume that we are in a finite-dimensional setting: E = C n , for some n ∈ N * . The linear forms L i : E → C are then matrices with one line: L i ∈ M 1,n (C). For any i, we set b i = |L i (x)|. The constraint |L i (x)| = b i can be rewritten as:

|L i (x)| = b i ⇐⇒ |L i (x)| 2 = b 2 i ⇐⇒ x * L * i L i x = b 2 i ⇐⇒ Tr (L * i L i xx * ) = b 2 i
The phase retrieval problem then becomes:

Find x s.t. ∀i, |L i (x)| = b i ⇐⇒ Find x s.t. ∀i, Tr (L * i L i xx * ) = b 2 i
With a change of variable X = xx * , this is equivalent to:

Find X ∈ M n (C) s.t. rank(X) = 1 and ∀i, Tr (L * i L i X) = b 2 i
This problem is not convex, because the set of matrices with rank 1 is not convex. However, it can be approximated by a convex problem, with the same method as used for matrix completion [START_REF] Candès | Exact matrix completion via convex optimization[END_REF]. This convex problem can be solved in a time which is polynomial in the dimension n and the number of measurements. [START_REF] Candès | Phaselift: exact and stable signal recovery from magnitude measurements via convex programming[END_REF]; [START_REF] Candès | Solving quadratic equations via phaselift when there are about as many equations as unknowns[END_REF] have shown that, when the linear forms L i are chosen according to random normal laws, then the convex approximation and the initial problem have the same solution, provided that:

Card I ≥ C dim E
(where C is a known constant).

Even if these correctness guarantees are limited to specific choices of measurements, empirical evaluations show that PhaseLift yields exact reconstructions in many cases.

However, the dimension of the reformulated problem is much higher than the dimension of the original problem: X has n 2 entries, while x is an n-dimensional vector. The algorithm thus has a high (although polynomial) complexity, worse than iterative methods. It prevents it from being used for applications where signals are necessarily of high dimension, as in audio processing.

1.1.3 A general phase retrieval algorithm, PhaseCut (chapter 2)

Chapter 2 of this thesis describes a new phase retrieval algorithm, called PhaseCut, belonging to the class of methods by convexification. It is similar to PhaseLift, but uses a different matricial reformulation. It yields an algorithm with a different complexity, more efficient than PhaseLift for some instances of phase retrieval problems.

Instead of directly reconstructing x ∈ R n , as in PhaseLift, we focus on reconstructing {L i (x)} i∈I (which, under mild assumptions, also allows to recover x). By assumption, the modulus |L i (x)| are known. So it suffices to reconstruct the phases, that is the complex numbers u i such that:

L i (x) = |L i (x)|u i and |u i | = 1
In the same way as, in PhaseLift, the constraints |L i (x)| = b i could be reformulated by introducing the matrix X = xx * , here, the constraints |u i | = 1 can be reformulated by introducing the matrix U = uu * , where:

u = u 1 . . . um
So, as in PhaseLift, we obtain a reformulation of the phase retrieval problem in terms of matrices, and this reformulation also has a convex approximation, which we call PhaseCut.

Almost the same correctness guarantees hold for PhaseLift and PhaseCut. We can actually show that PhaseLift yields exact reconstruction when and only when a slightly modified version of PhaseCut yields exact reconstruction. The same argument allows one to compare the stability to noise of both algorithms: PhaseCut is at least as stable to noise as a slightly modified version of PhaseLift, but can be more stable, especially when the set of measurements {L i (x)} i∈I is sparse.

The main advantage of PhaseCut over PhaseLift is its different complexity. At first sight, the complexity of PhaseCut seems worse, because the involved matrices are of higher dimension as in PhaseLift: they have m 2 entries, with m the number of measurements, while matrices in PhaseLift only have n 2 entries, with n the dimension of the unknown x (always smaller than m). However, it happens that the convex problem obtained with PhaseCut has the same form as convex relaxations of MaxCut-type problems, which have been widely studied since their introduction in [START_REF] Delorme | Laplacian eigenvalues and the maximum cut problem[END_REF][START_REF] Goemans | Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming[END_REF]. We can thus apply to PhaseCut optimization techniques developed for MaxCut, which do not apply to PhaseLift.

So the complexity of PhaseCut can be better or worse than the one of PhaseLift, depending on the optimization algorithm used to solve the convex problem. PhaseLift will then be more efficient for some problems, while PhaseCut will be better suited to others. Our numerical experiments indicate that PhaseCut is better for difficult phase retrieval problems, where the reconstruction is not very stable.

Phase retrieval for the wavelet transform

We now consider a specific phase retrieval problem, with important applications in audio processing: the case of the wavelet transform. We first define the wavelet transform, and give our motivations for studying the corresponding phase retrieval problem. Finally, we describe the theoretical and algorithmic contributions of this thesis.

Wavelet transform modulus

Wavelets have been introduced at the end of the eighties; introductions can be found in [START_REF] Cohen | Ondelettes et traitement numérique du signal[END_REF][START_REF] Daubechies | Ten Lectures on Wavelets[END_REF][START_REF] Mallat | A wavelet tour of signal processing[END_REF].

In this thesis, we call wavelet any function ψ : R → C, in L 1 ∩ L 2 (R) such that: R ψ(t)dt = 0

If such a function ψ is fixed, we can define by contraction or dilation of ψ a whole family of wavelets (ψ j ) j∈Z :

∀j ∈ Z, t ∈ R ψ j (t) = 2 -j ψ(2 -j t) ⇐⇒ ∀j ∈ Z, ω ∈ R ψj (ω) = ψ(2 j ω)
The wavelet transform W : L 2 (R) → (L 2 (R)) Z is then:

W : L 2 (R) → (L 2 (R)) Z f → {f ψ j } j∈Z
In general, the mother wavelet ψ is chosen so that it is well-localized around 0, and its Fourier transform ψ is well-localized around 1. For any j ∈ Z, ψ j can then be interpreted as a band-pass filter of characteristic frequency 2 -j and bandwidth proportional to 2 -j . Informally, the wavelet transform of a function f is thus a decomposition of f in (overlapping) frequency bands.

Figure 1.1 shows an example of a wavelet family, and Figure 1.2a an example of wavelet transform.

Wavelets are generally complex-valued, so too are thus the f ψ j . We denote the wavelet transform modulus by:

|W | : L 2 (R) → (L 2 (R)) Z f → {|f ψ j |} j∈Z
Informally, |f ψ j (t)| represents the energy of the signal f , around time t, in the frequency band centered at 2 -j . An example is on Figure 1.2b.

Interest of the phase retrieval problem

The modulus of the wavelet transform, sometimes called scalogram, is a common way to represent and analyze audio signals, relatively similar to the spectrogram. It has also been proposed as a model of the auditory cortex [START_REF] Yang | Auditory representations of acoustic signals[END_REF]. Indeed, it possesses the desirable properties mentioned at the beginning of this introduction [START_REF] Balan | On signal reconstruction without noisy phase[END_REF][START_REF] Risset | Exploration of timbre by analysis and synthesis[END_REF]:

• Audio signals which sound different to the human ear have different scalograms.

• Audio signals which are identical to the ear have almost the same scalograms, although they may not be equal.

To theoretically justify these properties, it is natural to consider the inverse phase retrieval problem: to what extent is it possible to recover an audio signal from the modulus of its wavelet transform? This problem has been investigated from the early eighties [START_REF] Griffin | Signal estimation from modified short-time fourier transform[END_REF][START_REF] Nawab | Signal reconstruction from short-time fourier transform magnitude[END_REF], but from an experimental, rather than theoretical, point of view. (a) Example of wavelet transform. On the left: a function f . On the right, from top to bottom: f ψ 1 , f ψ 0 , f ψ -1 (only real parts are displayed). Besides its theoretical motivations, the phase retrieval problem for the wavelet transform has concrete applications: in audio processing, some tasks require modifying the scalogram of a given signal, then reconstructing a new signal. It then amounts to solve a phase retrieval problem. Examples include blind source separation [START_REF] Virtanen | Monaural sound source separation by nonnegative matrix factorization with temporal continuity and sparseness criteria[END_REF] and audio texture synthesis [Bruna and Mallat, 2013b].

For the study of phase retrieval problems, the case of the wavelet transform is also interesting, because it has atypical properties, compared to other known examples:

• It is a non-random case, relevant for applications, where harmonic analysis properties allow to prove a uniqueness result, for a specific choice of the wavelet family.

• We can show that the reconstruction is not stable to noise in a strong sense. However, for a specific choice of the wavelet family, it satisfies a local stability property, which had not been observed until now.

1.2.3 Phase retrieval for the Cauchy wavelet transform (chapter 3)

In chapter 3, we study the phase retrieval problem from a theoretical point of view, for a specific choice of wavelets.

We use Cauchy wavelets (which will be defined in chapter 3). These wavelets satisfy a very particular property: for any function f ∈ L 2 (R), ∀j ∈ Z, t ∈ R, f ψ j (t) = c j F (t + i2 j )

where (c j ) j∈Z is a sequence of explicit constants, and F is the holomorphic extension of (a modification of) f to the complex upper plane. The phase retrieval problem can then be rephrased in terms of holomorphic functions, and analyzed with techniques similar to [START_REF] Akutowicz | On the determination of the phase of a Fourier integral[END_REF], by factorizing F into Blaschke products.

We have a uniqueness theorem:

Theorem (3.2). If f, g ∈ L 2 (R)
are two functions such that f (ω) = ĝ(ω) = 0 for any ω < 0, and, for at least two distinct values of j ∈ Z:

|f ψ j | = |g ψ j |
then f = e iφ g for some φ ∈ R.

This result is strong in the sense that it does not require the whole wavelet transforms of f and g to be equal in modulus: it suffices that only two components are equal.

|W |f -|W |g 2 < but ∀φ ∈ R, ||f -e iφ g|| 2 ≥ 1

where |W | still denotes the modulus of the wavelet transform. This holds for all wavelets, not especially Cauchy ones.

We nevertheless have a form of local stability. If |W |f ≈ |W |g, then our numerical experiments show that: ∀j ∈ Z, t ∈ R, f ψ j (t) ≈ e iφ j (t) g ψ j (t) (1.1) where (j, t) → φ j (t) is a phase whose variation is slow in j and t, except maybe at the points where f ψ j (t) is close to zero. Informally, Equation (1.1) expresses the fact that the wavelet transforms of f and g are approximately equal, up to a global phase, in the neighborhood of each point (j, t) of the time-frequency plane. Hence the term local stability. This numerical result can be formalized and proven for Cauchy wavelets (Theorems 3.17 and 3.18), although the statements are more technical than this simplified exposition.

1.2.4 Phase retrieval for wavelet transforms: a non-convex algorithm (chapter 4)

Results from Chapter 3 indicate that phase retrieval for the wavelet transform is a relatively well-posed problem, as reconstruction is unique and locally stable. However, generic phase retrieval algorithms tend to yield disappointing numerical results:

• Convexification methods like PhaseLift [START_REF] Chai | Array imaging using intensity-only measurements[END_REF][START_REF] Candès | Phase retrieval via matrix completion[END_REF] and Phase-Cut (Chapter 2) give very encouraging results for toy problems of small dimension [START_REF] Sun | Estimating a signal from a magnitude spectrogram via convex optimization[END_REF], Paragraph 2.4.3]. However, their complexity is too high for them to be used on real-size problems.

• Iterative methods for audio signals have been introduced by [START_REF] Griffin | Signal estimation from modified short-time fourier transform[END_REF] and subsequently refined [START_REF] Achan | Probabilistic inference of speech signals from phaseless spectrograms[END_REF][START_REF] Bouvrie | An incremental algorihtm for signal reconstruction from stft magnitude[END_REF]. However, as the problem is non-convex, they tend to get stuck in local optima, and they generally do not yield reconstruction results of high quality.

In Chapter 4, we propose a new iterative method, achieving the same precision as convexification methods on small-size problems, with a complexity roughly linear in the size of the signal, allowing it to be used on real audio signals.

The main tool of this algorithm is a reformulation of the phase retrieval problem. This reformulation uses the notion of holomorphic extension, as in Chapter 3, but applies to any family of (exponentially-decaying wavelets), and not only to Cauchy ones.

The reformulation yields a multiscale algorithm, reconstructing the unknown signal f frequency band by frequency band, starting with the low frequencies and ending with the high ones. At each step, the already recovered phase information is exploited to reconstruct one more frequency band.

Numerical experiments confirm that the algorithm performs well: the reconstruction is of high quality, and degrades proportionally to measurement noise. It is worth noting that, because the phase retrieval problem is locally but not globally stable, the signals that are reconstructed from the wavelet transform modulus may not be very close to the original signals. However, they have almost exactly the same wavelet transform, in modulus. In the case of audio signals, original signals and their reconstructions, although not necessarily equal, are in general indistinguishable to the ear.

We use our algorithm to experimentally validate the theoretical conclusions of Chapter 3. We in particular highlight the fact that reconstruction is more difficult and less stable when the wavelet transform modulus has a lot of values close to zero.

Scattering transforms

In the last two chapters of the thesis, we consider the integration of the wavelet transform modulus in a more sophisticated representation: the scattering transform.

The scattering operator, introduced in [START_REF] Mallat | Group invariant scattering[END_REF], is a deep representation, obtained by iterative application of the wavelet transform modulus. It has been defined so as to be invariant to translations of the input signal, and stable to small deformations. Since then, it has achieved important success in various data analysis tasks.

After a definition of the scattering (Paragraph 1.3.1), we discuss its main achievements and the theoretical questions it raises (Paragraph 1.3.2). In the frame of this thesis, the main question is to understand which properties of a signal are characterized by its scattering transform, especially which regularity properties. In Paragraphs 1. 3.3 and 1.3.4, we describe the results of Chapters 5 and 6 related to this question.

Definition of scattering

The scattering transform is defined as a cascade of wavelet transform modulus, followed by averages.

Starting from a function f ∈ L 2 (R), we compute its local temporal mean by convolving it with a smooth window function φ J , whose support has a characteristic size proportional to 2 J , for some J ∈ Z:

f → f φ J
This temporal mean f φ J (called scattering coefficient of order 0) has the good property of being almost invariant under translations of f :

f φ J ≈ f (. -τ ) φ J when |τ | 2 J
However, f φ J gives a very rough information on f : all high frequencies of f have been lost. Therefore, although it has the advantage of being invariant to translations, f φ J is not discriminative enough to be an interesting representation of f . We recover the high frequencies with the wavelet transform (whose very low frequencies can be discarded):

f ψ J , f ψ J-1 , f ψ J-2 , ...
After an application of the complex modulus, we average these components with φ J , to obtain scattering coefficients of order 1:

|f ψ J | φ J , |f ψ J-1 | φ J , |f ψ J-2 | φ J , ...
Again, these averaged functions are almost invariant under translations of f . Together with f φ J , they bring more information about f than f φ J alone, and thus form a more discriminative representation. However, the high frequencies of the |f ψ j | have been lost during the averaging. We recover them with a new application of the wavelet transform modulus, and so on.

Scattering coefficients of order n are thus defined as the temporal average of the wavelet transform modulus, n times composed with itself. The whole process is schematized on Figure 1.3. Scattering coefficients make up a representation of f that is invariant to translations (when J goes to infinity) and stable to small deformations of f [START_REF] Mallat | Group invariant scattering[END_REF].

This definition can be refined so as to be invariant or stable to other transformations than translations and deformations, for example rotations [START_REF] Sifre | Rotation, scaling and deformation invariant scattering for texture discrimination[END_REF][START_REF] Oyallon | Deep roto-translation scattering for object classification[END_REF].

Success and open question

Its invariance and stability properties allow the scattering transform to be successfully applied to various data analysis tasks, in particular classification tasks. Applications cover a high range of domains, such as images [Bruna and Mallat, 2013a], audio signals [START_REF] Andén | Multiscale scattering for audio classification[END_REF] or even quantum chemistry [START_REF] Hirn | Quantum energy regression using scattering transforms[END_REF].

Compared to learned classifiers, the scattering offers competitive performances, with the advantage of being a deterministic transform, that does not need to be trained from data. This gives a simple and generic framework, that can be efficiently implemented [Sifre et al., 2013], and analyzed from a theoretical point of view.

f |f ψ J-2 | |f ψ J-1 | |f ψ J | ||f ψ J | ψ J-1 | ||f ψ J | ψ J | f φ J ... |f ψ J-2 | φ J |f ψ J-1 | φ J |f ψ J | φ J ... ||f ψ J-1 | ψ J | φ J ... ||f ψ J | ψ J-1 | φ J ||f ψ J | ψ J | φ J ... ... ... ... ... ...
One of the main problems related to scattering is to determine which properties of a signal can be recovered from its scattering transform.

The scattering transform of a signal does not uniquely determine the signal. However, it characterizes some of its properties. For example, from the scattering transform of a sample of an auditive texture, it is not possible to exactly recover the sample. However, a perceptually plausible sample of the same texture can be reconstructed [Bruna and Mallat, 2013b].

In particular, we wonder to what extent the regularity of a signal can be characterized from its scattering transform. It is known that some regularity classes, like Besov spaces, can be characterized by the decay of wavelet coefficients [START_REF] Devore | Compression of wavelet decompositions[END_REF]. Are there similar results for scattering coefficients? An indication that this may be possible is that first and second-order scattering coefficients are already known to characterize some structural properties of stationary processes [START_REF] Bruna | Intermittent process analysis with scattering moments[END_REF].

Exponential decay of scattering coefficients (chapter 5)

In Chapter 5, we prove a result in this optic. We consider the decay of scattering coefficients as a function of their order; we show that it is controlled by the decay of the signal's Fourier transform. [START_REF] Mallat | Group invariant scattering[END_REF] proves that the scattering transform preserves the L 2 -norm, provided that the wavelets satisfy some hypotheses. For any real-valued function f ∈ L 2 (R):

n≥0 ||S n [f ]|| 2 2 = ||f || 2 2
where S n [f ] is the set of scattering coefficients of f with order n. However, this result gives no information on how ||S n (f )|| 2 decays with n. Theorem 5.2 partially solves this problem with an upper bound on ||S n (f )|| 2 . It indeed states that, under relatively general hypotheses on the wavelets, there exist constants r > 0, a > 1 such that, for any n ≥ 2 and any real-valued function f ∈ L 2 (R):

m≥n ||S m [f ]|| 2 2 ≤ R | f (ω)| 2 1 -e -( ω ra n ) 2 dω (1.2) The function ω → 1 -e -( ω ra n ) 2
is a high-pass filter, with a bandwidth proportional to ra n . Equation (1.2) shows the existence, in the scattering transform, of a phenomenon of energy propagation towards the low frequencies. The energy of f carried by the frequency band [-ra n ; ra n ] is output in the scattering coefficients with order smaller that n; the energy of scattering coefficients with order at least n comes almost only from higher frequency ranges.

Generalized scattering (chapter 6)

In Chapter 6, we introduce a generalization of the (finite-dimensional) scattering transform for stationary processes (as defined in [START_REF] Mallat | Group invariant scattering[END_REF]). We show that these generalized scattering coefficients characterize some simple properties of the underlying stationary processes.

The generalized scattering starts with a random process X, taking its values in R m . We iteratively define:

X 0 = X ∀n ∈ N, X n+1 = |W n X n -E(W n X n )|
where the W n are linear unitary operators and |.| denotes the coordinate-wise modulus.

The scattering coefficients are {E(W n X n )} n∈N . This definition generalizes the scattering, in the sense that the linear operators W n can represent transformations other than the wavelet transform; the expectation replaces the convolution with the low-pass filter φ J .

Its interest is to offer a mathematical framework for the study of deep learned representations, in particular the ones learned in an unsupervised way. Within this framework, we can indeed consider learning the linear operators W n , to adapt them to the law of X.

With no assumption on the ambient dimension, we simply show an energy preservation result (Theorem 6.2):

if E(X 2 ) < +∞ E(X 2 ) = +∞ n=0 ||E(W n X n )|| 2
In dimension one (that is, when the processes X n take their values in R instead of R m with general m), we can analyze the process with more precision. We show that scattering coefficients characterize the distribution tail of X (Theorem 6.4). More precisely, when X is not almost surely bounded and takes only positive values, then:

E(X n ) ∼ 2f (S n ) when n → +∞ where: -f (t) = E((X -t)1 X≥t ) characterizes the law of X -S n = k<n E(X k
) can be computed from the scattering coefficients The sequence (S n ) goes to infinity with n but S n+1 -S n → 0, so (E(X n )) n∈N precisely describes the asymptotic behavior of f , that is, the distribution tail of X.

Chapter 2

A general phase retrieval algorithm, PhaseCut

This chapter has be written in collaboration with Alexandre d 'Aspremont and Stéphane Mallat, and published in [Waldspurger et al., 2015].

We consider a generic phase retrieval problem, in finite dimension. So we aim at reconstructing an unknown vector x, from linear complex-valued measurements Ax. The phase of these measurements has been lost; we have only access to their modulus b = |Ax|. Such a problem can be ill-posed; the modulus |Ax| may for example not uniquely determine x. However, we are not concerned here with such issues: we assume the phase retrieval problem to be well-posed and focus on the design of a numerical algorithm.

Phase retrieval problems have traditionally been solved with iterative algorithms [START_REF] Gerchberg | A practical algorithm for the determination of phase from image and diffraction plane pictures[END_REF][START_REF] Fienup | Phase retrieval algorithms: a comparison[END_REF]. Despite their success in some cases [START_REF] Dainty | Phase retrieval and image reconstruction for astronomy[END_REF], these algorithms often stall in local minima, whose existence comes from the non-convexity of the problem. To overcome this difficulty, [START_REF] Chai | Array imaging using intensity-only measurements[END_REF] and [START_REF] Candès | Phase retrieval via matrix completion[END_REF] have proposed a convex relaxation for the phase retrieval problem, called PhaseLift. It has been proven in [START_REF] Candès | Phaselift: exact and stable signal recovery from magnitude measurements via convex programming[END_REF][START_REF] Candès | Solving quadratic equations via phaselift when there are about as many equations as unknowns[END_REF] that, with high probability, when the measurement matrix A has independent Gaussian entries, the relaxation is tight, meaning that it has the same solution as the original problem. So with high probability, the convex relaxation exactly reconstructs the unknown x, but, as it is convex, it can be solved with polynomial-time algorithms. It is also stable to noise.

The contribution of this chapter is to propose a different convex relaxation, called PhaseCut. Compared to PhaseLift, which directly reconstructs the unknown vector x, PhaseCut focuses on the reconstruction of the phase of Ax, which is an element of the complex unit torus. The convex relaxation to which we then arrive has the same form as semidefinite programs for solving MaxCut-type problems [START_REF] Delorme | Laplacian eigenvalues and the maximum cut problem[END_REF][START_REF] Goemans | Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming[END_REF]. Links had already been established between MaxCut and angular synchronization [START_REF] Singer | Angular synchronization by eigenvectors and semidefinite programming[END_REF] and maximum-likelihood channel detection [START_REF] Luo | An efficient quasi-maximum likelihood decoder for PSK signals[END_REF][START_REF] Kisialiou | Probabilistic analysis of semidefinite relaxation for binary quadratic minimization[END_REF]So, 2010], but it is the first time that a connection is drawn with phase retrieval.

We obtain correctness results for PhaseCut by showing (with the help of [START_REF] Voroninski | A comparison between the phaselift and phasecut algorithms[END_REF]) an equivalence with PhaseLift: in the noiseless case, the relaxation PhaseLift is tight if and only if a slightly modified version of PhaseCut is tight (the same equivalence exists between the unmodified PhaseCut and a slight modification of PhaseLift enjoying the same properties [START_REF] Demanet | Stable optimizationless recovery from phaseless linear measurements[END_REF][START_REF] Candès | Solving quadratic equations via phaselift when there are about as many equations as unknowns[END_REF]). We also compare the stability to noise of PhaseLift and PhaseCut: PhaseCut is at least as stable as a variant of PhaseLift, but empirically appears to be more stable in some cases, in particular when b = |Ax| is sparse.

Although the convex relaxation PhaseCut typically has a larger dimension thanPhaseLift, it has a simpler structure. We can then apply efficient algorithms designed for MaxCut, in particular a provably convergent block coordinate descent algorithm whose complexity per iteration is the same as in the iterative algorithm [START_REF] Gerchberg | A practical algorithm for the determination of phase from image and diffraction plane pictures[END_REF]. Depending on the used algorithm, the complexity of PhaseCut is thus different from the one of PhaseLift, and significantly better in some cases. Depending on the exact considered phase retrieval problem, one or the other algorithm suits better.

From the point of view of combinatorial optimization, showing an equivalence between phase recovery and MaxCut allows us to expose a new class of nontrivial problem instances where the semidefinite relaxation for a MaxCut-like problem is tight, together with explicit conditions for tightness directly imported from the matrix completion formulation of these problems (these conditions are also hard to check, but hold with high probability for some classes of random experiments).

The chapter is organized as follows. In Section 2.1, we explain how to factorize away the magnitude information in the phase retrieval problem, so as to reformulate it as a non-convex quadratic optimization problem on the phase variables. The convex relaxation is derived in Paragraph 2.1.4. In Section 2.2, we detail several algorithms for solving this problem. In Section 2.3, we prove the equivalence, in terms of tightness, between a variant of PhaseCut and PhaseLift. We compare the stability to noise and the complexity of both algorithms. In Section 2.4, we perform numerical experiments comparing PhaseLift, PhaseCut and an iterative baseline, for three different phase retrieval problems. Section 2.5 contains technical lemmas.

Notations

We write S p (resp. H p ) the cone of symmetric (resp. Hermitian) matrices of dimension p; S + p (resp. H + p ) denotes the set of positive symmetric (resp. Hermitian) matrices.

We write • p the Schatten p-norm of a matrix, that is the p-norm of the vector of its eigenvalues (in particular, • ∞ is the spectral norm). The notation A 1 refers to the sum of the modulus of the coefficients of A.

We write A † the (Moore-Penrose) pseudo-inverse of a matrix A. For x ∈ R p , we write diag (x) the matrix with diagonal x. When X ∈ H p however, diag (X) is the vector containing the diagonal elements of X. For X ∈ H p , X * is the Hermitian transpose of X, with X * = ( X) T . Finally, we write b 2 the vector with components b 2 i , i = 1, . . . , n.

Phase recovery

The phase recovery problem seeks to retrieve a signal x ∈ C p from the amplitude b = |Ax| of n linear measurements, solving find

x such that |Ax| = b, (2.1) in the variable x ∈ C p , where A ∈ C n×p and b ∈ R n .

Greedy optimization in the signal

Approximate solutions x of the recovery problem in (2.1) are usually computed from b = |Ax| using algorithms inspired from the alternating projection method in [START_REF] Gerchberg | A practical algorithm for the determination of phase from image and diffraction plane pictures[END_REF]. These algorithms compute iterates y k in the set F of vectors y ∈ C n such that |y| = b = |Ax|, which are getting progressively closer to the image of A.

The Gerchberg-Saxton algorithm projects the current iterate y k on the image of A using the orthogonal projector AA † , then it adjusts to b i the amplitude of each coordinate, so as to obtain a new element of F. We describe this method explicitly below.

Algorithm 1 Gerchberg-Saxton.

Input: An initial y 1 ∈ F, i.e. such that |y 1 | = b. 1: for k = 1, . . . , N -1 do 2:
Set

y k+1 i = b i (AA † y k ) i |(AA † y k ) i | , i = 1, . . . , n. (Gerchberg-Saxton)
Because F is not convex however, this alternating projection method usually converges to a stationary point y ∞ which does not belong to the intersection of F with the image of A, and hence |AA † y ∞ | = b. Several modifications proposed in [START_REF] Fienup | Phase retrieval algorithms: a comparison[END_REF] do not eliminate the existence of multiple stationary points. To guarantee convergence to a unique solution, which hopefully belongs to the intersection of F and the image of A, this non-convex optimization problem has recently been relaxed as a semidefinite program [START_REF] Chai | Array imaging using intensity-only measurements[END_REF][START_REF] Candès | Phase retrieval via matrix completion[END_REF], where phase recovery is formulated as a matrix completion problem (described in Section 2.3). Although the computational complexity of this relaxation is much higher than that of the Gerchberg-Saxton algorithm, it is able to recover x from |Ax| (up to a multiplicative constant) in a number of cases [START_REF] Chai | Array imaging using intensity-only measurements[END_REF][START_REF] Candès | Phase retrieval via matrix completion[END_REF].

Splitting phase and amplitude variables

As opposed to these strategies, we solve the phase recovery problem by explicitly separating the amplitude and phase variables, and by only optimizing the values of the phase variables. In the noiseless case, we can write Ax = diag (b)u where u ∈ C n is a phase vector, satisfying |u i | = 1 for i = 1, . . . , n. Given b = |Ax|, the phase recovery problem can thus be written as min

u∈C n , |u i |=1, x∈C p Ax -diag (b)u 2 2 ,
where we optimize over both variables u ∈ C n and x ∈ C p . In this format, the inner minimization problem in x is a standard least squares and can be solved explicitly by setting

x = A † diag (b)u,
which means that problem (2.1) is equivalent to the reduced problem min

|u i |=1 u∈C n AA † diag (b)u -diag (b)u 2 2 .
The objective of this last problem can be rewritten as follows

AA † diag (b)u -diag (b)u 2 2 = (AA † -I)diag (b)u 2 2 = u * diag (b T ) M diag (b)u.
where M = (AA † -I) * (AA † -I) = I -AA † . Finally, the phase recovery problem (2.1) becomes

minimize u * M u subject to |u i | = 1, i = 1, . . . n, (2.2)
in the variable u ∈ C n , where the Hermitian matrix

M = diag (b)(I -AA † )diag (b)
is positive semidefinite. The intuition behind this last formulation is that (I -AA † ) is the orthogonal projector on the orthogonal complement of the image of A (the kernel of A * ). So this last problem simply minimizes in the phase vector u the norm of the component of diag (b)u which is not in the image of A.

Greedy optimization in phase

Having transformed the phase recovery problem (2.1) in the quadratic minimization problem (2.2), we will be able to describe the associated convex relaxation. However, the reformulation (2.2) also yields a natural greedy optimization method; we begin with the latter.

Suppose that we are given an initial vector u ∈ C n , and focus on optimizing over a single component u i for i = 1, . . . , n. The problem is equivalent to solving

minimize ūi M ii u i + 2Re j =i ūj M ji u i subject to |u i | = 1, i = 1, . . . n,
in the variable u i ∈ C where all the other phase coefficients u j remain constant. Because |u i | = 1 this then amounts to solving min

|u i |=1 Re u i j =i M ji ūj which means u i = -j =i Mji u j j =i Mji u j (2.3)
for each i = 1, . . . , n, when u is the optimum solution to problem (2.2). We can use this fact to derive Algorithm 2, a greedy algorithm for optimizing the phase problem. This greedy algorithm converges to a stationary point u ∞ , but it is generally not a global solution of problem (2.2), and hence |AA † diag (u ∞ )b| = b. It has often nearly the same stationary points as the Gerchberg-Saxton algorithm. One can indeed verify that if u ∞ is a stationary point then y ∞ = diag (u ∞ )b is a stationary point of the Gerchberg-Saxton algorithm. Conversely if b has no zero coordinate and y ∞ is a stable stationary point of the Gerchberg-Saxton algorithm then u ∞ i = y ∞ i /|y ∞ i | defines a stationary point of the greedy algorithm in phase. If Ax can be computed with a fast algorithm using O(n log n) operations, which is the case for Fourier or wavelets transform operators for example, then each Gerchberg-Saxton iteration is Algorithm 2 Greedy algorithm in phase.

Input: An initial u ∈ C n such that |u i | = 1, i = 1, . . . , n. An integer N > 1.
1: for k = 1, . . . , N do 2:

for i = 1, . . . n do

3:

Set

u i = -j =i Mji u j j =i Mji u j 4:
end for 5: end for Output:

u ∈ C n such that |u i | = 1, i = 1, . . . , n.
computed with O(n log n) operations. The greedy phase algorithm above does not take advantage of this fast algorithm and requires O(n 2 ) operations to update all coordinates u i for each iteration k. However, we will see in Section 2.2.6 that a small modification of the algorithm allows for O(n log n) iteration complexity.

Complex MaxCut

In this paragraph, we describe the convex relaxation of (2.2). It is an optimization problem over the set of Hermitian matrices. In the second half of the paragraph, we explain how to reformulate it as a real semidefinite program.

Following the classical relaxation argument in [START_REF] Shor | Quadratic optimization problems[END_REF][START_REF] Lovász | Cones of matrices and set-functions and 0-1 optimization[END_REF][START_REF] Goemans | Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming[END_REF][START_REF] Nesterov | Semidefinite relaxation and nonconvex quadratic optimization[END_REF], we first write

U = uu * ∈ H n . Problem (2.2), written QP (M ) def = min. u * M u subject to |u i | = 1, i = 1, . . . n, in the variable u ∈ C n , is equivalent to min. Tr (U M ) subject to diag (U ) = 1 U 0, Rank(U ) = 1,
in the variable U ∈ H n . After dropping the (non-convex) rank constraint, we obtain the following convex relaxation

SDP (M ) def = min. Tr (U M ) subject to diag (U ) = 1, U 0, (PhaseCut)
which is a semidefinite program (SDP) in the matrix U ∈ H n and can be solved efficiently.

When the solution of problem PhaseCut has rank one, the relaxation is tight and the vector u such that U = uu * is an optimal solution of the phase recovery problem (2.2). If the solution has rank larger than one, a normalized leading eigenvector v of U is used as an approximate solution, and diag (U -vv T ) gives a measure of the uncertainty around the coefficients of v.

In practice, semidefinite programming solvers are rarely designed to directly handle problems written over Hermitian matrices and start by reformulating complex programs in H n as real semidefinite programs over S 2n based on the simple facts that follow. For Z, Y ∈ H n , we define T (Z) ∈ S 2n as in [START_REF] Goemans | Approximation algorithms for max-3-cut and other problems via complex semidefinite programming[END_REF]] are eigenvectors of T (Z), both with eigenvalue λ (depending on the normalization of z, one corresponds to (Re (z), Im (z)), the other one to (Re (i z), Im (i z)). This means in particular that Z 0 if and only if T (Z) 0. We can use these facts to formulate an equivalent semidefinite program over real symmetric matrices, written minimize Tr (T (M )X) subject to X i,i + X n+i,n+i = 2 X i,j = X n+i,n+j , X n+i,j = -X i,n+j , i, j = 1, . . . , n, X 0, in the variable X in S 2n . This last problem is equivalent to PhaseCut. In fact, because of symmetries in T (M ), the equality constraints enforcing symmetry can be dropped, and this problem is equivalent to a MaxCut-like problem in dimension 2n, which reads minimize Tr (T (M )X) subject to diag (X) = 1, X 0, (2.5) in the variable X in S 2n . As we will see below, formulating a relaxation to the phase recovery problem as a complex MaxCut-like semidefinite program has direct computational benefits.

T (Z) = Re (Z) -Im (Z) Im (Z) Re (Z) (2.

Algorithms

In the previous section, we have approximated the phase recovery problem (2.2) by a convex relaxation, written minimize Tr (U M ) subject to diag (U ) = 1, U 0, which is a semidefinite program in the matrix U ∈ H n . The dual, written

max w∈R n nλ min (M + diag (w)) -1 T w, (2.6)
is a minimum eigenvalue maximization problem in the variable w ∈ R n . Both primal and dual can be solved efficiently. When exact phase recovery is possible, the optimum value of the primal problem PhaseCut is zero and we must have λ min (M ) = 0, which means that w = 0 is an optimal solution of the dual.

Interior point methods

For small scale problems, with n ∼ 10 2 , generic interior point solvers such as SDPT3 [START_REF] Toh | Sdpt3 -a MATLAB software package for semidefinite programming[END_REF] solve problem (2.5) with a complexity typically growing as O (n 4.5 log(1/ )) where > 0 is the target precision [Ben-Tal and Nemirovski, 2001, §4.6.3]. Exploiting the fact that the 2n equality constraints on the diagonal in (2.5) are singletons, [START_REF] Helmberg | An interior-point method for semidefinite programming[END_REF] derive an interior point method for solving the MaxCut problem, with complexity growing as O (n 3.5 log(1/ )) where the most expensive operation at each iteration is the inversion of a positive definite matrix, which costs O(n 3 ) flops.

First-order methods

When n becomes large, the cost of running even one iteration of an interior point solver rapidly becomes prohibitive. However, we can exploit the fact that the dual of problem (2.5) can be written (after switching signs) as a maximum eigenvalue minimization problem. Smooth first-order minimization algorithms detailed in [START_REF] Nesterov | Smoothing technique and its applications in semidefinite optimization[END_REF]

Block coordinate descent

We can also solve the semidefinite program in PhaseCut using a block coordinate descent algorithm. While no explicit complexity bounds are available for this method in our case, the algorithm is particularly simple and has a very low cost per iteration (it only requires computing a matrix vector product). We write i c the index set {1, . . . , i -1, i + 1, . . . , n} and describe the method as Algorithm 3.

Block coordinate descent is widely used to solve statistical problems where the objective is separable (LASSO is a typical example) and was shown to efficiently solve semidefinite programs arising in covariance estimation [START_REF] Aspremont | First-order methods for sparse covariance selection[END_REF]. These results were extended by [START_REF] Wen | Block coordinate descent methods for semidefinite programming[END_REF] to a broader class of semidefinite programs, including MaxCut. We briefly recall its simple construction below, applied to a barrier version of the MaxCut relaxation PhaseCut, written minimize Tr (U M ) -µ log det(U ) subject to diag (U ) = 1 (2.7)

which is a semidefinite program in the matrix U ∈ H n , where µ > 0 is the barrier parameter.

As in interior point algorithms, the barrier enforces positive semidefiniteness and the value of µ > 0 precisely controls the distance between the optimal solution to (2.7) and the optimal set of PhaseCut. We refer the reader to [START_REF] Boyd | Convex optimization[END_REF] for further details. The key to applying coordinate descent methods to problems penalized by the log det(•) barrier is the following block-determinant formula

det(U ) = det(B) det(y -x T B -1 x), when U = B x x T y , U 0. (2.8)
This means that, all other parameters being fixed, minimizing the function det(X) in the row and column block of variables x, is equivalent to minimizing the quadratic form y -x T Z -1 x, arguably a much simpler problem. Solving the semidefinite program (2.7) row/column by row/column thus amounts to solving the simple problem (2.9) described in the following lemma.

Lemma 2.1. Suppose σ > 0, c ∈ R n-1 , and B ∈ S n-1 are such that b = 0 and B 0, then the optimal solution of the block problem

min x c T x -σ log(1 -x T B -1 x) (2.9)
is given by

x = σ 2 + γ -σ γ Bc
where γ = c T Bc.

Proof. As in [START_REF] Wen | Block coordinate descent methods for semidefinite programming[END_REF], a direct consequence of the first order optimality conditions for (2.9).

Here, we see problem (2.7) as an unconstrained minimization problem over the off-diagonal coefficients of U , and (2.8) shows that each block iteration amounts to solving a minimization subproblem of the form (2.9). Lemma 2.1 then shows that this is equivalent to computing a matrix vector product. Linear convergence of the algorithm is guaranteed by the result in [Boyd and Vandenberghe, 2004, §9.4.3] and the fact that the function log det is strongly convex over compact subsets of the positive semidefinite cone. So the complexity of the method is bounded by O log 1 but the constant in this bound depends on n here, and the dependence cannot be quantified explicitly.

Algorithm 3 Block Coordinate Descent Algorithm for PhaseCut.

Input: An initial X 0 = I n and ν > 0 (typically small). An integer N > 1.

1: for k = 1, . . . , N do 2:

Pick i ∈ [1, n]. 3: Compute x = X k i c ,i c M i c ,i and γ = x * M i c ,i 4:
If γ > 0, set

X k+1 i c ,i = X k+1 * i,i c = - 1 -ν γ x else X k+1 i c ,i = X k+1 * i,i c = 0.
5: end for Output: A matrix X 0 with diag (X) = 1.

Initialization & randomization

Suppose the Hermitian matrix U solves the semidefinite relaxation PhaseCut. As in [START_REF] Goemans | Approximation algorithms for max-3-cut and other problems via complex semidefinite programming[END_REF][START_REF] Ben-Tal | Extended matrix cube theorems with applications to mu-theory in control[END_REF][START_REF] Zhang | Complex quadratic optimization and semidefinite programming[END_REF][START_REF] Eldar | On approximating complex quadratic optimization problems via semidefinite programming relaxations[END_REF], we generate complex Gaussian vectors x ∈ C n with x ∼ N C (0, U ), and for each sample x, we form z ∈ C n such that

z i = x i |x i | , i = 1, . . . , n.
All the sample points z generated using this procedure satisfy |z i | = 1, hence are feasible points for problem (2.2). This means in particular that

QP (M ) ≤ E[z * M z].
In fact, this expectation can be computed almost explicitly, using

E[zz * ] = F (U ), with F (w) = 1 2 e i arg(w) π 0 cos(θ) arcsin(|w| cos(θ))dθ
where F (U ) is the matrix with coefficients F (U ij ), i, j = 1, . . . , n. We then get

SDP (M ) ≤ QP (M ) ≤ Tr (M F (U )) (2.10)
In practice, to extract good candidate solutions from the solution U to the SDP relaxation in PhaseCut, we sample a few points from N C (0, U ), normalize their coordinates and simply pick the point which minimizes z * M z. This sampling procedure also suggests a simple spectral technique for computing rough solutions to problem PhaseCut: compute an eigenvector of M corresponding to its lowest eigenvalue and simply normalize its coordinates (this corresponds to the simple bound on MaxCut by [START_REF] Delorme | Laplacian eigenvalues and the maximum cut problem[END_REF]). The information contained in U can also be used to solve a robust formulation [START_REF] Ben-Tal | Robust Optimization[END_REF] of problem (2.1) given a Gaussian model u ∼ N C (0, U ).

Approximation bounds

The semidefinite program in PhaseCut is a MaxCut-type graph partitioning relaxation whose performance has been studied extensively. Note however that most approximation results for MaxCut study maximization problems over positive semidefinite or nonnegative matrices, while we are minimizing in PhaseCut so, as pointed out in [START_REF] Kisialiou | Probabilistic analysis of semidefinite relaxation for binary quadratic minimization[END_REF][START_REF] So | Probabilistic analysis of semidefinite relaxation detectors for multipleinput, multiple-output[END_REF] for example, we do not inherit the constant approximation ratios that hold in the classical MaxCut setting.

Exploiting structure

In some instances, we have additional structural information on the solution of problems (2.1) and (2.2), which usually reduces the complexity of approximating PhaseCut and improves the quality of the approximate solutions. We briefly highlight a few examples below.

Alignment

In other instances, we might have prior knowledge that the phases of certain samples are aligned, i.e. that there is an index set I such that u i = u j , for all i, j ∈ I, this reduces to the symmetric case discussed above when the phase is arbitrary. W.l.o.g., we can also fix the phase to be one, with u i = 1 for i ∈ I, and solve a constrained version of the relaxation PhaseCut min.

Tr (U M ) subject to

U ij = 1, i, j ∈ I, diag (U ) = 1, U 0, which is a semidefinite program in U ∈ H n .

Fast Fourier transform

If the product M x can be computed with a fast algorithm in O(n log n) operations, which is the case for Fourier or wavelet transform operators, we significantly speed up the iterations of Algorithm 3 to update all coefficients at once. Each iteration of the modified Algorithm 3 then has cost O(n log n) instead of O(n 2 ).

Real valued signal

In some cases, we know that the solution vector x in (2.1) is real valued. Problem (2.1) can be reformulated to explicitly constrain the solution to be real, by writing it min

u∈C n , |u i |=1, x∈R p Ax -diag (b)u 2 2
or again, using the operator T (•) defined in (2.4) minimize

T (A) x 0 -diag b b Re (u) Im (u) 2 2 subject to u ∈ C n , |u i | = 1 x ∈ R p .
The optimal solution of the inner minimization problem in x is given by x = A † 2 B 2 v, where

A 2 = Re (A) Im (A) , B 2 = diag b b , and v = Re (u) Im (u)
hence the problem is finally rewritten

minimize (A 2 A † 2 B 2 -B 2 )v 2 2 subject to v 2 i + v 2 n+i = 1, i = 1, . . . , n,
in the variable v ∈ R 2n . This can be relaxed as above by the following problem minimize Tr (V M 2 ) subject to

V ii + V n+i,n+i = 1, i = 1, . . . , n, V 0,
which is a semidefinite program in the variable V ∈ S 2n , where

M 2 = (A 2 A † 2 B 2 -B 2 ) T (A 2 A † 2 B 2 - B 2 ) = B T 2 (I -A 2 A † 2 )B 2 .

Matrix completion & exact recovery conditions

In [START_REF] Chai | Array imaging using intensity-only measurements[END_REF][START_REF] Candès | Phase retrieval via matrix completion[END_REF], phase recovery (2.1) is cast as a matrix completion problem. In this section, we briefly review this approach and compare it with the semidefinite program in PhaseCut.

Given a signal vector b ∈ R n and a sampling matrix A ∈ C n×p , we look for a vector x ∈ C p satisfying |a * i x| = b i , i = 1, . . . , n, where the vector a * i is the i th row of A and x ∈ C p is the signal we are trying to reconstruct. The phase recovery problem is then written as minimize Rank(X) subject to Tr (a i a * i X) = b 2 i , i = 1, . . . , n X 0 in the variable X ∈ H p , where X = xx * when exact recovery occurs. This last problem can be relaxed as minimize Tr (X) subject to Tr (a i a *

i X) = b 2 i , i = 1, . . . , n X 0 (PhaseLift)
which is a semidefinite program (called PhaseLift by [START_REF] Candès | Phase retrieval via matrix completion[END_REF]) in the variable X ∈ H p . Recent results in [START_REF] Candès | Phaselift: exact and stable signal recovery from magnitude measurements via convex programming[END_REF][START_REF] Candès | Solving quadratic equations via phaselift when there are about as many equations as unknowns[END_REF] give explicit (if somewhat stringent) conditions on A and x under which the relaxation is tight (i.e. the optimal X in PhaseLift is unique, has rank one, with leading eigenvector x).

In Paragraph 2.3.1, we define a variant of PhaseLift, easier to compare with PhaseCut. In paragraph 2.3.2, we show that this variant and PhaseCut can be seen as projection problems. Paragraph 2.3.3 compares the tightness of PhaseLift and a slight modification of PhaseCut. Paragraph 2.3.4 discusses the stability to noise and paragraph 2.3.6 compares the complexities of both methods.

Weak formulation

We also introduce a weak version of PhaseLift, which is more directly related to PhaseCut and is easier to interpret geometrically. It was noted in [START_REF] Candès | Phaselift: exact and stable signal recovery from magnitude measurements via convex programming[END_REF] that, when I ∈ span{a i a * i } n i=1 , the condition Tr (a i a * i X) = b 2 i , i = 1, ..., n determines Tr (X), so in this case the trace minimization objective is redundant and PhaseLift is equivalent to

find X subject to Tr (a i a * i X) = b 2 i , i = 1, . . . , n X 0. (Weak PhaseLift) When I / ∈ span{a i a * i } n i=1
on the other hand, Weak PhaseLift and PhaseLift are not equivalent: solutions of PhaseLift solve Weak PhaseLift too but the converse is not true. Interior point solvers typically pick a solution at the analytic center of the feasible set of Weak PhaseLift which in general can be significantly different from the minimum trace solution.

However, in practice, the removal of trace minimization does not really seem to alter the performances of the algorithm. We will illustrate this affirmation with numerical experiments in Paragraph 2.4.4 and a formal proof is given in [START_REF] Demanet | Stable optimizationless recovery from phaseless linear measurements[END_REF][START_REF] Candès | Solving quadratic equations via phaselift when there are about as many equations as unknowns[END_REF] who showed that, in the case of Gaussian random measurements, the relaxation of Weak PhaseLift is tight with high probability under the same conditions as PhaseLift.

Phase recovery as a projection

We will see in what follows that phase recovery can be interpreted as a projection problem. These results will prove useful later to study stability. The PhaseCut reconstruction problem is written minimize Tr (U M ) subject to diag (U ) = 1, U 0,

with M = diag (b)(I -AA † )diag (b).
In what follows, we assume b i = 0, i = 1, ..., n, which means that, after scaling U , solving PhaseCut is equivalent to solving minimize Tr (V (I .11) In the following lemma, we show that this last semidefinite program can be understood as a projection problem on a section of the semidefinite cone using the trace (or nuclear) norm. We define

-AA † )) subject to diag (V ) = b 2 , V 0. ( 2
F = {V ∈ H n : x * V x = 0, ∀x ∈ Range (A) ⊥ } which is also F = {V ∈ H n : (I -AA † )V (I -AA † ) = 0}
, and we now formulate the objective of problem (2.11) as a distance.

Lemma 2.2. For all V ∈ H n such that V 0,

Tr (V (I -AA † )) = d 1 (V, F) (2.12)
where d 1 is the distance associated to the trace norm.

Proof. Let B 1 (resp. B 2 ) be an orthonormal basis of Range A (resp. (Range A) ⊥ ). Let T be the transformation matrix from canonical basis to orthonormal basis B 1 ∪ B 2 . Then

F = {V ∈ H n s.t. T -1 V T = S 1 S 2 S * 2 0 , S 1 ∈ H p , S 2 ∈ M p,n-p }
As the transformation X → T -1 XT preserves the nuclear norm, for every matrix V 0, if we write

T -1 V T = V 1 V 2 V * 2 V 3
then the orthogonal projection of V onto F is

W = T V 1 V 2 V * 2 0 T -1 , so d 1 (V, F) = V -W 1 = 0 0 0 V 3 1 . As V 0, V 1 V 2 V * 2 V 3 0 hence 0 0 0 V 3 0, so d 1 (V, F) = Tr 0 0 0 V 3 . Because AA † is the orthogonal projection onto Range (A), we have T -1 (I -AA † )T = ( 0 0 0 I ) hence d 1 (V, F) = Tr 0 0 0 V 3 = Tr ((T -1 V T )(T -1 (I -AA † )T )) = Tr (V (I -AA † ))
which is the desired result.

This means that PhaseCut can be written as a projection problem, i.e.

minimize d 1 (V, F) subject to V ∈ H + n ∩ H b (2.13) in the variable V ∈ H n , where H b = {V ∈ H n s.t. V i,i = b 2 i , i = 1, ..., n}. Moreover, with a i the i-th row of A, we have for all X ∈ H + p , Tr (a i a * i X) = a * i Xa i = diag (AXA * ) i , i = 1, . . . , n, so if we call V = AXA * ∈ F, when A is injective, X = A † V A † * and Weak PhaseLift is equivalent to find V ∈ H + n ∩ F subject to diag (V ) = b 2 .
First order algorithms for Weak PhaseLift will typically solve

minimize d(diag (V ), b 2 ) subject to V ∈ H + n ∩ F
minimize d s (V, H b ) subject to V ∈ H + n ∩ F (2.14)
which is another projection problem in V . Thus, PhaseCut and Weak PhaseLift are comparable, in the sense that both algorithms aim at finding a point of H + n ∩ F ∩ H b but PhaseCut does so by picking a point of H + n ∩ H b and moving towards F while Weak PhaseLift moves a point of H + n ∩ F towards H b . We can push the parallel between both relaxations much further. We will show in what follows that, in a very general case, PhaseLift and a modified version of PhaseCut are simultaneously tight. We will also be able to compare the stability of Weak PhaseLift and PhaseCut when measurements become noisy.

Tightness of the semidefinite relaxation

We will now formulate a refinement of the semidefinite relaxation in PhaseCut and prove that this refinement is equivalent, in terms of tightness, to the relaxation in PhaseLift under mild technical assumptions. Suppose u is the optimal phase vector, we know that the optimal solution to (2.1) can then be written x = A † diag (b)u, which corresponds to the matrix

X = A † diag (b)uu * diag (b)A † * in PhaseLift, hence Tr (X) = Tr (diag (b)A † * A † diag (b)uu * ).
Writing B = diag (b)A † * A † diag (b), when problem (2.1) is solvable, we look for the "minimum trace" solution among all the optimal points of relaxation PhaseCut by solving

SDP 2(M ) def = min. Tr (BU ) subject to Tr (M U ) = 0 diag (U ) = 1, U 0, (PhaseCutMod )
which is a semidefinite program in U ∈ H n . When problem (2.1) is solvable, then every optimal solution of the semidefinite relaxation PhaseCut is a feasible point of relaxation PhaseCutMod .

In practice, the semidefinite program SDP (M + γB), written minimize Tr ((M + γB)U ) subject to diag (U ) = 1, U 0, obtained by replacing M by M + γB in problem PhaseCut, will produce a solution to PhaseCut-Mod whenever γ > 0 is sufficiently small (this is essentially the exact penalty method detailed in [Bertsekas, 1998, §4.3] for example). This means that all algorithms (greedy or SDP) designed to solve the original PhaseCut problem can be recycled to solve PhaseCutMod with negligible effect on complexity. We now show that the PhaseCutMod and PhaseLift relaxations are simultaneously tight when A is injective. An earlier version of this text showed that PhaseLift tightness implies PhaseCutMod tightness, and the argument was reversed in [START_REF] Voroninski | A comparison between the phaselift and phasecut algorithms[END_REF] under mild additional assumptions.

Proposition 2.3. Assume that b i = 0 for i = 1, . . . , n, that A is injective and that there is a solution x to (2.1). The function

Φ : H p → H n X → Φ(X) = diag (b) -1 AXA * diag (b) -1
is a bijection between the feasible points of PhaseCutMod and those of PhaseLift.

Proof. Note that Φ is injective whenever b > 0 and A has full rank. We have to show that U is a feasible point of PhaseCutMod if and only if it can be written under the form Φ(X), where X is feasible for PhaseLift. We first show that

Tr (M U ) = 0, U 0, (2.15) is equivalent to U = Φ(X) (2.16)
for some X 0. Observe that Tr (U M ) = 0 means U M = 0 because U, M 0, hence Tr (M U ) = 0 in (2.15) is equivalent to

AA † diag (b)U diag (b) = diag (b)U diag (b) because b > 0 and M = diag (b)(I -AA † )diag (b). If we set X = A † diag (b)U diag (b)A † * , this last equality implies both AX = AA † diag (b)U diag (b)A † * = diag (b)U diag (b)A † * and AXA * = diag (b)U diag (b)A † * A * = diag (b)U diag (b)
which is U = Φ(X), and shows (2.15) implies (2.16). Conversely, if U = Φ(X) then:

diag (b)U diag (b) = AXA *
and, using

AA † A = A, we get AXA * = AA † AXA * = AA † diag (b)U diag (b) which means M U = 0, hence (2.15
) is in fact equivalent to (2.16) since U 0 by construction. Now, if X is feasible for PhaseLift, we have shown Tr (M Φ(X)) = 0 and φ(X) 0, moreover diag (Φ(X)

) i = Tr (a i a * i X)/b 2 i = 1, so U = Φ(X) is a feasible point of PhaseCutMod . Conversely, if U is feasible for PhaseCutMod , we have shown that there exists X 0 such that U = Φ(X) which means diag (b)U diag (b) = AXA * . We also have Tr (a i a * i X) = b 2 i U ii = b 2 i
, which means X is feasible for PhaseLift and concludes the proof.

We now have the following central corollary showing the equivalence between PhaseCutMod and PhaseLift in the noiseless case.

Corollary 2.4. If A is injective, b i = 0 for all i = 1, ..., n and if the reconstruction problem (2.1) admits an exact solution, then PhaseCutMod is tight (i.e. has a unique rank one solution) whenever PhaseLift is.

Proof. When A is injective, Tr (X) = Tr (BΦ(X)) and Rank(X) = Rank(Φ(X)).

This last result shows that in the noiseless case, the relaxations PhaseLift and PhaseCutMod are in fact equivalent. In the same way, we could have shown that Weak PhaseLift and Phase-Cut were equivalent. The performances of both algorithms may not match however when the information on b is noisy and perfect recovery is not possible.

Remark 2.5. Note that Proposition 2.3 and Corollary 2.4 also hold when the initial signal is real and the measurements are complex. In this case, we define the B in PhaseCutMod by

B = B 2 A † * 2 A † 2 B 2
(with the notations of paragraph 2.2.6). We must also replace the definition of Φ by Φ

(X) = B -1 2 A 2 XA * 2 B -1 2 .
Furthermore, all steps in the proof of Proposition 2.3 are still valid if we replace M by M 2 , A by A 2 and diag (b) by B 2 . The only difference is that now

1 b 2 i Tr (a i a * i X) = diag (Φ(X)) i + diag (Φ(X)) n+i .

Stability in the presence of noise

We now consider the case where the vector of measurements b is of the form b = |Ax 0 |+b noise . We first introduce a definition of C-stability for PhaseCut and Weak PhaseLift. The main result of this section is that, when the Weak PhaseLift solution in (2.14) is stable at a point, PhaseCut is stable too, with a constant of the same order. The converse does not seem to be true when b is sparse. Definition 2.6. Let x 0 ∈ C n , C > 0. The algorithm PhaseCut (resp. Weak PhaseLift) is said to be C-stable at x 0 iff for all b noise ∈ R n close enough to zero, every minimizer V of equation (2.13)

(resp. (2.14)) with b = |Ax 0 | + b noise , satisfies V -(Ax 0 )(Ax 0 ) * 2 ≤ C Ax 0 2 b noise 2 .
The following matrix perturbation result motivates this definition, by showing that a C-stable algorithm generates a O(C b noise 2 )-error over the signal it reconstructs.

Proposition 2.7. Let C > 0 be arbitrary. We suppose that Ax 0 = 0 and V -

(Ax 0 )(Ax 0 ) * 2 ≤ C Ax 0 2 b noise 2 ≤ Ax 0 2 2 /2. Let y be V 's main eigenvector, normalized so that (Ax 0 ) * y = Ax 0 2 . Then y -Ax 0 2 = O(C b noise 2 ),
and the constant in this last equation does not depend upon A, x 0 , C or b 2 .

Proof. We use [Karoui and d'Aspremont, 2010, Eq.10] for

u = Ax 0 Ax 0 2 v = y Ax 0 2 E = V -(Ax 0 )(Ax 0 ) * Ax 0 2 2
This result is based on [Kato, 1995, Eq. 3.29], which gives a precise asymptotic expansion of u-v.

For our purposes here, we only need the first-order term. See also [START_REF] Bhatia | Matrix analysis[END_REF], [START_REF] Stewart | Matrix perturbation theory[END_REF] or [START_REF] Stewart | Matrix algorithms[END_REF] among others for a complete discussion. We get [Karoui and d'Aspremont, 2010, Eq.10]. This implies

v -u = O( E 2 ) because if M = uu * , then R ∞ = 1 in
y -Ax 0 2 = Ax 0 2 u -v = O V -(Ax 0 )(Ax 0 ) * 2 Ax 0 2 = O(C b noise )
which is the desired result.

Note that normalizing y differently, we would obtain y -Ax 0 2 ≤ 4C b noise 2 . We now show the main result of this section, according to which PhaseCut is "almost as stable as" Weak PhaseLift. In numerical applications, the exact values of the stability constants have only a small importance, what matters is that they are of the same order.

Theorem 2.8. Let A ∈ C n×m , for all x 0 ∈ C n , C > 0, if Weak PhaseLift is C-stable in x 0 , then PhaseCut is (2C + 2 √ 2 + 1)-stable in x 0 . Proof. Let x 0 ∈ C n , C > 0 be such that Weak PhaseLift is C-stable in x 0 .
Ax 0 is a non-zero vector (because, with our definition, neither Weak PhaseLift nor PhaseCut may be stable in x 0 if Ax 0 = 0 and A = 0). We set D = 2C+2 √ 2+1 and suppose by contradiction that PhaseCut is not D-stable in x 0 . Let > 0 be arbitrary. Let b n,PC ∈ R n be such that b n,PC 2 ≤ max( Ax 0 2 , /2) and such that, for b = |Ax 0 | + b n,PC , the minimizer V P C of (2.13) verifies

V P C -(Ax 0 )(Ax 0 ) * 2 > D Ax 0 2 b n,PC 2 Such a V P C must exist or PhaseCut would be D-stable in x 0 . We call V P C the restriction of V P C to Range (A) (that is, the matrix such that x * (V P C )y = x * (V P C )y if x, y ∈ Range (A) and x * (V P C )y = 0 if x ∈ Range (A) ⊥ or y ∈ Range (A) ⊥ ) and V ⊥ P C the restriction of V P C to Range (A) ⊥ . Let us set b n,PL = V P C ii -|Ax 0 | ii for i = 1, ..., n. As V P C ∈ H + n ∩ F, V P C minimizes (2.14) for b = |Ax 0 | + b n,PL (because V P C ∈ H b )
. Lemmas 2.9 and 2.10 (proven in the appendix) imply that

V P C -(Ax 0 )(Ax 0 ) * 2 > C Ax 0 2 b n,PL 2 and b n,PL 2 ≤ . As is arbitrary, Weak PhaseLift is not C-stable in x 0 , which contradicts our hypotheses. Conse- quently, PhaseCut is (2C + 2 √ 2 + 1)-stable in x 0 .
The proof of this theorem is based on the fact that, when V P C solves (2.13), one can construct some V P L = V P C close to V P C , which is an approximate solution of (2.14). It is natural to wonder whether, conversely, from a solution V P L of (2.14), one can construct an approximate solution V P C of (2.13). It does not seem to be the case. One could for example imagine setting

V P C = diag (R)V P L diag (R), where R i = b i / √ V P L ii .
Then V P C would not necessarily minimize (2.13) but at least belong to H b . But V P C -V P L 2 might be quite large: (2.14) implies that diag (V P L ) -b 2 s is small but, if some coefficients of b are very small, some R i may still be huge, so diag (R) ≈ I. This does happen in practice (see Paragraph 2.4.5).

To conclude this section, we relate this definition of stability to the one introduced in [START_REF] Candès | Solving quadratic equations via phaselift when there are about as many equations as unknowns[END_REF]. Suppose that A is a matrix of random gaussian independant measurements such that E[|A i,j | 2 ] = 1 for all i, j. We also suppose that n ≥ c 0 p (for some c 0 independent of n and p). In the noisy setting, [START_REF] Candès | Solving quadratic equations via phaselift when there are about as many equations as unknowns[END_REF] showed that the minimizer X of a modified version of PhaseLift satisfies with high probability

||X -x 0 x * 0 || 2 ≤ C 0 || |Ax 0 | 2 -b 2 || 1 n (2.17)
for some C 0 independent of all variables. Assuming that the Weak PhaseLift solution in (2.14) behaves as PhaseLift in a noisy setting and that (2.17) also holds for Weak PhaseLift, then

||AXA * -(Ax 0 )(Ax 0 ) * || 2 ≤ ||A|| 2 ∞ ||X -x 0 x * 0 || 2 ≤ C 0 ||A|| 2 ∞ n || |Ax 0 | 2 -b 2 || 1 ≤ C 0 ||A|| 2 ∞ n (2||Ax 0 || 2 + ||b noise || 2 )||b noise || 2 Consequently, for any C > 2C 0 ||A|| 2 ∞ n , Weak PhaseLift is C-stable in all x 0 . With high probability, ||A|| 2 ∞ ≤ (1+1/8
)n (it is a corollary of [Candès and Li, 2014, Lemma 2.1]) so Weak PhaseLift (and thus also PhaseCut) is C-stable with high probability for some C independent of all parameters of the problem.

Perturbation results

We recall here sensitivity analysis results for semidefinite programming from [START_REF] Yildirim | Sensitivity analysis in linear programming and semidefinite programming using interior-point methods[END_REF]; [START_REF] Yildirim | An interior-point perspective on sensitivity analysis in semidefinite programming[END_REF], which produce explicit bounds on the impact of small perturbations in the observation vector b 2 on the solution V of the semidefinite program (2.11). Roughly speaking, these results show that if b 2 + b noise remains in an explicit ellipsoid (called Dikin's ellipsoid), then interior point methods converge back to the solution in one full Newton step, hence the impact on V is linear, equal to the Newton step. These results are more numerical in nature than the stability bounds detailed in the previous section, but they precisely quantify both the size and, perhaps more importantly, the geometry of the stability region.

Complexity comparisons

Both the relaxation in PhaseLift and that in PhaseCut are semidefinite programs and we highlight below the relative complexity of solving these problems depending on algorithmic choices and precision targets. Note that, in their numerical experiments, [START_REF] Candès | Phase retrieval via matrix completion[END_REF] solve a penalized formulation of problem PhaseLift, written min

X 0 n i=1 (Tr (a i a * i X) -b 2 i ) 2 + λTr (X) (2.18)
in the variable X ∈ H p , for various values of the penalty parameter λ > 0.

The trace norm promotes a low rank solution, and solving a sequence of weighted tracenorm problems has been shown to further reduce the rank in [START_REF] Fazel | Log-det heuristic for matrix rank minimization with applications to hankel and euclidean distance matrices[END_REF][START_REF] Candès | Phase retrieval via matrix completion[END_REF]. This method replaces Tr (X) by Tr (W k X) where W 0 is initialized to the identity I. Given a solution X k of the resulting semidefinite program, the weighted matrix is updated to W k+1 = (X k +ηI) -1 (see [START_REF] Fazel | Log-det heuristic for matrix rank minimization with applications to hankel and euclidean distance matrices[END_REF] for details). We denote by K the total number of such iterations, typically of the order of 10. Trace minimization is not needed for the semidefinite program (PhaseCut), where the trace is fixed because we optimize over a normalized phase vector.

However, weighted trace-norm iterations could potentially improve performance in PhaseCut as well.

Recall that p is the size of the signal and n is the number of measured samples with n = Jp in the examples reviewed in Section 2.4. In the numerical experiments in [START_REF] Candès | Phase retrieval via matrix completion[END_REF] as well as in this paper, J = 3, 4, 5. The complexity of solving the PhaseCut and PhaseLift relaxations in PhaseLift using generic semidefinite programming solvers such as SDPT3 [START_REF] Toh | Sdpt3 -a MATLAB software package for semidefinite programming[END_REF], without exploiting structure, is given by O J 4.5 p 4.5 log 1 and O K J 2 p 4.5 log 1

for PhaseCut and PhaseLift respectively [Ben-Tal and Nemirovski, 2001, § 6.6.3]. The fact that the constraint matrices have only one nonzero coefficient in PhaseCut can be exploited (the fact that the constraints a i a * i are rank one in PhaseLift helps, but it does not modify the principal complexity term) so we get O J 3.5 p 3.5 log 1 and O K J 2 p 4.5 log 1

for PhaseCut and PhaseLift respectively using the algorithm in [START_REF] Helmberg | An interior-point method for semidefinite programming[END_REF] for example. If we use first-order solvers such as TFOCS [START_REF] Becker | Tfocs v1.1. User guide[END_REF], based on the optimal algorithm in [START_REF] Nesterov | A method of solving a convex programming problem with convergence rate o(1/k 2 )[END_REF], the dependence on the dimension can be further reduced, to become

O J 3 p 3 and O KJ p 3
for a penalized version of the PhaseCut relaxation and the penalized formulation of PhaseLift in (2.18). While the dependence on the signal dimensions p is somewhat reduced, the dependence on the target precision grows from log(1/ ) to 1/ . Finally, the iteration complexity of the block coordinate descent Algorithm 3 is substantially lower and its convergence is linear, but no fully explicit bounds on the number of iterations are known in our case. The complexity of the method is then bounded by O log 1 but the constant in this bound depends on n here, and the dependence cannot be quantified explicitly. Algorithmic choices are ultimately guided by precision targets. If is large enough so that a first-order solver or a block coordinate descent can be used, the complexity of PhaseCut is not significantly better than that of PhaseLift. On the contrary, when is small, we must use an interior point solver, for which PhaseCut's complexity is an order of magnitude lower than that of PhaseLift because its constraint matrices are singletons. In practice, the target value for strongly depends on the sampling matrix A. For example, when A corresponds to the convolution by 6 Gaussian random filters (Paragraph 2.4.2), to reconstruct a Gaussian white noise of size 64 with a relative precision of η, we typically need ∼ 2.10 -1 η. For 4 Cauchy wavelets (Paragraph 2.4.3), it is twenty times less, with ∼ 10 -2 η. For other types of signals than Gaussian white noise, we may even need ∼ 10 -3 η.

Greedy refinement

If the PhaseCut or PhaseLift algorithms do not return a rank one matrix then an approximate solution of the phase recovery problem is obtained by extracting a leading eigenvector v. For PhaseCut and PhaseLift, x = A † diag (b)v and x = v are respectively approximate solutions of the phase recovery problem with |Ax| = b = |Ax|. This solution is then refined by applying the Gerchberg-Saxton algorithm initialized with x. If x is sufficiently close to x then, according to numerical experiments of Section 2.4, this greedy algorithm converges to λ x with |λ| = 1. These greedy iterations require much less operations than PhaseCut and PhaseLift algorithms, and thus have no significant contribution to the computational complexity.

Sparsity

Minimizing Tr (X) in the PhaseLift problem means looking for signals which match the modulus constraints and have minimum 2 norm. In some cases, we have a priori knowledge that the signal we are trying to reconstruct is sparse, i.e. Card(x) is small. The effect of imposing sparsity was studied in e.g. [START_REF] Moravec | Compressive phase retrieval[END_REF][START_REF] Shechtman | Sparsity-based single-shot sub-wavelength coherent diffractive imaging[END_REF][START_REF] Li | Sparse signal recovery from quadratic measurements via convex programming[END_REF].

Assuming n ≤ p, the set of solutions to Ax -diag (b)u 2 is written x = A † diag (b)u + F v where F is a basis for the nullspace of A. In this case, when the rows of A are independent, AA † = I and the reconstruction problem with a 1 penalty promoting sparsity is then written

minimize A † diag (b)u + F v 2 1 subject to |u i | = 1,
in the variables u ∈ C p and y ∈ C p-n . Using the fact that y 2 1 = yy * 1 , this can be relaxed as

minimize V U V * 1 subject to U 0, |U ii | = 1, i = 1, . . . , n, which is a semidefinite program in the (larger) matrix variable U ∈ H p and V = (A † diag (b), F ).
On the other hand, when n > p and A is injective, the matrix F disappears. We can take sparsity into account by adding an l 1 penalization to PhaseCut. As noted in [START_REF] Voroninski | A comparison between the phaselift and phasecut algorithms[END_REF] however, the effect of an 1 penalty on least-squares solutions is not completely clear.

Numerical results

In this section, we compare the numerical performance of the Gerchberg-Saxton, PhaseCut and PhaseLift algorithms on various phase recovery problems. As in [START_REF] Candès | Phase retrieval via matrix completion[END_REF], the PhaseLift problem is solved using the package in [START_REF] Becker | Tfocs v1.1. User guide[END_REF], with reweighting, using K = 10 outer iterations and 1000 iterations of the first order algorithm. The PhaseCut and Gerchberg-Saxton algorithms described here are implemented in a public software package available at http://www.cmap.polytechnique.fr/scattering/code/phaserecovery.zip

Other numerical experiments about PhaseCut have been conducted in the optic of applications to imaging problems, and can be found in [START_REF] Fogel | Phase retrieval for imaging problems[END_REF].

In our experiments, the phase recovery algorithms compute an approximate solution x from |Ax| and the reconstruction error is measured by the relative Euclidean distance up to a complex phase given by

(x, x) def = min c∈C,|c|=1 x -c x x . (2.19)
We also record the error over measured amplitudes, written

(|Ax|, |Ax|) def = |Ax| -|Ax| Ax .
(2.20)

Note that when the phase recovery problem either does not admit a unique solution or is unstable, we usually have (|Ax|, |Ax|) (x, x). In the next three subsections, we study these reconstruction errors for three different phase recovery problems, where A is defined as an oversampled Fourier transform, as multiple filterings with random filters, or as a wavelet transform. Numerical results are computed on three different types of test signals x: realizations of a complex Gaussian white noise, sums of complex exponentials a ω e iωm with random frequencies ω and random amplitudes a ω (the number of exponentials is random, around 6), and signals whose real and imaginary parts are scan-lines of natural images. Each signal has p = 128 coefficients. Figure 2.2 shows the real part of sample signals, for each signal type.

Oversampled Fourier transform

The discrete Fourier transform ŷ of a signal y of q coefficients is written In X-ray crystallography or diffraction imaging experiments, compactly supported signals are estimated from the amplitude of Fourier transforms oversampled by a factor J ≥ 2. The corresponding operator A computes an oversampled discrete Fourier transform evaluated over n = Jp coefficients. The signal x of size p is extended into x J by adding (J -1)p zeros and

ŷk = q-1 m=0 y m exp -i2πkm q .
(Ax) k = xJ k = p m=1 x m exp(- i2πkm n ).
For this oversampled Fourier transform, the phase recovery problem does not have a unique solution [START_REF] Akutowicz | On the determination of the phase of a Fourier integral[END_REF]. In fact, one can show [START_REF] Sanz | Mathematical considerations for the problem of fourier transform phase retrieval from magnitude[END_REF] that there are as many as 2 p-1 solutions x ∈ C p such that |Ax| = |Ax|. Moreover, increasing the oversampling factor J beyond 2 does not reduce the number of solutions.

Because of this intrinsic instability, we will observe that all algorithms perform similarly on this type of reconstruction problems and Table 2.1 shows that the percentage of perfect reconstruction is below 5% for all methods. The only signals which can be perfectly recovered are sums of few sinusoids. Because these test signals are very sparse in the Fourier domain, the number of signals having identical Fourier coefficient amplitudes is considerably smaller than in typical sample signals. As a consequence, there is a small probability (about 5%) of exactly reconstructing the original signal given an arbitrary initialization. None of the Gaussian random noises and image scan lines are exactly recovered. Note that we say that an exact reconstruction is reached when (x, x) < 10 -2 because a few iterations of the Gerchberg-Saxton algorithm from such an approximate solution x will typically converges to x. Numerical results are computed with 100 sample signals in each of the 3 signal classes.

Table 2.2 gives the average relative error (x, x) over signals that are not perfectly reconstructed, which is of order one here. Despite this large error, Table 2.3 shows that the relative error (|Ax|, |Ax|) over the Fourier modulus coefficients is below 10 -3 for all algorithms. This is due to the non-uniqueness of the phase recovery from Fourier modulus coefficients. Recovering a solution x with identical or nearly identical oversampled Fourier modulus coefficients as x does not guarantee that x is proportional to x. Overall, in this set of ill-posed Fourier experiments, recovery performance is very poor for all methods and the PhaseCut and PhaseLift relaxations do not improve much on the results of the faster Gerchberg-Saxton algorithm.

Multiple random illumination filters

To guarantee uniqueness of the phase recovery problem, one can add independent measurements by "illuminating" the object through J filters h j in the context of X-ray imaging or crystallography [START_REF] Candès | Phaselift: exact and stable signal recovery from magnitude measurements via convex programming[END_REF]

(Ax) k+pj = (xh j ) k = ( x ĥj ) k for 1 ≤ j ≤ J and 0 ≤ k < p,
where x ĥj is the circular convolution between x and ĥj . Candès et al. [2015]; Gross et al. [2015b] prove that, in a close setting to this one, PhaseLift is tight with high probability, for a number J of filters logarithmic in p. [START_REF] Candès | Phase retrieval via matrix completion[END_REF] empirically observe that, for signals of size p = 128, with J = 4 filters, perfect recovery is achieved in 100% of their experiments.

Table 2.1 confirms this behavior and shows that the PhaseCut algorithm achieves perfect recovery in all our experiments. As predicted by the equivalence results presented in the previous section, we observe that PhaseCut and PhaseLift have identical performance in these experiments. With 4 filters, the solutions of these two SDP relaxations are not of rank one but are "almost" of rank one, in the sense that their first eigenvector v has an eigenvalue much larger than the others, by a factor of about 5 to 10. Numerically, we observe that the corresponding approximate solutions, x = diag (v)b, yield a relative error (|Ax|, |Ax|) which, for scan-lines of images and especially for Gaussian signals, is of the order of the ratio between the largest and the second largest eigenvalue of the matrix U . The resulting solutions x are then sufficiently close to x so that a few iterations of the Gerchberg-Saxton algorithm started at x will converge to x.

Table 2.1 shows however that directly applying the Gerchberg-Saxton algorithm starting from a random initialization point yields perfect recovery in only about 50% of our experiments. This percentage decreases as the signal size p increases. The average error (x, x) on non-recovered signals in Table 2.2 is 1.3 whereas on the average error on the modulus (|Ax|, |Ax|) is 0.2.

Wavelet transform

Finally, we test the algorithm on the phase retrieval problem that will be discussed in the next two chapters of this thesis: the case of the wavelet transform.

To simplify experiments, we consider wavelets dilated by dyadic factors 2 j , which have a lower frequency resolution than audio wavelets. A discrete wavelet transform is computed by circular convolutions with discrete wavelet filters, i.e.

(Ax) k+jp = (x ψ j ) k = p m=1 x m ψ j k-m for 1 ≤ j ≤ J -1 and 1 ≤ k ≤ p
where ψ j m is a p periodic wavelet filter. It is defined by dilating, sampling and periodizing a complex wavelet ψ ∈ L 2 (C), with

ψ j m = ∞ k=-∞ ψ(2 j (m/p -k)) for 1 ≤ m ≤ p.
Numerical computations are performed with a Cauchy wavelet whose Fourier transform is, up to a scaling factor ψ(ω) = ω d e -ω 1 ω>0 , with d = 5. To guarantee that A is an invertible operator, the lowest signal frequencies are carried by a suitable low-pass filter φ and

(Ax) k+Jp = (x φ) k for 1 ≤ k ≤ p.
One can prove that x is always uniquely determined by |Ax|, up to a multiplication factor (this will be done in Chapter 3). We consider the case of real signals x; recall that the results of Paragraph 2.2.6 allow us to explicitly impose the condition that x is real in the PhaseCut recovery algorithm. For PhaseLift in [START_REF] Candès | Phase retrieval via matrix completion[END_REF], this condition is enforced by imposing that X = xx * is real. For the Gerchberg-Saxton algorithm, when x is real, we simply project at each iteration on the image of R p by A, instead of projecting on the image of C p by A.

Numerical experiments are performed on the real part of the complex test signals. Table 2.1 shows that Gerchberg-Saxton does not reconstruct exactly any test signal from the modulus of its wavelet coefficients. The average relative error (x, x) in Table 2.2 is 1.2 where the coefficient amplitudes have an average error (|Ax|, |Ax|) of 0.3 in Table 2.3.

PhaseLift reconstructs 62% of test signals, but the reconstruction rate varies with the signal type. The proportions of exactly reconstructed signals among random noises, sums of sinusoids and image scan-lines are 27%, 60% and 99% respectively. Indeed, image scan-lines have a large proportion of wavelet coefficients whose amplitudes are negligible. The proportion of phase coefficients having a strong impact on the reconstruction of x is thus much smaller for scan-line images than for random noises, which reduces the number of significant variables to recover. Sums of sinuoids of random frequency have wavelet coefficients whose sparsity is intermediate between image scan-lines and Gaussian white noises, which explains the intermediate performance of PhaseLift on these signals. The overall average error (x, x) on non-reconstructed signals is 0.5. Despite this relatively important error, x and x are usually almost equal on most of their support, up to a sign switch, and the importance of the error is precisely due to the number of sign switches.

The PhaseCut algorithm exactly reconstructs all test signals. Moreover, the recovered matrix U is always of rank one and it is therefore not necessary to refine the solution with Gerchberg-Saxton iterations. At first sight, this difference in performance between PhaseCut and PhaseLift may seem to contradict the equivalence results of Paragraph 2.3.3 (which are valid both when x is real and when x is complex). It can be explained however by the fact that 10 steps of reweighting and 1000 inner iterations per step are not enough to let PhaseLift fully converge. In these experiments, the precision required to get perfect reconstruction is very high and, consequently, the number of first-order iterations required to achieve it is too large (see Paragraph 2.3.6). With an interior-point-solver, this number would be much smaller but the time required per iteration would become prohibitively large. The much simpler structure of the PhaseCut relaxation allows us to solve these larger problems more efficiently.

Impact of trace minimization

We saw in Paragraph 2.3.1 that, in the absence of noise, PhaseCut was very similar to a simplified version of PhaseLift, Weak PhaseLift, in which no trace minimization is performed. Here, we confirm empirically that Weak PhaseLift and PhaseLift are essentially equivalent. Minimizing the trace is usually used as rank minimization heuristic, with recovery guarantees in certain settings [START_REF] Fazel | Log-det heuristic for matrix rank minimization with applications to hankel and euclidean distance matrices[END_REF][START_REF] Candès | Exact matrix completion via convex optimization[END_REF][START_REF] Chandrasekaran | The convex geometry of linear inverse problems[END_REF] but it does not seem to make much difference here. In fact, [START_REF] Demanet | Stable optimizationless recovery from phaseless linear measurements[END_REF]; [START_REF] Candès | Solving quadratic equations via phaselift when there are about as many equations as unknowns[END_REF] showed that in the setting where measurements are randomly chosen according to independant Gaussian laws, Weak PhaseLift has a unique (rank one) solution with high probability, i.e. the feasible set of PhaseLift is a singleton and trace minimization has no impact. Of course, from a numerical point of view, solving the feasibility problem Weak PhaseLift is about as hard as solving the trace minimization problem PhaseLift, so this result simplifies analysis but does not really affect numerical performance.

Figure 2.3 compares the performances of PhaseLift and Weak PhaseLift as a function of n (the number of measurements). We plot the percentage of successful reconstructions (left) and the percentage of cases where the relaxation was exact, i.e. the reconstructed matrix X was rank one (right). The plot shows clear phase transitions when the number of measurements increases. For PhaseLift, these transitions happen respectively at n = 155 ≈ 2.5p and n = 285 ≈ 4.5p, while for Weak PhaseLift, the values become n = 170 ≈ 2.7p and n = 295 ≈ 4.6p, so the transition thresholds are very similar. Note that, in the absence of noise, Weak PhaseLift and PhaseCut have the same solutions, up to a linear transformation (see Paragraph 2.3.2), so we can expect the same results when comparing PhaseCut with PhaseCutMod . No algorithm can overcome the first cause but good reconstruction methods will overcome the second one. In the following paragraphs, to complement the results in Paragraph 2.3.4, we will demonstrate empirically that PhaseCut is stable, and compare its performances with PhaseLift. We will observe in particular that PhaseCut appears to be more stable than PhaseLift when b is sparse. , which is the best we can hope for. Wavelet transform is a case where the underlying phase retrieval problem may present instabilities, therefore the reconstruction error (x, x) is sometimes much larger than (|Ax|, |Ax|). This remark applies especially to sums of sinusoids, which represent the most unstable case.

Reconstruction in the presence of noise

Wavelet transform

When all coefficients of Ax have approximately the same amplitude, PhaseLift and PhaseCut produce similar results, but when Ax is sparse, PhaseLift appears less stable. We gave a qualitative explanation of this behavior at the end of Paragraph 2.3.4 which seems to be confirmed by the results in Figure 2.4. Indeed, the performance of PhaseLift and PhaseCut are equivalent in the case of Gaussian random filters (where measurements are never sparse), they are a bit worse in the case of sinusoids (where measurements are sometimes sparse) and quite unsatisfactory for scan-lines of images (where measurements are always sparse).

Multiple random illumination filters

Candès and [START_REF] Candès | Solving quadratic equations via phaselift when there are about as many equations as unknowns[END_REF] prove that, if A is a Gaussian matrix, the reconstruction problem is stable with high probability, and PhaseLift reconstructs a x such that

(x, x) ≤ O b noise 2 Ax 2 .
The same result seems to hold for A corresponding to Gaussian random illumination filters (cf. Paragraph 2.4.2). Moreover, PhaseCut is as stable as PhaseLift. Actually, up to 20% of noise, when followed by some Gerchberg-Saxton iterations, PhaseCut and PhaseLift almost always reconstruct the same function. Figure 2.5 displays the corresponding empirical performance, confirming that both algorithms are stable. The relative reconstruction errors are approximately linear in the amount of noise, with

(|Ax|, |Ax|) ≈ 0.8 × b noise 2 Ax 2 and (x, x) ≈ 2 × b noise 2 Ax 2
in our experiments. The impact of the sparsity of b discussed in the last paragraph may seem irrelevant here: if A and x are independently chosen, Ax is never sparse. However, if we do not choose A and x independently, we may achieve partial sparsity. We performed tests for the case of five Gaussian random filters, where we chose x ∈ C 64 such that (Ax) k = 0 for k ≤ 60. This choice has no particular physical interpretation but it allows us to check that the influence of sparsity in |Ax| over PhaseLift is not specific to the wavelet transform. Figure 2.5 displays the relative error over the reconstructed matrix in the sparse and non-sparse cases. If we denote by X pl ∈ C p×p (resp. X pc ∈ C n×n ) the matrix reconstructed by PhaseLift (resp. PhaseCut), this relative error is defined by

= AX pl A * -(Ax)(Ax) * 2 (Ax)(Ax) * 2 (for PhaseLift) = diag (b)X pc diag (b) -(Ax)(Ax) * 2 (Ax)(Ax) * 2 (for PhaseCut)
In the non-sparse case, both algorithms yield very similar error ≈ 7 b noise 2 / Ax 2 (the difference for a relative noise of 10 -4 may come from a computational artifact). In the sparse case, there are less phases to reconstruct, because we do not need to reconstruct the phase of

Gaussian Sinusoids -3 -2.5 -2 -1.5 -1 -0.5 -3 -2.5 -2 -1.5 -1 -0.5 0 Amount of noise Relative error -3 -2.5 -2 -1.5 -1 -0.5 -3 -2.5 -2 -1.5 -1 -0.5 0

Amount of noise

Relative error

Image Scan-Lines

Amount of noise

Relative error

PhaseCut) null measurements. Consequently, the problem is better constrained and we expect the algorithms to be more stable. Indeed, the relative errors over the reconstructed matrices are smaller. However, in this case, the performance of PhaseLift and PhaseCut do not match anymore: ≈ 3 b noise 2 / Ax 2 for PhaseLift and ≈ 1.2 b noise 2 / Ax 2 for PhaseCut. This remark has no practical impact in our particular example here because taking a few Gerchberg-Saxton iterations would likely make both methods converge towards the same solution, but it confirms the importance of accounting for the sparsity of |Ax|.

-4 -3.5 -3 -2.5 -2 -1.5 -1 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.

Technical lemmas

We now prove two technical lemmas used in the proof of Theorem 2.8. Lemma 2.9. Under the assumptions and notations of Theorem 2.8, we have

V P C -(Ax 0 )(Ax 0 ) * 2 > 2C Ax 0 2 b n,PC 2
Proof. We first give an upper bound of V P C -V P C 2 . We use the Cauchy-Schwarz inequality: for every positive matrix X and all x, y, |x * Xy| ≤ √ x * Xx √ y * Xy. Let {f i } be an hermitian base of Range (A) diagonalizing V P C and {g i } an hermitian base of Range (A) ⊥ diagonalizing

V ⊥ P C . As {f i } ∩ {g i } is an hermitian base of C n , we have V P C -V P C 2 2 = i,i |f * i (V P C -V P C )f i | 2 + i,j |f * i (V P C -V P C )g j | 2 + i,j |g * j (V P C -V P C )f i | 2 + j,j |g * j (V P C -V P C )g j | 2 = 2 i,j |f * i (V P C )g j | 2 + i |g * i (V ⊥ P C )g i | 2 ≤ 2 i,j |f * i (V P C )f i g * j (V P C )g j | + ( i g * i (V ⊥ P C )g i ) 2 = 2Tr V P C Tr V ⊥ P C + (Tr V ⊥ P C ) 2 ≤ √ 2 Tr V P C Tr V ⊥ P C + Tr V ⊥ P C 2 (2.21)
Let us now bound Tr V ⊥ P C . We first note that Tr

V ⊥ P C = Tr ((I -AA † )V P C (I -AA † )) = Tr (V P C (I -AA † )) = d 1 (V P C , F) (according to Lemma 2.2). Let u ∈ C n be such that, for all i, |u i | = 1 and (Ax 0 ) i = u i |Ax 0 | i . We set b = |Ax 0 | + b n,PC and V = (b × u)(b × u) * . As V ∈ H + n ∩ H b and V P C minimizes (2.13), Tr V ⊥ P C = d 1 (V P C , F) ≤ d 1 (V, F) = d 1 ((Ax 0 + b n,PC u)(Ax 0 + b n,PC u) * , F) = d 1 ((b n,PC u)(b n,PC u) * , F) ≤ (b n,PC u)(b n,PC u) * 1 = Tr (b n,PC u)(b n,PC u) * = b n,PC 2 2
We also have Tr V P C = Tr V P C -Tr V ⊥ P C . This equality comes from the fact that, if {f i } is an hermitian base of Range (A) and {g i } an hermitian base of Range (A) ⊥ , then

Tr V P C = i f i V P C f * i + i g i V P C g * i = i f i V P C f * i + i g i V ⊥ P C g * i = Tr V P C + Tr V ⊥ P C As V ⊥ P C 0, Tr V P C ≤ Tr V P C = |Ax 0 | + b n,PC 2
2 and, by combining this with relations (2.21) and ( 2.22), we get

V P C -V P C 2 ≤ √ 2 |Ax 0 | + b n,PC 2 b n,PC 2 + b n,PC 2 2 ≤ √ 2 Ax 0 2 b n,PC 2 + (1 + √ 2) b n,PC 2 2
And, by reminding that we assumed b n,PC 2 ≤ Ax 0 2 ,

V P C -(Ax 0 )(Ax 0 ) * 2 ≥ V P C -(Ax 0 )(Ax 0 ) * 2 -V P C -V P C 2 > D Ax 0 2 b n,PC 2 - √ 2 Ax 0 2 b n,PC 2 -(1 + √ 2) b n,PC 2 2 ≥ (D -2 √ 2 -1) Ax 0 2 b n,PC 2 = 2C Ax 0 2 b n,PC 2
which concludes the proof.

Lemma 2.10. Under the assumptions and notations of Theorem 2.8, we have b n,PL 2 ≤ 2 b n,PC .

Proof. Let e i be the i-th vector of C n 's canonical base. We set e i = f i + g i where f i ∈ Range (A) and g i ∈ Range (A) ⊥ .

V P C ii = e * i V P C e i = f * i V P C f i + 2Re (f * i V P C g i ) + g * i V ⊥ P C g i = V P C ii + 2Re (f * i V P C g i ) + V ⊥ P C ii Because |f * i V P C g i | ≤ f * i V P C f i g * i V P C g i = V P C ii V ⊥ P C ii , ( V P C ii -V ⊥ P C ii ) 2 ≤ V P C ii ≤ ( V P C ii + V ⊥ P C ii ) 2 ⇒ V P C ii -V ⊥ P C ii ≤ V P C ii ≤ V P C ii + V ⊥ P C ii So |b n,PL,i | = | V P C ii -|Ax 0 | i | ≤ | V P C ii -V P C ii | + | V P C ii -|Ax 0 | i | ≤ V ⊥ P C ii + b n,PC,i
and, by (2.22),

b n,PL 2 ≤ V ⊥ P C ii i 2 + b n,PC 2 = Tr V ⊥ P C + b n,PC 2 ≤ 2 b n,PC 2
which concludes the proof.

Chapter 3

Phase retrieval for the Cauchy wavelet transform

After presenting an algorithm for solving generic phase retrieval problems in the previous chapter, we now consider a specific phase retrieval problem: the case of the wavelet transform. A wavelet family (ψ j ) j∈Z being fixed, the problem is:

reconstruct f ∈ L 2 (R) from {|f ψ j |} j∈Z
In this chapter, we study the well-posedness of this problem, in terms of uniqueness and stability: is any function f uniquely determined by its wavelet transform modulus? Is the reconstruction stable to noise?

We restrict ourselves to a specific class of wavelets, namely Cauchy wavelets, whose link with harmonic analysis makes it easier to study.

Besides its applications in audio processing, this problem has a theoretical interest in the general context of phase retrieval.

The wavelet transform is one of the only known natural1 operators for which the phase retrieval problem is now known to be well-posed (uniqueness of the reconstruction, and a form of stability). It in particular contrasts with the Fourier transform [START_REF] Walther | The question of phase retrieval in optics[END_REF][START_REF] Akutowicz | On the determination of the phase of a Fourier integral[END_REF] and the fractional Fourier transform [START_REF] Jaming | Uniqueness results in an extension of pauli's phase retrieval problem[END_REF], which can be studied with the same tools: in the first case, there is no uniqueness; in the second one, there is uniqueness, but there is no stability in a strong sense, and no result for a weaker form of stability is known.

Moreover, the notion of local stability which appears in the case of the wavelet transform is new to phase retrieval.

The uniqueness result is Corollary 3.2. It shows that two analytical signals whose wavelet transforms are identical in modulus are equal up to a global phase. A signal f is said to be analytical if f (ω) = 0 when ω < 0. This condition may seem restrictive, but it is actually not, because almost the same result holds for real-valued signals (Corollary 3.3), which covers all conceivable applications.

In Theorem 3.11, we prove that the reconstruction operator is continuous. This is a weak notion of stability to noise, and we show that no strong stability holds, in the sense that, for any > 0, there exist functions f, g such that:

{|f ψ j |} j -{|g ψ j |} j 2 < but ||f -g|| 2 ≥ 1
So two functions can have almost the same wavelet transform modulus without being close in the L 2 -norm sense. However, we have a form of local stability. Theorems 3.17 and 3.18 can be approximately summarized by the following informal assertion: if the wavelet transform modulus of two functions are close, then the wavelet transforms are close, up to a global phase, in the neighborhood of each point (j, t) of the time-frequency plane, except maybe at points where the wavelet transform is close to zero.

Our proof techniques naturally yield a reconstruction algorithm. We have implemented it; it yields precise reconstruction results and is relatively stable to noise. Because it only works for Cauchy wavelets, it has limited application. However, it allows us to empirically confirm our theoretical results. Section 3.1 is concerned with uniqueness results. Section 3.2 proves the weak stability theorem. Section 3.3 explains why there is no strong stability. The local stability results are proved in Section 3.4. Section 3.5 describes the algorithm and our numerical experiments. Finally, Section 3.6 proves useful technical lemmas.

The results of this chapter have been published in [START_REF] Mallat | Phase retrieval for the cauchy wavelet transform[END_REF].

Notations

For any f ∈ L 1 (R), we denote by f or F(f ) the Fourier transform of f :

f (ω) = R f (x)e -iωx dx ∀ω ∈ R
We extend this definition to L 2 by continuity.

We denote by F -1 : L 2 (R) → L 2 (R) the inverse Fourier transform and recall that, for any f ∈ L 1 ∩ L 2 (R):

F -1 (f )(x) = 1 2π R f (ω)e iωx dω
We denote by H the Poincaré half-plane:

H = {z ∈ C s.t. Im z > 0}.
3.1 Uniqueness of the reconstruction for Cauchy wavelets

Definition of the wavelet transform; comparison with Fourier

The most important phase retrieval problem, which naturally arises in several physical settings, is the case of the Fourier transform:

reconstruct f ∈ L 2 (R) from | f | Without additional assumptions over f , the reconstruction is clearly impossible: any choice of phase φ : R → R yield a signal g = F -1 (| f |e iφ ) ∈ L 2 (R) such that |ĝ| = | f |.
To avoid this problem, one may for example require that f is compactly supported. However, [START_REF] Akutowicz | On the determination of the phase of a Fourier integral[END_REF]; [START_REF] Walther | The question of phase retrieval in optics[END_REF] showed that, even with this constraint, the reconstruction is still not possible.

More precisely, their result is the following one. If f ∈ L 2 (R) is a compactly supported function, then its Fourier transform f admits a holomorphic extension F over all C:

F (z) = R f (x)e -izx dx. If g ∈ L 2 (
R) is another compactly supported function and G is this holomorphic extension of its Fourier transform, the equality | f | = |ĝ| happens to be equivalent to:

∀z ∈ C, F (z)F (z) = G(z)G(z)
This in turn is essentially equivalent to:

{z n } ∪ {z n } = {z n } ∪ {z n } (3.1)
where the (z n ) and (z n ) are the respective zeros of F and G over C, counted with multiplicity. This means that F and G must have the same zeros, up to symmetry with respect to the real axis.

Conversely, for every choice of {z n } satisfying (3.1), it is possible to find a compactly supported g such that the zeroes of G are the z n , which implies

| f | = |ĝ|.
A similar result can be established in the case where the function f ∈ L 2 (R) is assumed to be identically zero on the negative real line [START_REF] Akutowicz | On the determination of the phase of a Fourier integral[END_REF] instead of compactly supported.

Let us now define the wavelet transform and compare it with the Fourier transform. Let ψ ∈ L 1 ∩ L 2 (R) be a wavelet, that is a function such that R ψ(x)dx = 0. Let a > 1 be fixed; we call a the dilation factor. We define a family of wavelets by:

∀x ∈ R ψ j (x) = a -j ψ(a -j x) ⇔ ∀ω ∈ R ψj (ω) = ψ(a j ω)
The wavelet transform operator is:

f ∈ L 2 (R) → {f ψ j } j∈Z ∈ (L 2 (R)) Z
This operator is unitary if the so-called Littlewood-Paley condition is satisfied:

j | ψj (ω)| 2 = 1, ∀ω ∈ R ⇒ ||f || 2 2 = j ||f ψ j || 2 2 ∀f ∈ L 2 (R) (3.2)
The phase retrieval problem associated with this operator is:

reconstruct f ∈ L 2 (R) from {|f ψ j |} j∈Z
This problem may or may not be well-posed, depending on which wavelet family we use. The simplest case is the one where the wavelets are Shannon wavelets:

ψ = 1 [1;a] ⇒ ∀j ∈ Z, ψj (ω) = 1 [a -j ;a -j+1 ]
Reconstructing f amounts to reconstruct f 1 [a -j ;a -j+1 ] = f ψj for all j. For each j, we have only two pieces of information about f ψj : its support is included in [a -j ; a -j+1 ] and the modulus of its inverse Fourier transform is |f ψ j |. From the results of the Fourier transform case, it is not enough to determine uniquely f ψj . Thus, for Shannon wavelets, the phase retrieval problem is as ill-posed as for the Fourier transform.

In this example, the problem comes from the fact that the ψj have non-overlapping supports. Thus, reconstructing f is equivalent to reconstructing independently each f ψ j , and that is not possible.

However, in general, the ψj have overlapping supports and the f ψ j are not independent for different values of j. They satisfy the following relation:

(f ψ j ) ψ k = (f ψ k ) ψ j ∀j, k ∈ Z (3.3)
Thus, there is "redundancy" in the wavelet decomposition of f . We can hope that this redundancy compensates the loss of phase of |f ψ j |. In the following, we show that, at least for specific wavelets, it is the case.

Uniqueness theorem for Cauchy wavelets

In this paragraph, we consider wavelets of the following form:

ψ(ω) = ρ(ω)ω p e -ω 1 ω>0 (3.4) ψj (ω) = ψ(a j ω) ∀ω ∈ R where p > 0 and ρ ∈ L ∞ (R) is such that ρ(aω) = ρ(ω) for almost every ω ∈ R and ρ(ω) = 0, ∀ω.
The presence of ρ allows some flexibility in the choice of the family. In particular, if it is properly chosen, the Littlewood-Paley condition (3.2) may be satisfied. However, the proofs are the same with or without ρ.

When ρ = 1, the wavelets of the form (3.4) are called Cauchy wavelets of order p. The figure 3.1 displays an example of such wavelets. For these wavelets, the wavelet transform has the property to be a set of sections of a holomorphic function along horizontal lines.

If f ∈ L 2 (R), its analytic part f + is defined by:

f+ (ω) = 2 f (ω)1 ω>0 (3.5)
We define:

F (z) = 1 2π R ω p f+ (ω)e iωz dω ∀z s.t. Im z > 0 (3.6)
When f + is sufficiently regular, F is the holomorphic extension of its p-th derivative.

For each y > 0, if we denote by F (. + iy) the function x ∈ R → F (x + iy):

F (. + iy) = F -1 2ω p f (ω)1 ω>0 e -yω
Consequently, for each j ∈ Z:

a pj 2 F (. + ia j ) = f ψ j ∀j ∈ Z (3.7)
So f ψ j is the restriction of F to the horizontal line R + ia j . In this case, the relation (3.3) is equivalent to the fact that, for all j, k, f ψ j and f ψ k are the restrictions of the same holomorphic function to the lines R + ia j and R + ia k . Reconstructing f + from {|f ψ j |} j∈Z now amounts to reconstruct the holomorphic function F : H = {z ∈ C, Im z > 0} → C from its modulus on an infinite set of horizontal lines. The figure 3.2 shows these lines for a = 2. Our phase retrieval problem thus reduces to a harmonic analysis problem. Actually, knowing |F | on only two lines is already enough to recover F and one of the two lines may even be R, the boundary of H. We suppose that:

|F (x + iα)| = |G(x + iα)| for a.e. x ∈ R lim y→0 + |F (x + iy)| = lim y→0 + |G(x + iy)| for a.e. x ∈ R
Then, for some φ ∈ R:

F = e iφ G (3.9)
The proof is given in section 3.1.4.

Corollary 3.2. We consider wavelets (ψ j ) j∈Z of the form (3.4). Let f, g ∈ L 2 (R) be such that, for some j, k ∈ Z with j = k:

|f ψ j | = |g ψ j | and |f ψ k | = |g ψ k | (3.10)
We denote by f + and g + the analytic parts of f and g (as defined in (3.5))

There exists φ ∈ R such that:

f + = e iφ g + (3.11)
Another uniqueness result for the wavelet transform can be found in [Jaming, 2014, Thm 4.4]. However, it is of different nature, because it concerns wavelet transforms with continuous frequency parameter.

Proof. We may assume that j < k. We define F and G as in (3.6), with the additional ρ:

F (z) = 1 2π R ω p ρ(ω) f+ (ω)e iωz dω G(z) = 1 2π R ω p ρ(ω)ĝ + (ω)e iωz dω ∀z ∈ H For each y > 0, F (. + iy) = F -1 (2ω p ρ(ω)e -yω 1 ω>0 f (ω)). For y = a j and y = a k , it implies F (. + ia j ) = 2 a jp f ψ j and F (. + ia k ) = 2 a kp f ψ k . From (3.10): |F (. + ia j )| = 2 a pj |f ψ j | = 2 a pj |g ψ j | = |G(. + ia j )| |F (. + ia k )| = 2 a pk |f ψ k | = 2 a pk |g ψ k | = |G(. + ia k )|
So the functions F (. + ia j ) and G(. + ia j ) coincide in modulus on two horizontal lines: R and R + i(a k -a j ). From Theorem 3.1, they are equal up to a global phase. As ρ does not vanish, it implies that f + and g + are equal up to this global phase.

In order to be able to apply Theorem 3.1, we must verify that the condition (3.8) holds for F (. + ia j ) and G(. + ia j ). For any y > a j :

F (. + iy) = F -1 2ω p ρ(ω) f (ω)e -yω ⇒ ||F (. + iy)|| 2 2 = 1 2π ||2ω p ρ(ω) f (ω)e -yω 1 ω≥0 || 2 2 ≤ 1 2π ||2ω p ρ(ω) f (ω)e -a j ω 1 ω≥0 || 2 2 = 2 a jp 2 ||f ψ j || 2 2
The same inequality holds for G:

the condition (3.8) is true for M = 2 a jp 2 ||f ψ j || 2 2 .
We have just proven that the modulus of the wavelet transform uniquely determines, up to a global phase, the analytic part of a function, that is its positive frequencies. On the contrary, as wavelets are analytic ( ψj (ω) = 0 if ω < 0), the wavelet transform contains no information about the negative frequencies. In practice, signals are often real so negative frequencies are determined by positive ones and this latter limitation is not really important.

Corollary 3.3. Let f, g ∈ L 2 (R) be real-valued functions; f + and g + are their analytic parts. We assume that, for some j, k ∈ Z such that j = k:

|f ψ j | = |g ψ j | and |f ψ k | = |g ψ k |
Then, for some φ ∈ R: Before ending this section, let us note that, with a proof similar to the one of Corollary 3.2, Theorem 3.1 also implies the following result.

f + = e iφ g + ⇔ f =
Corollary 3.5. Let α > 0 be fixed. Let f, g ∈ L 2 (R) be such that f (ω) = ĝ(ω) = 0 for every ω < 0. If | f | = |ĝ| and | f (t)e -αt | = | g(t)e -αt |, then, for some φ ∈ R: f = e iφ g
This says that there is uniqueness in the phase retrieval problem associated to the masked Fourier transform, in the case where there are two masks, t → 1 and t → e -αt .

Discrete case

Naturally, the functions we have to deal with in practice are generally not in L 2 (R). They are instead discrete finite signals. In this section, we explain how to switch from the continuous to the discrete finite setting. As we will see, all results derived in the continuous case have a discrete equivalent but proofs become simpler because they use polynomials instead of holomorphic functions.

Let f ∈ C n be a discrete function. We assume n is even. The discrete Fourier transform of f is:

f [k] = n-1 s=0 f [s]e -2πisk n for k = - n 2 + 1, ..., n 2
The analytic part of f is f + ∈ C n such that:

f+ [k] = 0 if - n 2 + 1 ≤ k < 0 f+ [k] = f [k] if k = 0 or k = n 2 f+ [k] = 2 f [k] if 0 < k < n 2 When f is real, f = Re (f + ).
We consider wavelets of the following form, for p > 0 and a > 1: (3.12) where ρ : R + → C is such that ρ(ax) = ρ(x) for every x and ρ does not vanish.

ψj [k] = ρ(a j k)(a j k) p e -a j k 1 k≥0 for all j ∈ Z, k = - n 2 + 1, ..., n 2
As in the continuous case, the set {|f ψ j |} j∈Z almost uniquely determines f + . Naturally, the global phase still cannot be determined. The mean value of f + can also not be determined, because ψj [0] = 0 for all j. To determine the mean value and the global phase, we would need some additional information, for example the value of f φ for some low frequency signal φ.

Theorem 3.6 (Discrete version of 3.2). Let f, g ∈ C n be discrete signals and (ψ j ) j∈Z a family of wavelets of the form (3.12). Let j, l ∈ Z be two distinct integers. Then:

|f ψ j | = |g ψ j | and |f ψ l | = |g ψ l | (3.13)
if and only if, for some φ ∈ R, c ∈ C:

f + = e iφ g + + c
Proof. We first assume f + = e iφ g + + c. Taking the Fourier transform of this equality yields:

f [k] = e iφ ĝ[k] for all k = 1, ..., n 2
As ψj [k] = 0 for k = -n 2 + 1, ..., 0:

f [k] ψj [k] = e iφ ĝ[k] ψj [k] for all k = - n 2 + 1, ..., n 2 ⇒ (f ψ j = e iφ (g ψ j )) So |f ψ j | = |g ψ j | and, similarly, |f ψ l | = |g ψ l |.
We now suppose conversely that |f ψ j | = |g ψ j | and |f ψ l | = |g ψ l |. We define:

F (z) = 1 n n/2 k=1 f [k]ρ(k)k p z k G(z) = 1 n n/2 k=1 ĝ[k]ρ(k)k p z k ∀z ∈ C
These polynomials are the discrete equivalents of functions F and G used in the proof of 3.2. For all s = -n 2 + 1, ..., n 2 :

F (e -a j e

2πis n ) = 1 n n/2 k=1 f [k]ρ(k)k p e -a j k e 2πiks n = a -jp 1 n n/2 k=-n/2+1 f [k] ψj [k]e 2πiks n = a -jp (f ψ j [s])
Similarly, G(e -a j e 2πis n ) = a -jp (g ψ j [s]) for all s = -n 2 + 1, ..., n 2 . Thus, f ψ j and g ψ j can be seen as the restrictions of F and G to the circle of radius e -a j . This is similar to the continuous case, where f ψ j and g ψ j were the restrictions of functions F, G to horizontal lines.

The equality (3.13) implies:

F (e -a j e 2πis n ) 2 = G(e -a j e 2πis n ) 2 for all s = - n 2 + 1, ..., n 2 ⇔ F (e -a j e 2πis n )F (e -a j e -2πis n ) = G(e -a j e 2πis n )G(e -a j e -2πis n ) for all s = - n 2 + 1, ..., n 2
The functions z → F (e -a j z)F (e -a j 1 z ) and z → G(e -a j z)G(e -a j 1 z ) are polynomials of degree n-2 (up to multiplication by z n/2-1 ). They share n common values so they are equal. The same is true for l instead of j so:

F (e -a j z)F e -a j 1 z = G(e -a j z)G e -a j 1 z ∀z ∈ C (3.14) F (e -a l z)F e -a l 1 z = G(e -a l z)G e -a l 1 z ∀z ∈ C (3.15)
If we show that these equalities imply F = e iφ G for some φ ∈ R, the proof will be finished. Indeed, from the definition of F and G, we will then have f

[k] = e iφ ĝ[k] for all k = 1, ..., n 2 so f+ [k] = e iφ ĝ+ [k] for all k = 0. It implies f + = e iφ g + + c for c = 1 n f+ [0] -e iφ ĝ+ [0] .
It suffices to show that F and G have the same roots (with multiplicity) because then, they will be proportional and, from (3.14), (3.15), the proportionality constant must be of modulus 1.

For each z ∈ C, let µ F (z) (resp. µ G (z)) be the multiplicity of z as a root of F (resp. G). The polynomials of (3.14) are of respective degree n -2µ F (0) and n -2µ G (0) so µ F (0) = µ G (0).

For all z = 0, the multiplicity of e a j z as a zero of (3.14) is:

µ F (z) + µ F e -2a j z = µ G (z) + µ G e -2a j z
and the multiplicity of e 2a j -a l z as a zero of (3.15) is:

µ F (e 2(a j -a l ) z) + µ F e -2a j z = µ G (e 2(a j -a l ) z) + µ G e -2a j z
Subtracting this last equality to the previous one implies that, for all z:

µ F (z) -µ G (z) = µ F (e 2(a j -a l ) z) -µ G (e 2(a j -a l ) z)
By applying this equality several times, we get, for all n ∈ N:

µ F (z) -µ G (z) = µ F (e 2(a j -a l ) z) -µ G (e 2(a j -a l ) z) = µ F (e 4(a j -a l ) z) -µ G (e 4(a j -a l ) z) = ... = µ F (e 2n(a j -a l ) z) -µ G (e 2n(a j -a l ) z)
As F and G have a finite number of roots, µ F (e 2n(a j -a l ) z) -µ G (e 2n(a j -a l ) z) = 0 if n is large enough. So µ F (z) = µ G (z) for all z ∈ C.

As in the section 3.1.2, a very similar proof gives a uniqueness result for the case of the Fourier transform with masks, if the masks are well-chosen. Theorem 3.7 (Discrete version of 3.5). Let α > 0 be fixed. Let f, g ∈ C 2n-1 be two discrete signals with support in {0, ..., n -1}:

f [s] = g[s] = 0 for s = n, ..., 2n -2 If | f | = |ĝ| and | f [s]e -sα | = | g[s]e -sα |, then, for some φ ∈ R: f = e iφ g
Remark that this theorem describes systems of 4n -2 linear measurements whose moduli are enough to recover each complex signal of dimension n. It is known that 4n -4 generic measurements always achieve this property ( [START_REF] Balan | On signal reconstruction without noisy phase[END_REF][START_REF] Conca | Algebraic characterization of injectivity in phase retrieval[END_REF]). However, it is in general difficult to find deterministic systems for which it can be proven. We suppose that:

|F (x + iα)| = |G(x + iα)| for a.e. x ∈ R lim y→0 + |F (x + iy)| = lim y→0 + |G(x + iy)| for a.e. x ∈ R
Then, for some φ ∈ R:

F = e iφ G (3.16)
Proof of Theorem 3.1. This proof relies on the ideas used by [START_REF] Akutowicz | On the determination of the phase of a Fourier integral[END_REF].

If F = 0, the theorem is true: G is null over a whole line and, as G is holomorphic, G = 0. The same reasoning holds if G = 0. We now assume F = 0, G = 0.

The central point of the proof is to factorize the functions F, F (. + iα), G, G(. + iα) as in the following lemma.

Lemma 3.8. [START_REF] Kryloff | On functions which are regular in a half-plane[END_REF] 2 The function F admits the following factorization:

F (z) = e ic+iβz B(z)D(z)S(z)
Here, c and β are real numbers. The function B is a Blaschke product. It is formed with the zeros of F in the upper half-plane H. We call (z k ) these zeros, counted with multiplicity, with the exception of i. We call m the multiplicity of i as zero.

B(z) = z -i z + i m k |z k -i| z k -i |z k + i| z k + i z -z k z -z k (3.17)
This product converges over H, which is equivalent to:

k Im z k 1 + |z k | 2 < +∞ (3.18)
2 Non Russian speaking readers may also deduce this theorem from Rudin [1987, Thm 17.17]: functions over H may be turned into functions over D(0, 1) by composing them with the conformal application z ∈ D(0, 1) →

1-z 1+z i ∈ H. The main difficulty is to show that if H : H → C satisfies (3.8), then H : z ∈ D(0, 1) → H 1-z 1+z i ∈ C is of class H 2 and
Rudin's theorem can be applied.

The functions D and S are defined by:

D(z) = exp 1 πi R 1 + tz t -z log |F (t)| 1 + t 2 dt (3.19) S(z) = exp i π R 1 + tz t -z dE(t) (3.20)
In the first equation, |F (t)| is the limit of |F | on R. In the second one, dE is a positive bounded measure, singular with respect to Lebesgue measure. Both integrals converge absolutely for any z ∈ H.

The same factorization can be applied to F (. + iα), G and G(. + iα):

F (z) = e ic F +iβ F z B F (z)D F (z)S F (z) G(z) = e ic G +iβ G z B G (z)D G (z)S G (z) F (z + iα) = e ic F +i βF z BF (z) DF (z) SF (z) G(z + iα) = e ic G +i βG z BG (z) DG (z) SG (z)
As F (. + iα) and G(. + iα) are analytic on the real line, they actually have no singular part S. The proof may be found in [Garnett, 1981, Thm 6.3]; it is done for functions on the unit disk but also holds for functions on H. For all z ∈ H:

e ic F +iβ F (z+iα) B F (z + iα)D F (z + iα)S F (z + iα) e ic F +i βF z BF (z) DF (z) = F (z + iα) F (z + iα) = 1 = G(z + iα) G(z + iα) = e ic G +iβ G (z+iα) B G (z + iα)D G (z + iα)S G (z + iα) e ic G +i βG z BG (z) DG (z) ⇒ B F (z + iα) BG (z) B G (z + iα) BF (z) = e iC+iBz S G (z + iα) S F (z + iα) (3.22)
for some C, B ∈ R Equality (3.22) holds only for z ∈ H. It is a priori not even defined for z ∈ C -H. Before going on, we must show that (3.22) is meaningful and still valid over all C. This is the purpose of the two following lemmas, whose proofs may be found in Paragraph 3.6.1.

For z ∈ H, we denote by µ F (z) (resp. µ G (z)) the multiplicity of z as a zero of F (resp. G).

Lemma 3.9. There exists a meromorphic function B w : C → C such that:

B w (z) = B F (z + iα) BG (z) B G (z + iα) BF (z) ∀z ∈ H
Moreover, for every z ∈ H, the multiplicity of z -iα as a pole of B w is:

(µ F (z) -µ G (z)) -(µ F (z + 2iα) -µ G (z + 2iα)) (3.23)
Lemma 3.10. For every z ∈ H, S G (z+iα) S F (z+iα) = 1. Equation (3.22) and Lemmas 3.9 and 3.10 give, for all z ∈ H and thus all z ∈ C (because functions are meromorphic):

B w (z) = e iC+iBz ∀z ∈ C
The function e iC+iBz has no zero nor pole so, from (3.23), for all z ∈ H:

(µ F (z) -µ G (z)) -(µ F (z + 2iα) -µ G (z + 2iα)) = 0 So if µ F (z) = µ G (z)
for some z, we may by symmetry assume that µ F (z) > µ G (z) and, in this case, for all n ∈ N * :

µ F (z + 2niα) -µ G (z + 2niα) = ... = µ F (z + 2iα) -µ G (z + 2iα) = µ F (z) -µ G (z) > 0
In particular, z + 2niα is a zero of F for all n ∈ N * . But this is impossible because, if it is the case, Im(z+2niα) 1+|z+2niα| 2 ∼ 1 2nα and:

k Im z k 1 + |z k | 2 = +∞
where the (z k ) are the zeros of F over H. It is in contradiction with (3.18).

So for all z ∈ H, µ F (z) = µ G (z). This implies that B F = B G and BF = BG . So, for all z ∈ H:

F (z + iα) = e ic F +i βF z BF (z) DF (z) = e ic F +i βF z BG (z) DG (z) = e iγ+iδz G(z + iα)
with γ = cF -cG and δ = βF -βG

The functions F and G are meromorphic over H so the last equality actually holds over all {z ∈ C s.t. Im z > -α}.

| lim y→0 + F (x + iy)| = | lim y→0 + e iγ+iδ(x+iy-iα) G(x + iy)| = e δα | lim y→0 + G(x + iy)|
Consequently, because δ is real and α = 0, δ = 0. So:

F (z) = e iγ G(z) ∀z ∈ H

Weak stability of the reconstruction

In the previous section, we proved that the operator U : f → {|f ψ j |} was injective, up to a global phase, for Cauchy wavelets. So we can theoretically reconstruct any function f from U (f ). However, if we want the reconstruction to be possible in practice, we also need it to be stable to a small amount of noise:

(U (f 1 ) ≈ U (f 2 )) ⇒ (f 1 ≈ f 2 )
In this section, we show that it is, in some sense, the case: U -1 is continuous. Contrarily to the ones of the previous section, this result is not specific to Cauchy wavelets: it holds for all reasonable wavelets, as soon as U is injective.

Definitions

As in the previous section, we consider only functions without negative frequencies:

L 2 + (R) = {f ∈ L 2 (R) s.t. f (ω) = 0 for a.e. ω < 0}
As the reconstruction is always up to a global phase, we need to define the quotient

L 2 + (R)/S 1 : f = g in L 2 + (R)/S 1 ⇔ f = e iφ g for some φ ∈ R
The set L 2 + (R)/S 1 is equipped with a natural metric:

D 2 (f, g) = inf φ∈R ||f -e iφ g|| 2 Remark that D 2 (f, 0) = ||f || 2 .
We also define:

l 2 (Z, L 2 (R)) = (h j ) j∈Z ∈ L 2 (R) Z s.t. j ||h j || 2 2 < +∞ (h j ) -(h j ) 2 = j∈Z ||h j -h j || 2 2
for any (h j ), (h j ) ∈ l 2 (Z, L 2 (R))

We are interested in the operator U :

U : L 2 + (R)/S 1 → l 2 (Z, L 2 (R)) f → (|f ψ j |) j∈Z (3.24)
We require two conditions over the wavelets. They must be analytic: ψj (ω) = 0 for a.e. ω < 0, j ∈ Z (3.25)

and satisfy an approximate Littlewood-Paley inequality:

A ≤ j∈Z | ψj (ω)| 2 ≤ B
for a.e. ω > 0, for some A, B > 0 (3.26) This last inequality and the fact that D 2 (f, 0) = ||f || 2 imply:

∀f ∈ L 2 + (R)/S 1 , √ AD 2 (f, 0) ≤ ||U (f )|| 2 ≤ √ BD 2 (f, 0) (3.27)
In particular, it ensures the continuity of U .

Weak stability theorem

Theorem 3.11. We suppose that, for all j ∈ Z, ψ j ∈ L 1 (R) ∩ L 2 (R) and that (3.25) and (3.26) hold. We also suppose that U is injective. Then:

(i) The image of U , I U = {U (f ) s.t. f ∈ L 2 + (R)/S 1 } is closed in l 2 (Z, L 2 (R)). (ii) The application U -1 : I U → L 2 + (R)/S 1 is continuous.
Proof. What we have to prove is the following:

if (U (f n )) n∈N converges towards a limit v ∈ l 2 (Z, L 2 (R)), then v = U (g) for some g ∈ L 2 + (R)/S 1 and f n → g in L 2 + (R)/S 1 . So let (U (f n )) n∈N be a sequence of elements in I U , which converges in l 2 (Z, L 2 (R)). Let v = (h j ) j∈Z ∈ L 2
Z (R) be the limit. We show that v ∈ I U .

Lemma 3.12. For all j ∈ Z, {f n ψ j } n∈N is relatively compact in L 2 (R) (that is, the closure of this set in L 2 (R) is compact).

The proof of this lemma is given in Paragraph 3.6.2. It uses the Riesz-Fréchet-Kolmogorov theorem, which gives an explicit characterization of the relatively compact subsets of L 2 (R).

For every j ∈ Z, {f n ψ j } n∈N is thus included in a compact subset of L 2 (R). In a compact set, every sequence admits a convergent subsequence: there exists φ : N → N injective such that (f φ(n) ψ j ) n∈N converges in L 2 (R). Actually, we can choose φ such that (f φ(n) ψ j ) n converges for any j (and not only for a single one). We denote by l j the limits. Lemma 3.13 (Proof in Paragraph 3.6.2). There exists g ∈ L 2 + (R) such that l j = g ψ j for every j. Moreover,

f φ(n) → g in L 2 (R). As U is continuous, U (g) = lim n U (f φ(n) ) = v. So v belongs to I U .
The g such that U (g) = v is uniquely defined in L 2 + (R)/S 1 because U is injective (it does not depend on the choice of φ). We must now show that f n → g.

From Lemma 3.13, (f n ) n admits a subsequence (f φ(n) ) which converges to g. By the same reasoning, every subsequence (f ψ(n) ) n of (f n ) n admits a subsequence which converges to g. This implies that (f n ) n globally converges to g. Remark 3.14. The same proof gives a similar result for wavelets on R d , of the form (ψ j,γ ) j∈Z,γ∈Γ , for Γ a finite set of parameters.

The reconstruction is not uniformly continuous

Theorem 3.11 states that the operator U : f → {|f ψ j |} j∈Z has a continuous inverse U -1 , when it is invertible. However, U -1 is not uniformly continuous. Indeed, for any > 0, there exist g 1 , g 2 ∈ L 2 + (R)/S 1 such that:

||U (g 1 ) -U (g 2 )|| < but ||g 1 -g 2 || ≥ 1 (3.28)
In this section, we describe a way to construct such "unstable" pairs (g 1 , g 2 ): we start from any g 1 and modulate each g 1 ψ j by a low-frequency phase. We then (approximately) invert this modified wavelet transform and obtain g 2 .

This construction seems to be "generic" in the sense that it includes all the instabilities that we have been able to observe in practice.

A simple example

To begin with, we give a simple example of instabilities and relate it to known results about the stability in general phase retrieval problems.

In phase retrieval problems with (a finite number of) real measurements, the stability of the reconstruction operator is characterized by the following theorem ( [START_REF] Bandeira | Saving phase: Injectivity and stability for phase retrieval[END_REF], [START_REF] Balan | Invertibility and robustness of phaseless reconstruction[END_REF]).

Theorem 3.15. Let A ∈ R m×n be a measurement matrix. For any S ⊂ {1, ..., m}, we denote by A S the matrix obtained by discarding the rows of A whose indexes are not in S. We call λ 2 S the lower frame bound of A S , that is, the largest real number such that:

||A S x|| 2 2 ≥ λ 2 S ||x|| 2 2 ∀x ∈ R n
Then, for any x, y ∈ R n :

|| |Ax| -|Ay| || 2 ≥ min S λ 2 S + λ 2 S c . min(||x -y|| 2 , ||x + y|| 2 ) Moreover, min S λ 2 S + λ 2 S c is the optimal constant.
This theorem implies that, in the real case, the reconstruction operator has a Lipschitz constant exactly equal to 1/ min S λ 2 S + λ 2 S c . In the complex case, it is only possible to prove that the Lipschitz constant is at least 1/ min

S λ 2 S + λ 2 S c .
Theorem 3.16. Let A ∈ C m×n be a measurement matrix. There exist x, y ∈ C n such that:

|| |Ax| -|Ay| || 2 ≤ min S λ 2 S + λ 2 S c .min |η|=1 (||x -ηy|| 2 )
Consequently, if the set of measurements can be divided in two parts S and S c such that λ 2 S and λ 2 S c are very small, then the reconstruction is not stable. Such a phenomenon occurs in the case of the wavelet transform. We define: In each column, each graph corresponds to a specific frequency; the highest frequency is on top and the lowest one at bottom. For complex functions, only the real part is displayed.

S = {ψ j s.t. j ≥ 0} and S c = {ψ j s.t. j < 0} Let us fix a small > 0. We choose f 1 , f 2 ∈ L 2 (R) such that: f1 (x) = 0 if |x| < 1/ and f2 (x) = 0 if x / ∈ [-; ]
For every ψ j ∈ S, f 1 ψ j ≈ 0 because the characteristic frequency of ψ j is smaller than 1 and f is a very high frequency function. So:

|(f 1 + f 2 ) ψ j | ≈ |f 2 ψ j | = | -f 2 ψ j | ≈ |(f 1 -f 2 ) ψ j |
And similarly, for ψ j ∈ S c , f 2 ψ j ≈ 0 and:

|(f 1 + f 2 ) ψ j | ≈ |f 1 ψ j | ≈ |(f 1 -f 2 ) ψ j |
As a consequence:

{|(f 1 + f 2 ) ψ j |} j∈Z ≈ {|(f 1 -f 2 ) ψ j |} j∈Z Nevertheless, f 1 + f 2 and f 1 -f 2 may not be close in L 2 (R)/S 1 : g 1 = f 1 + f 2 and g 2 = f 1 -f satisfy (3.28).
The figure 3.3 displays an example of this kind.

A wider class of instabilities

We now describe the construction of more general "unstable" pairs (g 1 , g 2 ). Let g 1 ∈ L 2 (R) be any function. We aim at finding g 2 ∈ L 2 (R) such that, for all j ∈ Z:

(g 1 ψ j )e iφ j ≈ g 2 ψ j (3.29) for some real functions φ j .

In other words, we must find phases φ j such that (g 1 ψ j )e iφ j is approximately equal to the wavelet transform of some g 2 ∈ L 2 (R). Any phases φ j (t) which vary slowly both in t and in j satisfy this property.

Indeed, if the φ j (t) vary "slowly enough", we set:

g 2 = j∈Z (g 1 ψ j )e iφ j ψj
where { ψj } j∈Z are the dual wavelets associated to {ψ j }.

Then, for all k ∈ Z, t ∈ R:

g 2 ψ k (t) = j∈Z (g 1 ψ j )e iφ j ψj ψ k (t) = j∈Z R e iφ j (t-u) (g 1 ψ j )(t -u)( ψj ψ k )(u) du (g 1 ψ k (t))e iφ k (t) = e iφ k (t) j∈Z (g 1 ψ j ) ( ψj ψ k )(t) = j∈Z R e iφ k (t) (g 1 ψ j )(t -u)( ψj ψ k )(u) du So: g 2 ψ k (t) -(g 1 ψ k (t))e iφ k (t) = j∈Z R (e iφ j (t-u) -e iφ k (t) )(g 1 ψ j )(t -u)( ψj ψ k )(u) du (3.30)
The function ψj ψ k (u) is negligible if j is not of the same order as k or if u is too far away from 0. It means that, for some C ∈ N, U ∈ R (which may depend on k):

g 2 ψ k (t) -(g 1 ψ k (t))e iφ k (t) ≈ |j-k|≤C [-U ;U ] (e iφ j (t-u) -e iφ k (t) )(g 1 ψ j )(t -u)( ψj ψ k )(u) du If φ j (t -u) does not vary much over [k -C; k + C] × [-U ; U ],
it gives the desired relation:

g 2 ψ k (t) -(g 1 ψ k (t))e iφ k (t) ≈ 0 which is (3.29).
To summarize, we have described a way to construct g 1 , g 2 ∈ L 2 (R) such that |g 1 ψ j | ≈ |g 2 ψ j | for all j. The principle is to multiply the wavelet transform of g 1 by any set of phases {e iφ j (t) } j∈Z whose variations are slow enough in j and t.

How slow the variations must be depends on g 1 . Indeed, at the points (j, t) where g 1 ψ j (t) is small, the phase may vary more rapidly because, then, the presence of g 1 ψ j (t -u) in (3.30) compensates for a bigger (e iφ j (t-u) -e iφ k (t) ).

All instabilities g 1 , g 2 that we were able to observe in practice were of the form we described: each time, the wavelet transforms of g 1 and g 2 were equal up to a phase whose variation was slow in j and t, except at the points where g 1 ψ j was small.

Local stability result

The goal of this section is to give a partial formal justification to the fact that has been non-rigorously discussed in section 3.3.2: when two functions g 1 , g 2 satisfy |g 1 ψ j | ≈ |g 2 ψ j | for all j, then the wavelet transforms {g 1 ψ j (t)} j and {g 2 ψ j (t)} j are equal up to a phase whose variation is slow in t and j, except eventually at the points where |g 1 ψ j (t)| is small.

In the whole section, we consider f (1) , f (2) two non-zero functions. We denote by F (1) , F (2) the holomorphic extensions defined in (3.6). We recall that, for all j ∈ Z:

f ψ j (x) = a pj 2 F (x + ia j ) ∀x ∈ R (3.31)
We define:

N j = sup x∈R,s=1,2 |f (s) ψ j (x)|

Main principle

From |f ψ j |, one can calculate |f ψ j | 2 and thus, from (3.31), |F (x+ia j )| 2 , for all x ∈ R. But this last function coincides with G j (z) = F (z + ia j )F (z + ia j ) on the horizontal line Im z = 0. As G j is holomorphic, it is uniquely determined by its values on one line. Consequently, G j is uniquely determined from |f ψ j |.

Combining the functions G j for different values of j allows to write explicit reconstruction formulas. The stability of these formulas can be studied, to obtain relations of the following form, for K > 0: (2) ψ j+K ) ∀j ∈ Z These relations imply that, for each j, the phases of f (1) ψ j and f (2) ψ j are approximately equal up to multiplication by the phase of

|f (1) ψ k | ≈ |f (2) ψ k | ∀k ∈ Z ⇒ (f (1) ψ j )(f (1) ψ j+K ) ≈ (f (2) ψ j )(f
f (1) ψ j+K f (2) ψ j+K
. If K is not too small, this last phase is low-frequency, compared to the phase of f (1) ψ j and f (2) ψ j .

The results we obtain are local, in the sense that if the approximate equality |f (1) ψ k | ≈ |f (2) ψ k | only holds on a (large enough) interval of R, the equality (f (1) ψ j )(f (1) ψ j+K ) ≈ (f (2) ψ j )(f (2) ψ j+K ) still holds (also on an interval of R).

Our main technical difficulty was to handle properly the fact that the G j 's may have zeros (which is a problem because we need to divide by G j in order to get reconstruction formulas). We know that, when the wavelet transform has a lot of zeros, the reconstruction becomes unstable. On the other hand, if they are only a few isolated zeros, the reconstruction is stable and this must appear in our theorems.

They are several ways to write reconstruction formulas, which give different stability results. In the dyadic case (a = 2), there is a relatively simple method. We present it first. Then we handle the case where a < 2. We do not consider the case where a > 2. Indeed, it has less practical interest for us. Moreover, when the value of a increases, the reconstruction becomes much less stable.

Case a = 2

In the dyadic case, we only assume that two consecutive moduli are approximately known, on an interval of R: |f ψ j | and |f ψ j+1 |. We also assume that, on this interval, the moduli are never too close to 0. Then we show these moduli determine in a stable way:

f ψ j+2 f ψ j+1 Theorem 3.17. Let , c, λ ∈]0; 1[, M > 0 be fixed, with c ≥ .
We assume that, for all x ∈ [-M 2 j ; M 2 j ]: and:

|f (1) ψ j (x)| 2 -|f (2) ψ j (x)| 2 ≤ N 2 j |f (1) ψ j+1 (x)| 2 -|f (2) ψ j+1 (x)| 2 ≤ N 2
|f (1) ψ j (x)| 2 , |f (2) ψ j (x)| 2 ≥ cN 2 j |f (1) ψ j+1 (x)| 2 , |f (2) ψ j+1 (x)| 2 ≥ cN 2 j+1
Then, for all x ∈ [-λ 2 M 2 j ; λ 2 M 2 j ]:

f (1) ψ j+2 f (1) ψ j+1 (x) - f (2) ψ j+2 f (2) ψ j+1 (x) ≤ A c N j-1 N j+1 4/3 (1/3-α M )(4/5-α M )
if 1/3 -α M > 0 and 4/5 -α M > 0, where:

• A is a constant which depends only on p.

• α M , α M → 0 exponentially when M → +∞.

Principle of the proof. Here, we only give a broad outline of the proof. A rigorous one is given in Paragraph 3.6.3, with all the necessary technical details.

As explained in the paragraph 3.4.1, |f (1) ψ j+1 | uniquely determines the values of z → F (1) (z + i2 j+1 )F (1) (z + i2 j+1 ) on the line Im z = 0. Thus, it uniquely determines all the values (because the function is holomorphic) and in particular (for z = x + i2 j ):

F (1) (x + i3.2 j )F (1) (x + i2 j ) ∀x ∈ R
Moreover, this determination is a stable operation:

|f (1) ψ j+1 (x)| 2 ≈ |f (2) ψ j+1 (x)| 2 ∀x ∈ R ⇒ F (1) (x + i3.2 j )F (1) (x + i2 j ) ≈ F (2) (x + i3.2 j )F (2) (x + i2 j ) ∀x ∈ R
If we divide this last expression by |F (1) (x + i2 j )| 2 ≈ |F (2) (x + i2 j )| 2 (whose values we know from |f ψ j | 2 ):

F (1) (x + i3.2 j ) F (1) (x + i2 j ) ≈ F (2) (x + i3.2 j ) F (2) (x + i2 j ) for x ∈ R
As previously, using the holomorphy of F allows to replace, in the last expression, the real number x by x + i2 j :

F (1) (x + i2 j+2 ) F (1) (x + i2 j+1 ) ≈ F (2) (x + i2 j+2 ) F (2) (x + i2 j+1 ) for x ∈ R
By (3.31), this is the same as:

f (1) ψ j+2 f (1) ψ j+1 ≈ f (2) ψ j+2 f (2) ψ j+1
From this theorem, if f (s) ψ j+2 has no small values either on [-λ 2 M 2 j ; λ 2 M 2 j ], then:

phase(f (1) ψ j+1 ) -phase(f (2) ψ j+1 ) ≈ phase(f (1) ψ j+2 ) -phase(f (2) ψ j+2 )
If more than two consecutive components of the wavelet transform have almost the same modulus (and all these components do not come close to 0), one can iterate this approximate equality. It gives:

phase(f (1) ψ j+1 ) -phase(f (2) ψ j+1 ) ≈ phase(f (1) ψ j+K ) -phase(f (2) ψ j+K )

This holds for any K ∈ N * but with an approximation error that becomes larger and larger as K increases.

When K is large enough, this means that f (1) ψ j+1 and f (2) ψ j+1 are equal up to a low-frequency phase.

Case a < 2

For this section, we fix:

• j ∈ Z: the frequency of the component whose phase we want to estimate • K ∈ N * even: the number of components of the wavelet transform whose modulus are approximately equal

• , κ ∈]0; 1[: they will control the difference between |f (1) ψ j | and |f (2) ψ j |, as well as the minimal value of those functions.

• M > 0: we will assume that the approximate equality between the modulus holds on [-M a j+K ; M a j+K ].

• k ∈ N * such that a -k < 2 -a: this number will control the stability with which one can derive information about f ψ l-1 from |f ψ l |. Typically, for a ≤ 1.5, we may take k = 3.

We define:

• J ∈ [j + K -1; j + K] such that a J = 2
a+1 a j+K + a-1 a+1 a j : we will prove that f (1) ψ j and f (2) ψ j are equal up to a phase which is concentrated around a J in frequencies (that is, a much lower-frequency phase than the phase of f ψ j ).

• c = 1 -a-1 1-a -k ∈]0; 1[ and d M = c -4 e -πM/(K+2)
1-e -πM/(K+2) , which converges exponentially to c when M K goes to ∞.

Theorem 3.18. We assume that κ ≥2(1-c) . We assume that, for x ∈ [-M a j+K ; M a j+K ] and l = j + 1, ..., j + K:

|f (1) ψ l (x)| 2 -|f (2) ψ l (x)| 2 ≤ N 2 l (3.32) |f (1) ψ l (x)| 2 , |f (2) ψ l (x)| 2 ≥ κN 2 l (3.33)
Then, for any x ∈ -M a j+K

2

; M a j+K 2 , as soon as d M < 1:

1 N J N j f (1) ψ J (x) f (1) ψ j (x) -f (2) ψ J (x) f (2) ψ j (x) ≤ C K κ K/4 d M (3.34)
where

C K = 6 1- √ κ K/2-1 s=0 a p(k-1) N ns-1-k N ns-2
As in the dyadic case a = 2, this theorem shows that, if two functions f (1) and f (2) have their wavelet transforms almost equal in moduli, then, for each j, f (1) ψ j ≈ f (2) ψ j up to multiplication by a low-frequency function.

In contrast to the dyadic case, we are not able to show directly that:

f (1) ψ j f (2) ψ j ≈ f (1) ψ j+1 f (2) ψ j+1
Because of that, the inequality we get is less good than in the dyadic case: the bound in (3.34) is exponential in K instead of being proportional to K. With a slightly different method, we could have obtained a better bound, proportional to K. This better bound would have been valid for any a > 1, but under the condition that f ψ l does not come close to 0 for some explicit non-integer values of l, which would have been rather unsatisfactory because, in practice, these values of l do not seem to play a particular role.

Principle of the proof. The full proof may be found in Paragraph 3.6.4. Its principle is to show, by induction over s = 0, ..., K/2, that:

(f (1) ψ Js )(f (1) ψ j+K-2s ) ≈ (f (2) ψ Js )(f (2) ψ j+K-2s ) (3.35)
where J s is an explicit number in the interval [j + K -1; j + K].

For s = 0, we set J s = j + K and (3.35) just says:

f (1) ψ j+K 2 ≈ f (2) ψ j+K
Then, to go from s to s + 1, we use the fact that:

(f (1) ψ j+K-2s )(f (1) ψ l ) ≈ (f (2) ψ j+K-2s )(f (2) ψ l ) (3.36)
if we choose l such that a l = 2a j+K-2s-1 -a j+K-2s : we can check that, up to multiplication by a constant, (f (r) ψ j+K-2s )(f (r) ψ l ) is the evaluation on the line a j+K-2s -a j+K-2s-1 of the holomorphic extension of |f (r) ψ j+K-2s-1 | 2 . The holomorphic extension is a stable transformation (in a sense that has to be made precise). As |f (1) ψ j+K-2s-1 | 2 ≈ |f (2) ψ j+K-2s-1 | 2 , this implies (3.36).

Multiplying (3.35) and ( 3.36) and dividing by |f (1) ψ j+K-2s | 2 ≈ |f (2) ψ j+K-2s | 2 yields:

(f (1) ψ Js )(f (1) ψ l ) ≈ (f (2) ψ Js )(f (2) ψ l ) (3.37) If J s+1 is suitably chosen, (f (r) ψ J s+1 )(f (r) ψ j+K-2(s+1)
) may be seen as the restriction to a line of the holomorphic extension of (f (r) ψ Js )(f (r) ψ l ). Because, again, taking the holomorphic extension is relatively stable, the relation (3.37) implies the recurrence hypothesis (3.35) at order s + 1. For s = K/2, the recurrence hypothesis is equivalent to the stated result.

Numerical experiments

In the previous section, we proved a form of stability for the phase retrieval problem associated to the Cauchy wavelet transform. The proof implicitly relied on the existence of an explicit reconstruction algorithm. In this section, we describe a practical implementation of this algorithm and its performances.

The main goal of our numerical experiments is to investigate the issue of stability. Theorems 3.17 and 3.18 prove that the reconstruction is, in some sense, stable, at least when the wavelet transform does not have small values. Are these results confirmed by the implementation? To what extent does the presence of small values make the reconstruction unstable?

As we will see, our algorithm can fail when large parts of the wavelet transform are close to zero. In all other cases, it seems to succeed and to be stable to noise, even when the amount of noise over the wavelet transform is relatively high (∼ 10%). The presence of a small number of zeroes in the wavelet transform is not a problem.

In practical applications, the wavelet transforms of the signals of interest (mostly audio signals) always have a lot of small values. The algorithm that we present is thus mostly a theoretical tool. Without modifications, it is not intended for real applications. Nevertheless, the results it gives for audio signals are better than expected so, with some more work, it could be suited to practical applications in audio processing. This will be the subject of future work.

The code is available at http://www.di.ens.fr/ ~waldspurger/cauchy_phase_retrieval. html, along with examples of reconstruction for audio signals. It only handles the dyadic case a = 2 but could easily be extended to other values of a.

Description of the algorithm

In practice, we must restrict our wavelet transform to a finite number of components. So we only consider the |f ψ j | for j ∈ {J min , ..., J max }. To compensate for the loss of the |f ψ j | with j > J max , we give to our algorithm an additional information about the low-frequency, under the form of f φ Jmax , where φJmax is negligible outside an neighborhood of 0 of size ∼ a -Jmax .

The algorithm takes as input the functions |f ψ J min |, |f ψ J min +1 |, ..., |f ψ Jmax |, f φ Jmax , for some unknown f , and tries to reconstruct f . The input functions may be contaminated by some noise. To simplify the implementation, we have assumed that the probability distribution of the noise was known.

For any real numbers j, k 1 , k 2 such that j ∈ Z and 2.a j = a k 1 + a k 2 , it comes from the reasoning of the previous section that |f ψ j | uniquely determines (f ψ k 1 ).(f ψ k 2 ). More precisely, we have, for all ω ∈ R:

(f ψ k 1 ).(f ψ k 2 )(ω) = |f ψ j | 2 (ω)e (a k 2 -a j )ω a k 1 +k 2 a 2j (3.38)
The algorithm begins by fixing real numbers k J min -1 , k J min , ..., k Jmax such that:

k J min -1 < J min < k J min < J min + 1 < ... < J max < k Jmax (3.39) ∀j, 2.a j = a k j-1 + a k j
Then, for all j, it applies (3.38) to determine g j def = (f ψ k j-1 ).(f ψ k j ). Because of the exponential function present in (3.38), the g j may take arbitrarily high values in the frequency band {(a k 2 -a j )ω 1}. To avoid this, we truncate the high frequencies of g j . The function f ψ k Jmax may be approximately determined from f φ Jmax . From this function and the g j , the algorithm estimates all the f ψ k j . As this estimation involves divisions by functions which may be close to zero at some points, it is usually not very accurate. In particular, the estimated set {f ψ k j } j do not generally satisfy the constraint that it must belong to the range of the function

f ∈ L 2 (R) → {f ψ k j } J min -1≤j≤Jmax .
Thus, in a second step, the algorithm refines the estimation. To do this, it attempts to minimize an error function which takes into account both the fact that (f ψ k j-1 ).(f ψ k j ) is known for every j and the fact that {f ψ k j-1 } J min -1≤j≤Jmax must belong to the range of f ∈ L 2 (R) → {f ψ k j } J min -1≤j≤Jmax . The minimization is performed by gradient descent, using the previously found estimations as initialization.

Finally, we deduce f from the f ψ k j-1 and refine this estimation one more time by a few steps of the classical Gerchberg-Saxton algorithm ( [START_REF] Gerchberg | A practical algorithm for the determination of phase from image and diffraction plane pictures[END_REF]). This final refinement step is useful, because the Gerchberg-Saxton algorithm converges much faster than the gradient descent. According to our tests, the performances of the algorithm would be approximately the same with more gradient descent iterations and no final refinement. However, the execution time would be much longer.

The principle of the algorithm is summarized by the pseudo-code 4.

Algorithm 4 Reconstruction algorithm

Input: {|f ψ j |} J min ≤j≤Jmax and f φ Jmax 1: Choose k J min -1 , ..., k Jmax as in (3.39).

2: for all j do 3:

Determine g j = (f ψ k j-1 ).(f ψ k j ) from |f ψ j | 2 . 4: end for 5: Determine f ψ k Jmax from f φ Jmax . 6: for all j do 7:

Estimate h j ≈ f ψ k j . 8: end for 9: Refine the estimation with a gradient descent. 10: Deduce f from {f ψ k j } J min -1≤j≤Jmax . 11: Refine the estimation of f with the Gerchberg-Saxton algorithm. Output: f

Input signals

We study the performances of this algorithm on three classes of input signals with finite size n. The figure 3.4 shows an example for each of these three classes.

The first class contains realizations of Gaussian processes with renormalized frequencies. More precisely, the signals f of this class satisfy:

f [n] = X n √ n + 2
where the X n are independent realizations of a Gaussian random variable X ∼ N (0, 1). The normalization 1 √ n+2 ensures that all dyadic frequency bands contain approximately the same amount of energy. The second class consists in sums of a few sinusoids. The amplitudes, phases and frequencies of the sinusoids are randomly chosen. In each dyadic frequency band, there is approximately the same mean number of sinusoids (slightly smaller than 1).

The signals of the third class are random lines extracted from real images. They usually are structured signals, with smooth regular parts and large discontinuities at a small number of points.

To study the influence of the size of the signals on the reconstruction, we perform tests for signals of size N = 128, N = 1024 and N = 8192. For each N , we used log 2 (N ) -1 Cauchy wavelets of order p = 3. Our low-pass filter is a Gaussian function of the form φ[k] = exp(-αk 2 /2), with α independent of N .

Noise

The inputs that are provided to the algorithm are not exactly {|f ψ j |}, f φ Jmax but {|f ψ j | + n ψ,j }, f φ Jmax + n φ . The n ψ,j and the n φ represent an additive noise. In all our experiments, this noise is white and Gaussian.

We measure the amplitude of the noise in relative l 2 -norm:

relative noise = ||n φ || 2 2 + j ||n ψ,j || 2 2 ||f φ Jmax || 2 2 + j ||f ψ j || 2 2

Results

The results are displayed on the figure 3.5.

The x-axis displays the relative error induced by the noise over the input and the y-axis represents the reconstruction error, both over the reconstructed function and over the modulus of the wavelet transform of the reconstructed function.

For an input signal f and output f rec , we define the relative error between f and f rec by:

function error = ||f -f rec || 2 ||f || 2
and the relative error over the modulus of the wavelet transform by:

modulus error = ||f φ Jmax -f rec φ Jmax || 2 2 + j || |f ψ j | -|f rec ψ j | || 2 2 ||f φ Jmax || 2 2 + j ||f ψ j || 2 2
The modulus error describes the capacity of the algorithm to reconstruct a signal whose wavelet transform is close, in modulus, to the one which has been provided as input. The function error, on the other hand, quantifies the intrinsic stability of the phase retrieval problem. If the modulus error is small but the function error is large, it means that there are several functions whose wavelet transforms are almost equal in moduli and the reconstruction problem is ill-posed.

An ideal reconstruction algorithm would yield a small modulus error (that is, proportional to the noise over the input). Nevertheless, the function error could be large or small, depending on the well-posedness of the phase retrieval problem.

We expect that our algorithm may fail when the input modulus contain very small values (because the algorithm performs divisions, which become very unstable in presence of zeroes).

For almost each of the signals that we consider, there exist x's such that f ψ k j (x) ≈ 0 but the number of such points vary greatly, depending on which class the signal belongs. As an example, the wavelet transforms of the three signals of the figure 3.4 are displayed in 3.6.

For Gaussian signals, there are generally not many points at which the wavelet transform vanishes. The positions of these points do not seem to be correlated in either space or frequency.

For piecewise regular signals, there are more of this points but they are usually distributed in such a way that if f ψ j (x) ≈ 0, then f ψ k (x) ≈ 0 for all wavelets ψ k of higher frequencies than ψ j . This distribution makes the reconstruction easier.

When the signals are sums of sinusoids, it often happens that some components of the wavelet transform are totally negligible: for some j, f ψ j (x) ≈ 0 for any x. The negligible frequencies may be either high, low or intermediate.

From the results shown in 3.5, it is clear that the number of zeros influences the reconstruction, but also that isolated zeroes do not prevent reconstruction. The algorithm performs well on

Relative noise

Relative error

Gaussian signals (Gerchberg-Saxton) moduli ( 128) function ( 128) moduli ( 1024) function ( 1024) moduli ( 8192 128) function ( 128) moduli ( 1024) function ( 1024) moduli ( 8192 Gaussian or piecewise regular signals. The distance in modulus between the wavelet transform of the reconstructed signal and of the original one is proportional to the amount of noise (and generally significantly smaller). This holds up to large levels of noise (10%). By comparison, the classical Gerchberg-Saxton algorithm is much less efficient.

However, the algorithm often fails when the input signal is a sum of sinusoids. Not surprisingly, the most difficult signals in this class are the ones for which the sinusoids are not equally distributed among frequency bands and the wavelet transform has a lot of zeroes. The relative error over the modulus of the wavelet transform is then often of several percent, even when the relative error induced by the noise is of the order of 0.1%.

In the section 3.3, we explained why, for any function f , it is generally possible to construct g such that f and g are not close but their wavelet transform have almost the same modulus. This construction holds provided that the time and frequency support of f is large enough.

Increasing the time and frequency support of f amounts here to increase the size N of the signals. Thus, we expect the function error to increase with N . It is indeed the case but this effect is very weakly perceptible on Gaussian signals. It is stronger on piecewise regular functions, probably because the wavelet transforms of these signals have more zeroes; their reconstruction is thus less stable.

In the case of the sums of sinusoids, because of the failure of the algorithm, we can not draw firm conclusions regarding the stability of the reconstruction. We nevertheless suspect that this class of signals is the least stable of all and that these instabilities are the cause of the incorrect behavior of our algorithm.

3.6 Technical lemmas 3.6.1 Lemmas of the proof of Theorem 3.1

Proof of Lemma 3.9. We recall equation (3.22):

B F (z + iα) BG (z) B G (z + iα) BF (z) = e iC+iBz S G (z + iα) S F (z + iα) (3.22)
We want to show that the left part of this equality admits a meromorphic extension to C. We also want this meromorphic extension to have the same poles (with multiplicity) than it would if all four functions B F , B G , BF and BG were meromorphically defined over all C. We first remark that BF and BG admit meromorphic extensions to C. Indeed, if the (z k ) k are the zeros of F (. + iα) in H, this set has no accumulation point in H: if z ∞ was an accumulation point, z ∞ + iα ∈ H would be an accumulation point of the zeros of F and, as F is holomorphic, it would be the null function. From the classical properties of Blaschke products, BF converge over C and so does BG .

On the contrary, B F and B G may not admit meromorphic extensions over C. But their quotient B F /B G does.

We define:

B F (z) = z -i z + i m F k |z F k -i| z F k -i |z F k + i| z F k + i z -z F k z -z F k
where the (z F k )'s are the zeros of F , each z F k being counted, not with multiplicity µ F (z F k ), but with multiplicity max(0, µ F (z F k ) -µ G (z F k )) (and m F is still the multiplicity of i as a zero of F ). Similarly:

B G (z) = z -i z + i m G k |z G k -i| z G k -i |z G k + i| z G k + i z -z G k z -z G k where the (z G k )'s are the zeros of G counted with multiplicity max(0, µ G (z G k ) -µ F (z G k )). We define: B F,G (z) = k |z F,G k -i| z F,G k -i |z F,G k + i| z F,G k + i z -z F,G k z -z F,G k where the z F,G k are the zeros of F or G, counted with multiplicity min(µ F (z F,G k ), µ G (z F,G k ))
. The function B F,G corresponds to the "common part" of B F and B G , which we may factorize in the quotient B F /B G .

The products B F , B G , B F,G converge over H and, for all z ∈ H:

B F (z) = B F (z)B F,G (z) B G (z) = B G (z)B F,G (z)
So for all z ∈ H:

B F (z + iα) BG (z) B G (z + iα) BF (z) = B F (z + iα) BG (z) B G (z + iα) BF (z)
If we show that B F and B G converge over C, we can take B w (z) =

B F (z+iα) BG (z)
B G (z+iα) BF (z) . It will be meromorphic over C.

To prove this, we first establish a relation between the zeros of F and G.

Let z be such that 0 < Im z ≤ α. The zeros of B F are the zeros of F in H, counted with multiplicity. Thus, z -iα is a zero of B F (.+iα) with multiplicity µ F (z). It is a zero of B G (.+iα) with multiplicity µ G (z).

Because Im (z -iα) ≤ 0, it is not a zero of BF (resp. BG ) but may be a pole. As a pole, its multiplicity is the multiplicity of z -iα = z + iα as a zero of F (. + iα) (resp. G(.

+ iα)): it is µ F (z + 2iα) (resp. µ G (z + 2iα)).
The right part of (3.22), e iC+iBz S G (z+iα) S F (z+iα) has no zero neither pole over {z ∈ C s.t. Im z > -α} (from the definition of S G and S F given in (3.20)). So neither does the left part. In particular, z -iα is not a zero and is not a pole:

µ F (z) -µ G (z) -µ G (z + 2iα) + µ F (z + 2iα) = 0 (3.40)
We now explain why B F converges over C. The same result will hold for B G . From the properties of Blaschke products, B F converges over C if (z F k ) has no accumulation point in R. By contradiction, we assume that some subsequence of (z F k ), denoted by (z

F φ(k) ), converges to λ ∈ R. Because the z F k 's appear in B F with multiplicity max(0, µ F (z F k ) -µ G (z F k )), we must have: µ F (z F φ(k) ) -µ G (z F φ(k) ) > 0 ∀k ∈ N We can assume that, for all k, 0 < Im z F φ(k) ≤ α. From (3.40): µ G (z F φ(k) + 2iα) -µ F (z F φ(k) + 2iα) = µ F (z F φ(k) ) -µ G (z F φ(k) ) > 0 Consequently, z F φ(k) + 2iα is a zero of G for all k. As z F φ(k) → λ ∈ R, λ + 2iα ∈
H is an accumulation point of the zeros of G. This is impossible because G is holomorphic over H and we have assumed that it was not the null function.

To conclude, we have to prove Equation (3.23). For any z ∈ H, the multiplicity of z -iα as a pole of B F (. + iα) is the multiplicity of z as a zero of B F , that is max(0, µ F (z) -µ G (z)). Its multiplicity as a pole of B G (. + iα) is max(0, µ G (z) -µ F (z)). As a pole of BF (resp. BG ), it is µ F (z + 2iα) (resp. µ G (z + 2iα)).

The multiplicity of z -iα as a pole of B w is then, as required:

max(0, µ F (z) -µ G (z)) -max(0, µ G (z) -µ F (z)) -µ F (z + 2iα) + µ G (z + 2iα) = (µ F (z) -µ G (z)) -(µ F (z + 2iα) -µ G (z + 2iα))
Proof of Lemma 3.10. We call dE F and dE G the singular measures appearing in the definitions of S F and S G (see (3.20)).

From equation (3.22) and Lemma 3.9, for any z ∈ H:

exp i π R 1 + tz t -z (dE G -dE F )(t) = S G (z) S F (z) = B w (z -iα)e -iC-iB(z-iα)
The function z → B w (z -iα)e -iC-iB(z-iα) is meromorphic over C. From the following lemma, dE G -dE F must then be the null measure, so S G = S F over H. Lemma 3.19. Let dE be a real bounded measure, singular with respect to Lebesgue measure. We define:

S(z) = exp i π R 1 + tz t -z dE(t) ∀z ∈ H
If S admits a meromorphic extension in the neighborhood of each point of R, then dE = 0.

Proof. Let s(z) = -log |S(z)| for all z ∈ H. This is well-defined and:

s(x + iy) = 1 π R y (t -x) 2 + y 2 (1 + t 2 )dE(t) ∀x, y ∈ R s.t. y > 0
This is the Poisson integral of (1 + t 2 )dE(t). So, as dE is bounded, (1 + t 2 )dE(t) is the limit, in the sense of distributions, of s(t + iy)dt when y → 0 + . The principle of the proof will then be to show that s(. + iy) also converges to

-log |S |R |, where S |R is the extension of S to R, so dE = -log |S(t)|dt 1+t 2
. The singularity of dE will imply log |S |R | = 0 and dE = 0. We still denote by S(t) the meromorphic extension of S to a neighborhood of H. Let {r k } be the zeros or poles of S.

When y → 0 + , s(. + iy) tends to -log |S| almost everywhere. On every compact of R -{r k }, the convergence is uniform, and thus in L 1 . Let r k be any zero or pole and > 0 be such that S admits a meromorphic extension over a neighborhood of [r k -; r k + ] × [-; ] and r j / ∈ [r k -; r k + ] for all j = k. There exist h : [r k -; r k + ] × [-; ] → C holomorphic and m ∈ Z such that:

S(z) = (z -r k ) m h(z) ∀z ∈ [r k -; r k + ] × [-; ] and h(r k ) = 0
For all y ∈]0; [:

r k + r k - |s(t + iy) + log |S(t)||dt = r k + r k - m log |t -r k + iy| + log |h(t + iy)| -m log |t -r k | -log |h(t)| dt ≤ m r k + r k - log |t -r k + iy| -log |t -r k | dt + r k + r k - log |h(t + iy)| -log |h(t)| dt (3.41) As log |h| is continuous, log |h(. + iy)| converges uniformly to log |h |R | over [r k -; r k + ]: r k + r k - log |h(t + iy)| -log |h(t)| dt → 0 when y → 0 + As log |. -r k + iy| converges to log |. -r k | in L 1 ([r k -; r k + ]): r k + r k - log |t -r k + iy| -log |t -r k | dt → 0 So, by (3.41), s(. + iy) converges in L 1 to t ∈ R → -log |S(t)|, over [r k -; r k + ]. As the sequence (r k ) has no accumulation point in R, s(. + iy) → -log |S R | (in L 1 ) over each compact set of R. For every f ∈ C 0 c (R): R f (t)(1 + t 2 )dE(t) = lim y→0 + R s(t + iy)f (t)dt = - R log |S(t)|f (t)dt We deduce that dE(t) = -log |S(t)|dt 1+t 2
. As dE is singular with respect to Lebesgue measure, we must have log |S(t)| = 0 for all t ∈ R and dE = 0.

Lemmas of the proof of Theorem 3.11

Proof of Lemma 3.12. We first recall the Riesz-Fréchet-Kolmogorov theorem.

Theorem (Riesz-Fréchet-Kolomogorov). Let p ∈ [1; +∞[. Let F be a subset of L p (R). The set F is relatively compact if and only if:

(i) F is bounded.

(ii) For every > 0, there exists some compact K ⊂ R such that:

sup f ∈F ||f || L p (R-K) ≤
(iii) For every > 0, there exists δ > 0 such that:

sup f ∈F ||f (. + h) -f || p ≤ ∀h ∈ [-δ; δ]
We want to apply this theorem to p = 2 and

F = {f n ψ j } n∈N . First of all, F is bounded: actually, from (3.27), (f n ) n∈N itself is bounded (because (U (f n )) n converges and thus is bounded). It implies that {f n ψ j } n is bounded because ||f n ψ j || 2 ≤ ||f n || 2 ||ψ j || 1 (by Young's inequality).
Let us now prove (ii). Let any > 0 be fixed. The sequence

(|f n ψ j |) n converges in L 2 (R) (to h j , because U (f n ) → (h j ) j∈Z in l 2 (Z, L 2 (R))). So {|f n ψ j |} n is relatively compact in L 2 (R)
. By the Riesz-Fréchet-Kolmogorov theorem, there exists K ⊂ R a compact set such that:

sup n∈N || |f n ψ j | || L 2 (R-K) ≤ But, for all n, || |f n ψ j | || L 2 (R-K) = ||f n ψ j || L 2 (R-K) so (ii) holds: sup n∈N ||f n ψ j || L 2 (R-K) ≤
We finally check (iii). Let > 0 be fixed. For any h ∈ R:

||(f n ψ j )(. + h) -(f n ψ j )|| 2 = ||f n (ψ j (. -h) -ψ j )|| 2 ≤ ||f n || 2 ||ψ j (. -h) -ψ j || 1 As sup n ||f n || 2 < +∞ and lim h→0 ||ψ j (. -h) -ψ j || 1 = 0 (this property holds for any L 1 function),
we have, for δ > 0 small enough:

sup n ||(f n ψ j )(. + h) -(f n ψ j )|| 2 ≤ ∀h ∈ [-δ; δ]
Proof of Lemma 3.13. We want to find g ∈ L 2 + (R) such that lj = ĝ ψj for every j ∈ Z. If ω ≤ 0, we set ĝ(ω) = 0. Then, for each j, we set ĝ = lj / ψj on the support of ψj , which we denote by Supp ψj . This definition is correct in the sense that:

if j 1 = j 2 , lj 1 ψj 1 = lj 2 ψj 2
a.e. on Supp ψj 1 ∩ Supp ψj 2

Indeed, for all n, (f φ(n) ψ j 1 ) ψ j 2 = (f φ(n) ψ j 2 ) ψ j 1 so, by taking the limit in n, l j 1 ψ j 2 = l j 2 ψ j 1 and lj 1 ψj 2 = lj 2 ψj 1 . We can note that, for all j, ĝ ψj = lj . It is true on Supp ψj , by definition. And, on R-Supp ψj , lj = 0 = ĝ ψj because lj is the L 2 -limit of fφ(n) ψj and fφ(n) ψj = 0 on R -Supp ψj .

The ĝ we just defined belongs to L 2 (R). Indeed, by (3.26):

||ĝ|| 2 2 ≤ 1 B R + |ĝ| 2 j | ψj | 2 = 1 B R + j | lj | 2 = 1 B j ||l j || 2 2
As f φ(n) ψ j goes to l j when n goes to ∞ and

U (f φ(n) ) = {|f φ(n) ψ j |} j goes to (h j ) j∈Z ∈ l 2 (Z, L 2 (R)), we must have |l j | = h j for each j. So 1 B j ||l j || 2 2 = 1 B j ||h j || 2 2 = 1 B ||(h j ) j∈Z || 2 2 < +∞ and ĝ belongs to L 2 (R).
As ĝ ∈ L 2 (R), it is the Fourier transform of some g ∈ L 2 (R). For all j ∈ Z, as ĝ ψj = lj , we have g ψ j = l j .

We now show that f φ(n) → g when n → ∞.

For every J, n ∈ N:

|j|>J ||f φ(n) ψ j || 2 2 = |j|>J || U (f φ(n) ) j || 2 2 ≤ |j|>J ||U (f φ(n) ) j -h j || 2 2 + |j|>J ||h j || 2 2 ≤ ||U (f φ(n) ) -(h j )|| 2 + |j|>J ||h j || 2 2 So lim sup n |j|>J ||f φ(n) ψ j || 2 2 ≤ |j|>J ||h j || 2 2 and: lim sup n j∈Z ||f φ(n) ψ j -g ψ j || 2 2 ≤ lim sup n   |j|≤J ||f φ(n) ψ j -g ψ j || 2 2   94 + lim sup n   |j|>J ||f φ(n) ψ j -g ψ j || 2 2   = lim sup n   |j|>J ||f φ(n) ψ j -g ψ j || 2 2   ≤ |j|>J ||h j || 2 2
This last quantity may be as small as desired, for J large enough, so

j∈Z ||f φ(n) ψ j -g ψ j || 2 2 → 0. By (3.26): j∈Z ||f φ(n) ψ j -g ψ j || 2 2 = R fφ(n) -ĝ 2 ( j | ψj | 2 ) ≥ A R fφ(n) -ĝ 2 = A 2π ||f φ(n) -g|| 2 2 so ||f φ(n) -g|| 2 → 0.
3.6.3 Proof of Theorem 3.17

In this section, we prove Theorem 3.17, which gives a stability result for the case of dyadic wavelets.

For all y > 0, we define:

N (y) = sup x∈R,s=1,2 |F (s) (x + iy)|
The following lemma is not necessary to our proof but we will use it to progressively simplify our inequalities. Lemma 3.20. For all y 1 , y 2 ∈ R * + , if y 1 < y 2 :

N (y 1 ) ≥ N (y 2 ) (3.42)
and for all y 3 ∈ [y 1 ; y 2 ]:

N (y 3 ) ≤ N (y 1 ) y 2 -y 3 y 2 -y 1 N (y 2 ) y 3 -y 1 y 2 -y 1 (3.43)
Proof. The second inequality comes directly from Theorem 3.24, applied to functions F (1) and F (2) on the band {z ∈ C s.t. y 1 < Im z < y 2 }.

The first inequality may be derived from (3.43). The function N (y) is bounded when y → +∞.

Keeping y 1 and y 3 fixed in (3.43) and letting y 2 go to +∞ then gives:

N (y 3 ) ≤ N (y 1 )
Remark 3.21. When y → +∞, then N (y) → 0 because, from (3.6) and the Hölder inequality:

|F (s) (x + iy)| = 1 2π R ω p f+ (ω)e iω(x+iy) dω ≤ 1 2π || f+ || 2 ||ω → ω p e iω(x+iy) || 2 ≤ 1 2π || f+ || 2 ||ω → ω p e -ωy || 2
which decreases geometrically to zero when y → +∞.

We can now prove the theorem.

Proof of Theorem 3.17. From the relation (3.31) between F (s) and the f (s) ψ j and from the hypotheses, the following inequalities hold for all x ∈ [-M 2 j ; M 2 j ]:

|F (1) (x + i2 j )| 2 -|F (2) (x + i2 j )| 2 ≤ N (2 j ) 2 |F (1) (x + i2 j+1 )| 2 -|F (2) (x + i2 j+1 )| 2 ≤ N (2 j+1 ) 2 |F (1) (x + i2 j )| 2 , |F (2) (x + i2 j )| 2 ≥ cN (2 j ) 2 |F (1) (x + i2 j+1 )| 2 , |F (2) (x + i2 j+1 )| 2 ≥ cN (2 j+1 ) 2
Let us set, for all z such that -2 j+1 < Im z < 2 j+1 :

G(z) = F (1) (z + i2 j+1 )F (1) (z + i2 j+1 ) -F (2) (z + i2 j+1 )F (2) (z + i2 j+1 )
For all z such that Im z = 0:

|G(z)| = |F (1) (z + i2 j+1 )| 2 -|F (2) (z + i2 j+1 )| 2 ≤ N (2 j+1 ) 2 if |Re z| ≤ M 2 j ≤ N (2 j+1 ) 2 if |Re z| > M 2 j
and for all z such that Im z = 3.2 j-1 :

|G(z)| = |F (1) (Re z + 7.2 j-1 i)F (1) (Re z + 2 j-1 i) -F (2) (Re z + 7.2 j-1 i)F (2) (Re z + 2 j-1 i)| ≤ 2 N (7.2 j-1 )N (2 j-1 )
We apply Lemma 3.25 for a = 0, b = 3.2 j-1 , t = 2/3, A = N (2 j+1 ) 2 , B = 2 N (7.2 j-1 )N (2 j-1 ). It implies that, for all x ∈ [-λM 2 j ; λM 2 j ]:

|G(x + i2 j )| ≤ 2 2/3 1/3-α M N (2 j+1 ) 2/3 N (2 j-1 ) 2/3 N (7.2 j-1 ) 2/3
where

α M = 4 3 exp(-2π 3 (1-λ)M ) 1-exp(-2π 3 (1-λ)M )
.

Replacing G by its definition gives, for all x ∈ [-λM 2 j ; λM 2 j ]:

|F (1) (x + i3.2 j )F (1) (x + i2 j ) -F (2) (x + i3.2 j )F (2) (x + i2 j )| ≤ 2 2/3 1/3-α M N (2 j+1 ) 2/3 N (2 j-1 ) 2/3 N (7.2 j-1 ) 2/3 ≤ 2 1/3-α M N (2 j+1 ) 4/3 N (2 j-1 ) 2/3
We used Equation (3.42) to obtain the last inequality. So, for all x ∈ [-λM 2 j ; λM 2 j ]: 2) (x + i3.2 j )F (2) (x + i2 j )|.|F (2) (x + i2 j )F (2) (x + i2 j )| + |F (2) (x + i3.2 j )F (2) (x + i2 j )||F (2) (x + i2 j )F (2)

F (1) (x + i3.2 j )F (1) (x + i2 j )F (2) (x + i2 j )F (2) (x + i2 j ) -F (2) (x + i3.2 j )F (2) (x + i2 j )F (1) (x + i2 j )F (1) (x + i2 j ) ≤ |F (1) (x + i3.2 j )F (1) (x + i2 j ) -F (
(x + i2 j ) -F (1) (x + i2 j )F (1) (x + i2 j )| ≤ 2 1/3-α M N (2 j+1 ) 4/3 N (2 j-1 ) 2/3 |F (2) (x + i2 j )| 2 + N (2 j ) 2 |F (2) (x + i3.2 j )F (2) (x + i2 j )|
Dividing by |F (1) (x + i2 j )F (2) (x + i2 j )| gives:

|F (1) (x + i3.2 j )F (2) (x + i2 j ) -F (2) (x + i3.2 j )F (1) (x + i2 j )| ≤ 2 1/3-α M N (2 j+1 ) 4/3 N (2 j-1 ) 2/3 |F (2) (x + i2 j )| |F (1) (x + i2 j )| + N (2 j ) 2 |F (2) (x + i3.2 j )| |F (1) (x + i2 j )|
For each x ∈ [-λM 2 j ; λM 2 j ], this relation also holds if we switch the roles of F (1) and F (2) . Thus, we can assume that |F (2)

(x+i2 j )| ≤ |F (1) (x+i2 j )|. Using also the fact that |F (1) (x+i2 j )| ≥ √ cN (2 j
) yields (always for x ∈ [-λM 2 j ; λM 2 j ]):

|F (1) (x + i3.2 j )F (2) (x + i2 j ) -F (2) (x + i3.2 j )F (1) (x + i2 j )| ≤ 2 1/3-α M N (2 j+1 ) 4/3 N (2 j-1 ) 2/3 + √ c N (2 j )N (3.2 j ) = 2N (2 j )N (3.2 j ) N (2 j+1 ) 4/3 N (2 j-1 ) 2/3 N (2 j )N (3.2 j ) 1/3-α M + 2 √ c ≤ 2N (2 j )N (3.2 j ) N (2 j-1 ) N (2 j+1 ) 2/3 1/3-α M + 2 √ c ≤ 3N (2 j )N (3.2 j ) N (2 j-1 ) N (2 j+1 ) 2/3 1/3-α M (3.44)
In the middle, we used Equation (3.43):

N (2 j+1 ) ≤ N (2 j ) 1/2 N (3.2 j ) 1/2 .
For the last inequality, we used the fact that c ≥ so

2 √ c ≤ √ 2 ≤ 1/3-α M 2 ≤ N (2 j-1 ) N (2 j+1 ) 2/3 1/3-α M 2 .
For all z such that Im z > -2 j , we set:

H(z) = F (1) (z + i3.2 j )F (2) (z + i2 j ) -F (2) (z + i3.2 j )F (1) (z + i2 j )
From (3.44):

|H(z)| ≤ 2N (2 j )N (3.2 j ) if Im z = 0 and |Re z| > λM 2 j ≤ 2N (2 j )N (3.2 j ) min 1, 3 2
N (2 j-1 ) N (2 j+1 ) 2/3 1/3-α M if Im z = 0 and |Re z| ≤ λM 2 j ≤ 2N (2 j+3 )N (6.2 j ) if Im z = 5.2 j
We may apply Lemma 3.25 again. For all x ∈ [-λ 2 M 2 j ; λ 2 M 2 j ]:

|H(x + i2 j )| ≤ 2 min 1, 3 2
N (2 j-1 ) N (2 j+1 ) 2/3 1/3-α M 4/5-α M × N (2 j ) 4/5 N (3.2 j ) 4/5 N (2 j+3 ) 1/5 N (6.2 j ) 1/5 ≤ 2 min 1, 3 2
N (2 j-1 ) N (2 j+1 ) 2/3 1/3-α M 4/5-α M N (2 j ) 4/5 N (2 j+1 ) 6/5
where α

M = 2 5 exp(-π 5 λ(1-λ)M )) 1-exp(-π 5 λ(1-λ)M )) .
Replacing H by its definition and dividing by |F (1) (x+i2 j+1 )F (2) (x+i2 j+1 )| (which is greater that cN (2 j+1 ) 2 ) gives:

F (1) (x + i2 j+2 ) F (1) (x + i2 j+1 ) - F (2) (x + i2 j+2 ) F (2) (x + i2 j+1 ) 98 ≤ 2 c min 1, 3 2
N (2 j-1 ) N (2 j+1 ) 2/3 1/3-α M 4/5-α M N (2 j ) N (2 j+1 ) 4/5
As soon as 4/5 -α M > 0 and 1/3 -α M > 0:

F (1) (x + i2 j+2 ) F (1) (x + i2 j+1 ) - F (2) (x + i2 j+2 ) F (2) (x + i2 j+1 ) ≤ 3 c N (2 j-1 ) N (2 j+1 ) 8/15 N (2 j ) N (2 j+1 ) 4/5 (1/3-α M )(4/5-α M ) ≤ 3 c N (2 j-1 ) N (2 j+1 ) 4/3 (1/3-α M )(4/5-α M ) = 3 c N j-1 N j+1 2 2p 4/3 (1/3-α M )(4/5-α M ) So: f (1) ψ j+2 (x) f (1) ψ j+1 (x) - f (2) ψ j+2 (x) f (2) ψ j+1 (x) ≤ 3 c 2 11p 3 N j-1 N j+1 4/3 (1/3-α M )(4/5-α M )
which is the desired result with A = 3.2 11p 3 .

Proof of Theorem 3.18

In this whole section, as in the paragraph 3.4.3, k is assumed to be a fixed integer such that:

a -k < 2 -a
and we define: c = 1 -a -1 1 -a -k Lemma 3.22. Let the following numbers be fixed:

∈]0; 1[ M > 0 µ ∈ [0; M [ j ∈ Z
We assume that, for all x ∈ [-M a j ; M a j ]:

|F (1) (x + ia j )| 2 -|F (2) (x + ia j )| 2 ≤ N (a j ) 2
Then, for all x ∈ [-(M -µ)a j ; (M -µ)a j ]:

F (1) (x + i(2a j -a j+1 ))F (1) (x+ ia j+1 ) -F (2) (x + i(2a j -a j+1 ))F (2) (x + ia j+1 ) ≤ N (a j ) 2c 2N (a j+1 )N (a j-k ) 1-c c-α
where:

α = 2 e -πµ 1 -e -πµ
Proof. We set:

H(z) = F (1) (z + ia j )F (1) (z + ia j ) -F (2) (z + ia j )F (2) (z + ia j ) When y = 0, |H(x + iy)| = |F (1) (x + ia j )| 2 -|F (2) (x + ia j )| 2 . So: |H(x + iy)| ≤ N (a j ) 2 if x ∈ [-M a j ; M a j ] ≤ N (a j ) 2 if x / ∈ [-M a j ; M a j ]
When y = a j -a j-k :

|H(x + iy)| = |F (1) (x + ia j-k )F (1) (x + i(2a j -a j-k )) -F (2) (x + ia j-k )F (2) (x + i(2a j -a j-k ))| ≤ 2N (2a j -a j-k )N (a j-k ) (∀x ∈ R)
We apply Lemma 3.25 to H, restricted to the band {z ∈ C s.t. Im z ∈ [0; a j -a j-k ]}. From this lemma, when y = a j+1 -a j and x ∈ [-µM a j ; µM a j ]:

|H(x + iy)| ≤ f (x+iy) N (a j ) 2c 2N (2a j -a j-k )N (a j-k ) 1-c
where c = 1 -a-1 1-a -k and:

f (x + iy) ≥ c -2 a -1 1 -a -k e -π M a j -|x| a j -a j-k 1 -e -π M a j -|x| a j -a j-k
Because of the definition of k, a-1 1-a -k ≤ 1. Moreover, M a j -|x| a j -a j-k ≥ µ 1-a -k ≥ µ, so:

f (x + iy) ≥ c -2 e -πµ 1 -e -πµ = c -α
Replacing H by its definition yields:

F (1) (x + i(2a j -a j+1 ))F (1) (x + ia j+1 )-F (2) (x + i(2a j -a j+1 ))F (2) (x + ia j+1 ) = |H(x + i(a j+1 -a j ))| ≤ c-α N (a j ) 2c 2N (2a j -a j-k )N (a j-k ) 1-c
To conclude, it suffices to note that, because of the way we chose k, 2a j -a j-k ≥ a j+1 so, from 3.20, N (2a j -a j-k ) ≤ N (a j+1 ).

Theorem 3.23. Let the following numbers be fixed:

, κ ∈]0; 1[ with κ ≥ 2(1-c) M > 0 µ ∈ [0; M [ j ∈ Z K ∈ N
We assume that, for any n ∈ {j + 1, ..., j + K} and x ∈ [-M a j+K ; M a j+K ]:

|F (1) (x + ia n )| 2 -|F (2) (x + ia n )| 2 ≤ N (a n ) 2 (3.45) |F (1) (x + ia n )| 2 , |F (2) (x + ia n )| 2 ≥ κN (a n ) 2 (3.46)
We define recursively:

n 0 = j + K w 0 = a j+K ∀l ∈ N n l+1 = n l -2 w l+1 = w l -(a -1) 2 a n l+1
We define:

D l = l-1 s=0 N (a ns-1-k ) N (a ns-2 ) and c l = c -2 1 + 2 a a 2 -1 a + 2 l-1 s=0 a -2s e -πµ 1 -e -πµ
For any l ≥ 0 such that n l ≥ j and M -(l + 1)µ > 0, we have, provided that c l < 1:

1 N (w l )N (a n l ) F (1) (x + iw l )F (1) (x + ia n l )-F (2) (x + iw l )F (2) (x + ia n l ) ≤ 3D l 2κ -l/2 -κ -(l-1)/2 -1 1 - √ κ c l ∀x ∈ [-(M -(l + 1)µ)a j+K ; (M -(l + 1)µ)a j+K ] (3.47)
Proof. We proceed by induction over l.

For l = 0, (3.47) is a direct consequence of (3.45). Indeed,

w 0 = a n 0 , D 0 = 1, c 0 < 1 so, for x ∈ [-M a j+K ; M a j+K ]: 1 N (a n 0 ) 2 |F (1) (x + ia n 0 )| 2 -|F (2) (x + ia n 0 )| 2 ≤ ≤ 3D 0 c 0
We now suppose that (3.47) holds for l and prove it for l + 1. We proceed in two parts. First, we use the induction hypothesis to bound the function

F (1) (x + iw l )F (1) (x + i(2a n l -1 -a n l )) -F (2) (x + iw l )F (2) (x + i(2a n l -1 -a n l )) .
We then use this bound to obtain the desired result.

First part: by triangular inequality,

F (1) (x + iw l ) F (1) (x + i(2a n l -1 -a n l )) -F (2) (x + iw l )F (2) (x + i(2a n l -1 -a n l )) ≤ F (1) (x + iw l )F (1) (x + ia n l ) -F (2) (x + iw l )F (2) (x + ia n l ) (3.48) × F (1) (x + i(2a n l -1 -a n l )) F (1) (x + ia n l ) + F (2) (x + ia n l ) F (1) (x + ia n l ) - F (1) (x + ia n l ) F (2) (x + ia n l ) (3.49) × F (2) (x + iw l )F (1) (x + i(2a n l -1 -a n l )) + F (1) (x + ia n l )F (1) (x + i(2a n l -1 -a n l )) -F (2) (x + ia n l )F (2) (x + i(2a n l -1 -a n l )) (3.50) × F (2) (x + iw l ) F (2) (x + ia n l )
By the induction hypothesis, for x ∈ [-(M -2lµ)a j+K ; (M -2lµ)a j+K ], (3.48) is bounded by:

F (1) (x + iw l )F (1) (x + ia n l ) -F (2) (x + iw l )F (2) (x + ia n l ) ≤ 3D l 2κ -l/2 -κ -(l-1)/2 -1 1 - √ κ N (w l )N (a n l ) c l
Because of (3.45) and (3.46) (for n = n l ), (3.49) is bounded by:

F (2) (x + ia n l ) F (1) (x + ia n l ) - F (1) (x + ia n l ) F (2) (x + ia n l ) = |F (2) (x + ia n l )| 2 -|F (1) (x + ia n l )| 2 F (1) (x + ia n l )F (2) (x + ia n l ) ≤ κ
Finally, from Lemma 3.22 applied to j = n l -1, (3.50) is bounded by:

F (1) (x + ia n l )F (1) (x + i(2a n l -1 -a n l )) -F (2) (x + ia n l )F (2) (x + i(2a n l -1 -a n l )) ≤ N (a n l -1 ) 2c (2N (a n l )N (a n l -1-k )) 1-c c-α for all x ∈ [-M a j+K + µa j ; M a j+K -µa j ] ⊃ [-(M -(l + 1)µ)a j+K ; (M -(2l + 1)µ)a j+K ].
We insert these bounds into the triangular inequality. We also use the fact that |F (1) (x + ia n l )|, |F (2)

(x+ia n l )| ≥ √ κN (a n l ). We get, for any x ∈ [-(M -(l+1)µ)a j+K ; (M -(l+1)µ)a j+K ]: F (1) (x + iw l ) F (1) (x + i(2a n l -1 -a n l )) -F (2) (x + iw l )F (2) (x + i(2a n l -1 -a n l )) ≤ 1 √ κ 3D l 2κ -l/2 -κ -(l-1)/2 -1 1 - √ κ N (w l )N (2a n l -1 -a n l ) c l + κ N (w l )N (2a n l -1 -a n l ) + 2 1-c √ κ N (w l ) N (a n l ) c N (a n l -1 ) 2c N (a n l -1-k ) 1-c c-α
We must now simplify this inequality. First, 2a

n l -1 -a n l = ca n l -1 + (1 -c)a n l -1-k so, from Lemma 3.20, N (2a n l -1 -a n l ) ≤ N (a n l -1 ) c N (a n l -1-k ) 1-c . So: F (1) (x + iw l ) F (1) (x + i(2a n l -1 -a n l )) -F (2) (x + iw l )F (2) (x + i(2a n l -1 -a n l )) ≤ N (w l )N (a n l -1 ) c N (a n l -1-k ) 1-c × 1 √ κ 3D l 2κ -l/2 -κ -(l-1)/2 -1 1 - √ κ c l + κ + 2 1-c √ κ N (a n l -1 ) c N (a n l ) c c-α Now we note that 1 ≤ N (a n l -1 ) c N (a n l ) c
(from Lemma 3.20 again, because

a n l -1 ≤ a n l ). Because κ ≥ 2(1-c) , we also have κ ≤ c √ κ ≤ c-α √ κ .
And as c -α ≥ c l , c-α ≤ c l . This gives:

F (1) (x + iw l ) F (1) (x + i(2a n l -1 -a n l )) -F (2) (x + iw l )F (2) (x + i(2a n l -1 -a n l )) ≤ c l √ κ N (w l ) N (a n l -1 ) 2c N (a n l -1-k ) 1-c N (a n l ) c 3D l 2κ -l/2 -κ -(l-1)/2 -1 1 - √ κ + 1 + 2 1-c
If we bound 2 1-c by 2 and notice that D l ≥ 1 (because, from 3.20, it is a product of terms bigger that 1), we have:

F (1) (x + iw l ) F (1) (x + i(2a n l -1 -a n l )) -F (2) (x + iw l )F (2) (x + i(2a n l -1 -a n l )) ≤ c l √ κ 3D l N (w l ) N (a n l -1 ) 2c N (a n l -1-k ) 1-c N (a n l ) c 2κ -l/2 -κ -(l-1)/2 -1 1 - √ κ + 1 ≤ 3 c l D l N (w l ) N (a n l -1 ) 2c N (a n l -1-k ) 1-c N (a n l ) c 2κ -(l+1)/2 -κ -l/2 -1 1 - √ κ
Finally, from 3.20, we have N (a n l -1 ) ≤ N (a n l ) 1/2 N (2a n l -1 -a n l ) 1/2 so:

F (1) (x + iw l ) F (1) (x + i(2a n l -1 -a n l )) -F (2) (x + iw l )F (2) (x + i(2a n l -1 -a n l )) ≤ 3 c l D l N (w l )N (2a n l -1 -a n l ) N (a n l -1-k ) N (2a n l -1 -a n l ) 1-c 2κ -(l+1)/2 -κ -l/2 -1 1 - √ κ y l ≥ 2a j+K a + 1 -2a n l -1 + a n l ≥ 2a j+K a + 1 -2a j+K-1 + a j+K = (a -1)(a + 2) (a + 1) a j+K-1
From this we deduce:

f (z) ≥ 1 -4 a 2 -1 a + 2 a -2l-1   e -π (M -(l+1)µ)a j+K -|Re z| a j+K 1 -e -π (M -(l+1)µ)a j+K -|Re z| a j+K   So, when |Re z| ≤ (M -(l + 2)µ)a j+K , f (z) ≥ 1 -4 a 2 -1 a+2 a -2l-1 e -πµ 1-e -πµ . As B ≥ 1 and f (z) ≤ 1, B f (z) ≤ B. Moreover, c l ≤ 1 so c l f (z) ≥ c l -(1-f (z)) if 1-f (z) ≥ 0. Equation (3.51) thus gives: |H(z)| ≤ 2N (w l )N (2a n l -1 -a n l )B c l -4 a 2 -1 a+2 a -2l-1 e -πµ 1-e -πµ = 2N (w l )N (2a n l -1 -a n l )B c l+1 = 3D l N (w l )N (2a n l -1 -a n l ) c N (a n l -1-k ) 1-c 2κ -(l+1)/2 -κ -l/2 -1 1 - √ κ c l+1 Because w l ≥ w l+1 and 2a n l -1 -a n l ≥ a n l -1-k , we have N (w l ) ≤ N (w l+1 ) and N (2a n l -1 -a n l ) ≤ N (a n l -1-k ). Thus: |H(z)| ≤ 3D l N (w l+1 )N (a n l -2 ) N (a n l -1-k ) N (a n l -2 ) 2κ -(l+1)/2 -κ -l/2 -1 1 - √ κ c l+1 = 3D l+1 N (w l+1 )N (a n l -2 ) 2κ -(l+1)/2 -κ -l/2 -1 1 - √ κ c l+1 So, for any x ∈ [-(M -(l + 2)µ)a j+K ; (M -(l + 2)µ)a j+K ]: 1 N (w l+1 )N (a n l+1 ) F (1) (x + iw l+1 )F (1) (x + ia n l+1 ) -F (2) (x + iw l+1 )F (2) (x + ia n l+1 ) = |H(x + i(a -1) 2 a n l -2 )| ≤ 3D l+1 2κ -(l+1)/2 -κ -l/2 -1 1 - √ κ c l+1
This is exactly the induction hypothesis at the order l + 1.

Proof of Theorem 3.18. We will obtain the desired theorem as a corollary of the previous one (3.23).

The conditions (3.45) and (3.46) in the statement of Theorem 3.23 are equivalent to (3.32) and (3.33), required in Theorem 3.18.

Thus, if we fix µ ∈ [0; M [, we have that, for any l ≥ 0 such that n l ≥ j and M -(l + 1)µ > 0, under the condition that c l < 1:

1 N (w l )N (a n l ) F (1) (x + iw l )F (1) (x + ia n l )-F (2) (x + iw l )F (2) (x + ia n l ) ≤ 3D l 2κ -l/2 -κ -(l-1)/2 -1 1 - √ κ c l ∀x ∈ [-(M -(l + 1)µ)a j+K ; (M -(l + 1)µ)a j+K ]
where the constants are defined as in 3.23. We can check that, for any l, w l = a j+K a+1 2 + (a -1)a -2l . We take l = K/2. We then have w l = 2 a+1 a j+K + a-1 a+1 a j = a J and n l = j. For this l, the previous inequality is equivalent to:

1 N J N j f (1) ψ J (x)f (1) ψ j (x) -f (2) ψ J (x) f (2) ψ j (x) ≤ 3D l 2κ -K/4 -κ -(K-2)/4 -1 1 - √ κ c l We observe that c l ≥ lim l→∞ c l = c-2 1 + 2 a a+2 e -πµ 1-e -πµ ≥ c -4 e -πµ
1-e -πµ and 2κ -K/4 -κ -(K-2)/4 -1

1- √ κ ≤ 2κ -K/4 1- √ κ . So, for any x ∈ [-(M -µ(1 + K/2))a j+K ; (M -µ(1 + K/2))a j+K ]: 1 N J N j f (1) ψ J (x)f (1) ψ j (x) -f (2) ψ J (x) f (2) ψ j (x) ≤ 6D l κ -K/4 1 - √ κ c-4 e -πµ
1-e -πµ

From Equation (3.31):

D l = K/2-1 s=0 N (a ns-1-k ) N (a ns-2 ) = K/2-1 s=0 a p(k-1) N ns-1-k N ns-2
For µ = M K+2 , our last inequality is exactly the desired result.

Bounds for holomorphic functions

In the proofs of the section 3.4, we often have to consider holomorphic functions defined on a band of the complex plane. We want to obtain information about their values inside the band from their values on the boundary of the band. This is the purpose of the three theorems contained in this section.

In the whole section, a, b are fixed real numbers such that a < b. We write B a,b = {z ∈ C s.t. a < Im z < b}. We consider a holomorphic function W : B a,b → C which satisfies the following properties:

(i) W is bounded on B a,b .
(ii) W admits a continuous extension over B a,b , which we still denote by W .

The first theorem we need is a well-known fact. We recall its proof because it is very short and relies on the same idea that will also be used in the other proofs. Theorem 3.24. We suppose that, for some A, B > 0:

|W (z)| ≤ A if Im z = a |W (z)| ≤ B if Im z = b
Then, for all t ∈]0; 1[ and all z ∈ C such that Im z = (1 -t)a + tb:

|W (z)| ≤ A 1-t B t
Proof. For every > 0 and z ∈ B a,b : Letting go to 0 implies: Let A, B, > 0 be fixed real numbers, with ≤ 1. We assume that:

L(z) = log(|W (z)|) - (b -Im z) log(A) + (Im z -a) log(B) b -a -log |z + i(1 -a)| is subharmonic on B a
log(|W (z)|) ≤ (b -Im z) log(A) + (Im z -a) log(B) b -a ∀z ∈ B a,b ⇒ |W (z)| ≤ A b-Im z b-a B Im z-
|W (z)| ≤ B if Im z = b |W (z)| ≤ A if Im z = a and Re z / ∈ [-M ; M ] |W (z)| ≤ A if Im z = a and Re z ∈ [-M ; M ] Then, for all z such that a < Im z < b, if t ∈ [0; 1] is such that Im z = (1 -t)a + tb: |W (z)| ≤ f (z) A 1-t B t
where: b-a) -e π(z-ia)/(b-a) e -πM/(b-a) -e π(z-ia)/(b-a)

f (z) = 1 π arg e πM/(
and this function satisfies, when |Re z| ≤ M :

f (z) ≥ (1 -t) -2t e -π M -|Re z| b-a 1-e -π M -|Re z| b-a .
Proof. The function f may be continuously extended to B a,b -{-M + ia; M + ia}. By looking at the figure 3.7, one sees that:

f (x + ia) = 0 for all x ∈ R -[-M ; M ] = 1 for all x ∈] -M ; M [ f (x + ib) = 0 for all x ∈ R We set: f (-M + ia) = f (M + ia) = 1
This definition makes the extension of f upper semi-continuous on B a,b (because f ≤ 1 on all B a,b ).

For any η > 0, the following function is subharmonic on B a,b :

L(z) = log(|W (z)|) -log( )f (z) - (b -Im z) log(A) + (Im z -a) log(B) b -a -η log |z + i(1 -a)|
It is upper semi-continuous on B a,b and tends to -∞ when Re z → ±∞. Thus, this function admits a local maximum over B a,b . This maximum is attained on ∂B a,b , because L is subharmonic.

From the hypotheses, one can check that L(z) ≤ 0 for all z ∈ ∂B a,b . The function L is thus negative on the whole band B a,b . Letting η go to zero gives, for all z ∈ B a,b such that Im z = (1 -t)a + tb:

|W (z)| ≤ f (z) A 1-t B t
We are only left to show that

f (z) ≥ (1 -t) -2t e -π M -|Re z| b-a 1-e -π M -|Re z| b-a when Im z = (1 -t)a + tb.
If we write x = Re (z), we have:

f (z) = 1 π arg -e -iπt 1 -e π(x-M )/(b-a) e πit ) 1 -e -π(M +x)/(b-a) e -πit = (1 -t) + 1 π arg 1 -e π(x-M )/(b-a) e πit ) 1 -e -π(M +x)/(b-a) e -πit
We note that:

arg 1 -e π x-M b-a e iπt ≤ tan 1 -e π x-M b-a e iπt = |sin(πt)| e π x-M b-a 1 -e π x-M b-a cos(πt) ≤ |sin(πt)| e π x-M b-a 1 -e π x-M b-a ≤ πt e π x-M b-a 1 -e π x-M b-a ≤ πt e -π M -|Re z| b-a 1 -e -π M -|Re z| b-a
And the same inequality holds for arg 1 -e -π M +x b-a e -iπt . This implies the result.

The proof of the third result is similar to the proof of the second one. We do not reproduce it.

Lemma 3.26. Let M, A, > 0 be fixed real numbers, with ≤ 1. We assume that:

|W (x + ia)| ≤ A |W (x + ib)| ≤ A ∀x ∈ R -[-M ; M ] |W (x + ia)| ≤ A |W (x + ib)| ≤ A ∀x ∈ [-M ; M ]
Then, for all z such that a < Im z < b:

|W (z)| ≤ f (z) A
where: -a) .

f (z) = 1 π arg e πM/(b-a) -e π(z-ia)/(b-a) e -πM/(b-a) -e π(z-ia)/(b
-e -πM/(b-a) -e π(z-ia)/(b-a) -e πM/(b-a) -e π(z-ia)/(b-a)

and this function satisfies, when |Re z| ≤ M :

f (z) ≥ 1 -4t e -π M -|Re z| b-a 1-e -π M -|Re z| b-a , for t = Im z-a b-a .
Chapter 4

Phase retrieval for wavelet transforms: a non-convex algorithm

In Chapter 3, we proved that, at least for a specific choice of wavelets, the phase retrieval problem for the wavelet transform is well-posed: any function is uniquely determined by the modulus of its wavelet transform (sometimes called scalogram in the context of audio signals), and the reconstruction has a form of stability to noise. In this chapter, we propose an algorithm to numerically solve the problem.

Algorithms used to reconstruct audio signals from their scalogram (or spectrogram, which is similar) are divided into the same two classes as generic phase retrieval problems: iterative and convexified methods.

Iterative algorithms have been introduced for the spectrogram in [START_REF] Griffin | Signal estimation from modified short-time fourier transform[END_REF]. While simple and relatively fast, they tend to produce distinct auditive artifacts. Improvements have been achieved in particular in [START_REF] Bouvrie | An incremental algorihtm for signal reconstruction from stft magnitude[END_REF] (by applying the algorithm to small temporal windows, and not to the whole signal at once) and in [START_REF] Achan | Probabilistic inference of speech signals from phaseless spectrograms[END_REF][START_REF] Eldar | Sparse phase retrieval from short-time fourier measurements[END_REF] (in the case where additional information is available about the signal). The Douglas-Rachford method [START_REF] Fienup | Phase retrieval algorithms: a comparison[END_REF][START_REF] Bauschke | Phase retrieval, error reduction algorithm, and fienup variants: a view from convex optimization[END_REF] can also yield significant improvements; however, from our experiments, it does not remove all artifacts1 .

Methods by convexification seem to perform very well on small signals [START_REF] Sun | Estimating a signal from a magnitude spectrogram via convex optimization[END_REF] or Paragraph 2.4.3 of this thesis]. Nevertheless, their high complexity prevents them to be used on real audio signals.

In this chapter, we propose a new iterative algorithm, combining the advantages of both families. Its complexity is roughly linear in the signal size, up to logarithmic factors, so that it can be used on real-size problems. However, the quality of reconstructed signals is as good as with a convexified method. This new algorithm is multiscale: it reconstructs the signal frequency band by frequency band, starting with the low frequencies. Its main tool is a reformulation of the phase retrieval problem, which extends to more general wavelets the reconstruction algorithm presented in the previous chapter (Section 3.5) for the case of Cauchy wavelets. This reformulation has two advantages. First, it gives a simple method to propagate the phase information reconstructed in low frequency bands towards higher frequency bands. Second, it naturally yields a local optimization algorithm to refine approximate solutions; this local optimization method, although non-convex, seems to be robust to the problem of local minima.

In Section 4.1, we describe our reformulation and prove its equivalence with the original problem. We explain its advantages. In Section 4.2, we describe the resulting algorithm, including a new multigrid error correction step. In Section 4.3, we discuss the superiority of multiscale algorithms over non-multiscale ones. Finally, Section 4.4 is devoted to numerical results. It shows that our algorithm is both precise and stable to noise. Moreover, it uses the algorithm to numerically investigate the intrinsic stability of the phase retrieval problem, in line with the theoretical results of Chapter 3.

Definitions and assumptions

All signals f [n] are of finite length N . Their discrete Fourier transform is defined by:

f [k] = N -1 s=0 f [n]e -2πi kn N k = 0, ..., N -1
and the convolution always refers to the circular convolution.

We define a family of wavelets (ψ j ) 0≤j≤J by:

ψj [k] = ψ(a j k) k = 0, ..., N -1
where the dilation factor a can be any number in (1; +∞) and ψ : R → C is a fixed mother wavelet. We assume that J is sufficiently large so that ψJ is negligible outside a small set of points. An example is shown in Figure 4.1.

The wavelet transform is defined by:

∀f ∈ R N , W f = {f ψ j } 0≤j≤J
The problem we consider here consists in reconstructing functions from the modulus of their wavelet transform: Reconstruct f from {|f ψ j |} 0≤j≤J Multiplying a function by a unitary complex does not change the modulus of its wavelet transform, so we only aim at reconstructing functions up to multiplication by a unitary complex, that is up to a global phase.

All signals are assumed to be analytic:

f [k] = 0 when N/2 < k ≤ N -1 (4.1)
Equivalently, we could assume the signals to be real but set the ψj [k] to zero for N/2 < k ≤ N -1.

Reformulation of the phase retrieval problem

In the first part of this section, we reformulate the phase retrieval problem for the wavelet transform, by introducing two auxiliary wavelet families.

We then describe the two main advantages of this reformulation. First, it allows to propagate the phase information from the low-frequencies to the high ones, and so enables us to perform the reconstruction scale by scale. Second, from this reformulation, we can define a natural objective function to locally optimize approximate solutions. Although non-convex, this function has few local minima; hence, the local optimization algorithm is efficient.

Introduction of auxiliary wavelets and reformulation

Let us fix r ∈]0; 1[ and define:

∀k = 0, ..., N -1, ψlow j [k] = ψj [k]r k ψhigh j [k] = ψj [k]r -k
This definition is illustrated by Figure 4.2. The wavelet ψlow j has a lower characteristic frequency than ψj and ψhigh j a higher one. The following theorem explains how to rewrite a condition on the modulus of f ψ j as a condition on f ψ low j and f ψ high j .

Theorem 4.1. Let j ∈ {0, ..., J} and g j ∈ (R + ) N be fixed. Let Q j be the function whose Fourier transform is:

Qj [k] = r k g 2 j [k] (4.2) ∀k = N 2 -N + 1, ..., N 2
For any f ∈ C N satisfying the analycity condition (4.1), the following two properties are equivalent:

1. |f ψ j | = g j 2. (f ψ low j )(f ψ high j ) = Q j
Proof. The proof consists in showing that the second inequality is the analytic extension of the first one, in a sense that will be precisely defined.

For any function h : {0, ..., N -1} → C, let P (h) be:

∀z ∈ C P (h)(z) = N 2 k= N 2 -N +1 ĥ[k]z k
Up to a change of coordinates, P (|f ψ j | 2 ) and P (g 2 j ) are equal to P ((f ψ low j )(f ψ high j

)) and P (Q j ): Lemma 4.2. For any f satisfying the analycity condition (4.1):

∀z ∈ C P (|f ψ j | 2 )(rz) = P ((f ψ low j )(f ψ high j ))(z)
and P (g 2 j )(rz) = P (Q j )(z) This lemma is proved in the appendix 4.5.2. It implies the result because then:

|f ψ j | = g j ⇐⇒ |f ψ j | 2 = g 2 j ⇐⇒ ∀z, P (|f ψ j | 2 )(z) = P (g 2 j )(z) ⇐⇒ ∀z, P (|f ψ j | 2 )(rz) = P (g 2 j )(rz) ⇐⇒ ∀z, P ((f ψ low j )(f ψ high j ))(z) = P (Q j )(z) ⇐⇒ (f ψ low j )(f ψ high j ) = Q j
By applying simultaneously Theorem 4.1 to all indexes j, we can reformulate the phase retrieval problem |f ψ j | = g j , ∀j in terms of the f ψ low j 's and f ψ high j 's.

Corollary 4.3 (Reformulation of the phase retrieval problem). Let (g j ) 0≤j≤J be a family of signals in (R + ) N . For each j, let Q j be defined as in (4.2). Then the following two problems are equivalent:

Find f satisfying (4.1) such that:

∀j, |f ψ j | = g j ⇐⇒ Find f satisfying (4.1) such that: ∀j, (f ψ low j )(f ψ high j ) = Q j (4.3)

Phase propagation across scales

This new formulation yields a natural multiscale reconstruction algorithm, in which one reconstructs f frequency band by frequency band, starting from the low frequencies.

Indeed, once f ψ J , ..., f ψ j+1 have been reconstructed, it is possible to estimate f ψ low j by deconvolution. This deconvolution is stable to noise because, if r is sufficiently small, then the frequency band covered by ψ low j is almost included in the frequency range covered by ψ J , ..., ψ j+1 (see figure 4.2). From f ψ low j , one can reconstruct f ψ high j , using (4.3):

f ψ high j = Q j f ψ low j (4.4)
Finally, one reconstructs f ψ j from f ψ high j and f ψ low j . The classical formulation of the phase retrieval problem does not allow the conception of such a multiscale algorithm. Indeed, from f ψ J , ..., f ψ j+1 , it is not possible to directly estimate f ψ j : it would require performing a highly unstable deconvolution. The introduction of the two auxiliary wavelet families is essential.

Local optimization of approximate solutions

From the reformulation (4.3), we can define a natural objective function for the local optimization of approximate solutions to the phase retrieval problem. This is also possible from the classical formulation but the objective function then has numerous local minima, which make it difficult to globally minimize. Empirically, the objective function associated to the reformulation suffers dramatically less from this drawback.

The objective function has 2J + 3 variables: (h low j ) 0≤j≤J , (h high j ) 0≤j≤J and f . The intuition is that f is the signal we aim at reconstructing and the h low j , h high j correspond to the f ψ low j 's and f ψ high j 's. The objective function is:

obj(h low J , ..., h low 0 , h high J , ..., h high 0 , f ) = J j=0 ||h low j h high j -Q j || 2 2 + λ J j=0 ||f ψ low j -h low j || 2 2 + ||f ψ high j -h high j || 2 2 (4.5)
We additionally constrain the variables (h low j ) 0≤j≤J and (h high j ) 0≤j≤J to satisfy:

∀j = 0, ..., J -1 h low j ψ high j+1 = h high j+1 ψ low j (4.6)
The first term of the objective ensures that Equalities (4.3) are satisfied, while the second term and the additional constraint (4.6) enforce the fact that the h low j 's and h high j 's must be the wavelet transforms of the same function f .

The parameter λ is a positive real number. In our implementation, we choose a small λ, so that the first term dominates over the second one.

A similar objective function can also be derived directly from the classical formulation. However, empirically, it appears to have much more local minima than the function (4.5); hence, it is more difficult to efficiently minimize. A possible explanation is that the set of zeroes of the first term of (4.5) (which dominates the second one) has a smaller dimension when the reformulation is used, thus reducing the number of local minima it contains.

Description of the algorithm

In this section, we describe our implementation of the multiscale reconstruction algorithm introduced in Section 4.1. We explain the general organization in Paragraph 4.2.1. We then describe our exhaustive search method for solving phase retrieval problems of very small size (paragraph 4.2.2), which our algorithm uses to initialize the multiscale reconstruction. In Paragraph 4.2.3, we describe an additional multigrid correction step.

Organization of the algorithm

We start by reconstructing f ψ J from |f ψ J | and |f ψ J-1 |. We use an exhaustive search method, described in the next paragraph 4.2.2, which takes advantage of the fact that ψJ and ψJ-1 have very small supports.

We then reconstruct the components of the wavelet transform scale by scale, as described in Section 4.1.

At each scale, we reconstruct f ψ low j by propagating the phase information coming from f ψ J , ..., f ψ j+1 (as explained in Paragraph 4.1.2). This estimation can be imprecise, so we refine it by local optimization, using the objective function defined in Paragraph 4.1.3, from which we drop all the terms with higher scales than j. The local optimization algorithm we use in the implementation is L-BFGS ( [START_REF]Updating quasi-newton matrices with limited storage[END_REF]), a low-memory approximation of a second order method.

We then reconstruct f ψ high j by Equation (4.4).

At the end of the reconstruction, we run a few steps of the classical Gerchberg-Saxton algorithm to further refine the estimation.

The pseudo-code 5 summarizes the structure of the implementation.

Algorithm 5 overview of the algorithm

Input: {|f ψ j |} 0≤j≤J 1: Initialization: reconstruct f ψ J by exhaustive search 2: for all j = J : (-1) : 0 do

Reconstruction by exhaustive search for small problems

In this paragraph, we explain how to reconstruct f ψ j from |f ψ j | and |f ψ j-1 | by exhaustive search, in the case where the support of ψj and ψj-1 is small. This is the method we use to initialize our multiscale algorithm. It is also useful for the multigrid error correction step described in the next paragraph 4.2.3. Lemma 4.4. Let m ∈ R N and K ∈ N * be fixed. We consider the problem:

Find g ∈ C N s.t. |g| = m
and Supp(ĝ) ⊂ {1, ..., K}

This problem has at most 2 K-1 solutions, up to a global phase, and there exist a simple algorithm which, from m and N , returns the list of all possible solutions.

Proof. This lemma is a consequence of classical results about the phase retrieval problem for the Fourier transform. It can for example be derived from [START_REF] Hayes | The reconstruction of a multidimensional sequence from the phase or magnitude of its fourier transform[END_REF]. We give a proof in the appendix 4.5.1.

We apply this lemma to m = |f ψ j | and |f ψ j-1 |, and construct the lists of all possible f ψ j 's and of all possible f ψ j-1 's. The true f ψ j and f ψ j-1 are the only pair in these two lists which satisfy the equality:

(f ψ j ) ψ j-1 = (f ψ j-1 ) ψ j

Usefulness of the error correction step

The error correction step does not perfectly correct the errors, but greatly reduces the amplitude of large ones. Figure 4.4 shows an example of this phenomenon. It deals with the reconstruction of a difficult audio signal, representing a human voice saying "I'm sorry". Figure 4.4a shows f ψ low 7 after the multiscale reconstruction at scale 7, but before the error correction step. The reconstruction presents large errors. Figure 4.4b shows the value after the error correction step. It is still not perfect but much closer to the ground truth.

So the error correction step must be used when large errors are susceptible to occur, and turned off otherwise: it makes the algorithm faster without reducing its precision.

Figure 4.5 illustrates this affirmation by showing the mean reconstruction error for the same audio signal as previously. When 200 iterations only are allowed at each local optimization step, there are large errors in the multiscale reconstruction; the error correction step significantly reduces the reconstruction error. When 2000 iterations are allowed, all the large errors can be corrected during the local optimization steps and the error correction step is not useful.

Multiscale versus non-multiscale

Our reconstruction algorithm has very good reconstruction performances, mainly because it uses the reformulation of the phase retrieval problem introduced in Section 4.1. However, the 120 quality of its results is also due to its multiscale structure. It is indeed known that, for the reconstruction of functions from their spectrogram or scalogram, multiscale algorithms perform better than non-multiscale ones [START_REF] Bouvrie | An incremental algorihtm for signal reconstruction from stft magnitude[END_REF][START_REF] Bruna | Scattering representations for recognition[END_REF].

In this section, we propose two justifications for this phenomenon (paragraph 4.3.1). We then introduce a multiscale version of the classical Gerchberg-Saxton algorithm, and numerically verify that it yields better reconstruction results than the usual non-multiscale version (paragraph 4.3.2).

Advantages of the multiscale reconstruction

At least two factors can explain the superiority of multiscale methods, where the f ψ j 's are reconstructed one by one, and not all at the same time.

First, they can partially remedy the possible ill-conditioning of the problem. In particular, if the f ψ j 's have very different norms, then a non-multiscale algorithm will be more sensitive to the components with a high norm. It may neglect the information given by |f ψ j |, for the values of j such that this function has a small norm. With an multiscale algorithm where all the |f ψ j |'s are successively considered, this happens less frequently.

Second, iterative algorithms, like Gerchberg-Saxton, are very sensitive to the choice of their starting point (hence the care given to their initialization in the literature [START_REF] Netrapalli | Phase retrieval using alternating minimization[END_REF]Candès et al., 2015]). If all the components are reconstructed at the same time and the starting point is randomly chosen, the algorithm almost never converges towards the correct solution: it gets stuck in a local minima. In a multiscale algorithm, the starting point at each scale can be chosen so as to be consistent with the values reconstructed at lower scales; it yields much better results.

Multiscale Gerchberg-Saxton

To justify the efficiency of the multiscale approach, we introduce a multiscale version of the classical Gerchberg-Saxton algorithm [START_REF] Gerchberg | A practical algorithm for the determination of phase from image and diffraction plane pictures[END_REF] (by alternate projections) and compare its performances with the non-multiscale algorithm.

The multiscale algorithm reconstructs f ψ J by exhaustive search (paragraph 4.2.2). Then, for each j, once f ψ J , ..., f ψ j+1 are reconstructed, an initial guess for f ψ j is computed by deconvolution. The frequencies of f ψ j for which the deconvolution is too unstable are set to zero. The regular Gerchberg-Saxton algorithm is then simultaneously applied to f ψ J , ..., f ψ j .

We test this algorithm on realizations of Gaussian random processes (see Section 4.4.2 for details), of various lengths. On None of the algorithms is able to perfectly reconstruct the signals, in particular when their size increases. However, the multiscale algorithm clearly yields better results, with a mean error approximately twice smaller.

Numerical results

In this section, we describe the behavior of our algorithm. We compare it with Gerchberg-Saxton and with PhaseLift. We show that it is much more precise than Gerchberg-Saxton. It is comparable with PhaseLift in terms of precision, but significantly faster, so it allows to reconstruct larger signals.

The performances strongly depend on the type of signals we consider. The main source of difficulty for our algorithm is the presence of small values in the wavelet transform, especially in the low frequencies.

Indeed, the reconstruction of f ψ high j by Equation (4.4) involves a division by f ψ low j . When f ψ low j has small values, this operation is unstable and induces errors. As we will see in Section 4.4.3, the signals whose wavelet transform has many small values are also the signals for which the phase retrieval problem is the least stable (in the sense that two functions can have wavelet transforms almost equal in modulus without being close in l 2 -norm). This suggests that this class of functions is intrinsically the most difficult to reconstruct; it is 123 not an artifact of our algorithm.

Experimental setting

At each reconstruction trial, we choose a signal f and compute its wavelet transform {|f ψ j |} 0≤j≤J . We corrupt it with a random noise n j :

h j = |f ψ j | + n j (4.7)
We measure the amplitude of the noise in l 2 -norm, relatively to the l 2 -norm of the wavelet transform:

amount of noise = j ||n j || 2 2 j ||f ψ j || 2 2 (4.8)
We run the algorithm on the noisy wavelet transform {h j } 0≤j≤J . It returns a reconstructed signal f rec . We quantify the reconstruction error by the difference, in relative l 2 -norm, between the modulus of the wavelet transform of the original signal f and the modulus of the wavelet transform of the reconstructed signal f rec :

reconstruction error = j || |f ψ j | -|f rec ψ j | || 2 2 j ||f ψ j || 2 2 (4.9)
Alternatively, we could measure the difference between f and f rec :

error on the signal = ||f -f rec || 2 ||f || 2 (4.10)
But we know that the reconstruction of a function from the modulus of its wavelet transform is not stable to noise (Section 3.3 of this document). So we do not hope the difference between f and f rec to be small. We just want the algorithm to reconstruct a signal f rec whose wavelet transform is close to the wavelet transform of f , in modulus. Thus, the reconstruction error (4.9) is more relevant to measure the performances of the algorithm.

In all the experiments, unless otherwise specified, we use dyadic Morlet wavelets, to which we subtract Gaussian functions of small amplitude so that they have zero mean:

ψ(ω) = exp(-p(ω -1) 2 ) -β exp(-pω 2 )
where β > 0 is chosen so that ψ(0) = 0 and the parameter p is arbitrary (it controls the frequency bandwidth of the wavelets). For N = 256, our family of wavelets contains eight elements, which are plotted on Figure 4.18a. The performances of the algorithm strongly depend on the choice of the wavelet family; this is discussed in Paragraph 4.4.4. The maximal number of iterations per local optimization step is set to 10000 (with an additional stopping criterion, so that the 10000-th iteration is not always reached). We study the influence of this parameter in Paragraph 4.4.4. The noises are realizations of Gaussian white noises. The error correction step described in Paragraph 4.2.3 is always turned on.

Gerchberg-Saxton is applied in a multiscale fashion, as described in Paragraph 4.3.2, which yields better results than the regular implementation.

We use PhaseLift [START_REF] Candès | Phase retrieval via matrix completion[END_REF] with ten steps of reweighting, followed by 2000 iterations of the Gerchberg-Saxton algorithm. In our experiments with PhaseLift, we only consider signals of size N = 256. Handling larger signals is difficult with a straightforward Matlab implementation.

Results

We describe four classes of signals, whose wavelet transforms have more or less small values. For each class, we plot the reconstruction error of our algorithm, Gerchberg-Saxton and PhaseLift as a function of the noise error.

Realizations of Gaussian random processes

We first consider realizations of Gaussian random processes. A signal f in this class is defined by: f where X 1 , ..., X N/2 are independent realizations of complex Gaussian centered variables. The role of the √ k + 1 is to ensure that all components of the wavelet transform approximately have the same l 2 -norm (in expectation). An example is displayed on Figure 4.7, along with the modulus of its wavelet transform.

[k] = X k √ k + 1 if k ∈ {1,
The wavelet transforms of these signals have few small values, disposed in a seemingly random pattern. This is the most favorable class for our algorithm.

The reconstruction results are shown in Figure 4.8. Even for large signals (N = 10000), the mean reconstruction error is proportional to the input noise (generally 2 or 3 times smaller); this is the best possible result. The performances of PhaseLift are exactly the same, but Gerchberg-Saxton often fails.

Lines from images

The second class consists in lines randomly extracted from photographs. These signals have oscillating parts (corresponding to the texture zones of the initial image) and smooth parts, with large discontinuities in between. Their wavelet transforms generally contain a lot a small values, but, as can be seen in Figure 4.9, the distribution of these small values is particular. They are more numerous at high frequencies and the non-small values tend to concentrate on vertical lines of the time-frequency plane.

This distribution is favorable to our algorithm: small values in the wavelet transform are mostly a problem when they are in the low frequencies and prevent the correct initialization of the reconstruction at medium or high frequencies. Small values at high frequencies are not a problem.

Indeed, as in the case of Gaussian signals, the reconstruction error is proportional to the input noise (figure 4.10). This is also the case for PhaseLift but not for Gerchberg-Saxton.

Sums of a few sinusoids

The next class of signals contains sums of a few numbers of sinusoids, multiplied by a window function w to avoid boundary effects. Formally, a signal in this class is of the form:

f [n] =   N/2 k=1 α k exp i 2πkn N   × w[n]
where the α k are zero with high probability and realizations of complex Gaussian centered variables with small probability.

The wavelet transforms of these signals often have components of very small amplitude, which may be located at any frequential scale (figure 4.11). This can prevent the reconstruction.

The results are on Figure 4.12. Our algorithm performs much better than Gerchberg-Saxton but the results are not as good as for the two previous classes of signals.

In most reconstruction trials, the signal is correctly reconstructed, up to an error proportional to the noise. But, with a small probability, the reconstruction fails. The same phenomenon occurs for PhaseLift.

The probability of failure seems a bit higher for PhaseLift than for our algorithm. For example, when the signals are of size 256 and the noise has a relative norm of 0.01%, the reconstruction error is larger than the noise error 20% of the time for PhaseLift and only 10% of the time for our algorithm. However, PhaseLift has a smaller mean reconstruction error because, in these failure cases, the result it returns, although not perfect, is more often close to the truth: the mean reconstruction error in the failure cases is 0.2% for PhaseLift versus 1.7% for our algorithm.

Audio signals

Finally, we test our algorithm on real audio signals. These signals are difficult to reconstruct because they do not contain very low frequencies (as the human ear cannot hear them, these frequencies are not included in the recordings), so the first components of their wavelet transforms are very small.

The reconstruction results may vary from one audio signal to the other. We focus here on two representative examples.

The first signal is an extract of five seconds of a musical piece played by an orchestra (the Flight of the Bumblebee, by Rimsky-Korsakov). Figure 4.13a shows the modulus of its wavelet transform. It has 16 components and 9 of them (the ones with lower characteristic frequencies) seem negligible, compared to the other ones. However, its non-negligible components have a moderate number of small values.

The second signal is a recording of a human voice saying "I'm sorry" (figure 4.13b). The lowfrequency components of its wavelet transform are also negligible, but even the high-frequency components tend to have small values, which makes the reconstruction even more difficult.

The results are presented in Figures 4.14 and 4.15. For relatively high levels of noise (0.5% or higher), the results, in the sense of the l 2 -norm, are satisfying: the reconstruction error is smaller or equal to the amount of noise.

In the high precision regime (that is, for 0.1% of noise or less), the lack of low frequencies does not allow a perfect reconstruction. Nevertheless, the results are still good: the reconstruction error is of the order of 0.1% or 0.2% when the noise error is below 0.1%. More iterations in the optimization steps can further reduce this error. By comparison, the reconstruction error with Gerchberg-Saxton is always several percent, even when the noise is small.

Stability of the reconstruction

In this section, we use our reconstruction algorithm to investigate the stability of the reconstruction. From Section 3.3, we know that the reconstruction is not globally stable to noise: the reconstruction error (4.9) can be small (the modulus of the wavelet transform is almost exactly reconstructed), even if the error on the signal (4.10) is not small (the difference between the initial signal and its reconstruction is large).

We show that this phenomenon can occur for all classes of signals, but is all the more frequent when the wavelet transform has a lot of small values, especially in the low frequency components.

We also experimentally show that, when this phenomenon happens, the original and reconstructed signals have their wavelet transforms {f ψ j (t)} j∈Z,t∈R equal up to multiplication by a phase {e iφ j (t) } j∈Z,t∈R , which varies slowly in both j and t, except maybe at the points where f ψ j (t) is close to zero. For Cauchy wavelets, we have proven an approximation of this assertion in Section 3.4; we conjecture this result to be valid for more general wavelets.

We perform a large number of reconstruction trials, with various reconstruction parameters. This gives us a large number of pairs (f, f rec ), such that ∀j, t, |f ψ j (t)| ≈ |f rec ψ j (t)|. For each one of these pairs, we compute: The results are plotted on Figure 4.16, where each point corresponds to one reconstruction trial. The x-coordinate represents the error on the modulus and the y-coordinate the error on the signal.

We always have: error on the modulus ≤ C × (error on the function) with C a constant of the order of 1. This is not surprising because the modulus of the wavelet transform is a Lipschitz operator, with a constant close to 1.

As expected, the converse inequality is not true: the error on the function can be significantly larger than the error on the modulus. For each class, an important number of reconstruction trials yield errors such that: error on the signal ≈ 30 × error on the modulus For realizations of Gaussian random processes or for lines extracted from images (figures 4.16a and 4.16b), the ratio between the two errors never exceeds 30 (except for one outlier). But for sums of a few sinusoids (4.16c) or audio signals (4.16d), we may even have: error on the signal ≥ 100 × error on the modulus So instabilities appear in the reconstruction of all kinds of signals, but are stronger for sums of sinusoids and audio signals, that is for the signals whose wavelet transforms have a lot of small values, especially in the low frequencies.

These results have a theoretical justification. Let us recall that, in Section 3.3, we have explained how, from any signal f , it is possible to construct g such that |f ψ j | ≈ |g ψ j | for all j but f ≈ g in the l 2 -norm sense.

The principle of the construction is to multiply each f ψ j (t) by a phase e iφ j (t) . The function (j, t) → e iφ j (t) must be chosen so that it varies slowly in both j and t, except maybe at the points (j, t) where f ψ j (t) is small. Then there exist a signal g such that (f ψ j (t))e iφ j (t) ≈ g ψ j (t) for any j, t. Taking the modulus of this approximate equality yields:

∀j, t |f ψ j (t)| ≈ |g ψ j (t)|
However, we may not have f ≈ g. This construction works for any signal f (unless the wavelet transform is very localized in the time frequency domain), but the number of possible {e iφ j (t) } j,t is larger when the wavelet transform of f has a lot of small values, because the constraint of slow variation is relaxed at the points where the wavelet transform is small (especially when the small values are in the low frequencies). This is probably why instabilities occur for all kinds of signals, but more frequently when the wavelet transforms have a lot of zeroes. Figure 4.17: (a) modulus of the wavelet transform of a signal (b) phase difference between the wavelet transform of this signal and of its reconstruction (black points correspond to places where the modulus is too small for the phase to be meaningful)

From our experiments, it seems that the previous construction describes all the instabilities: when the wavelet transforms of f and f rec have almost the same modulus and f is not close to f rec , then the wavelet transforms of f and f rec are equal up to slow-varying phases {e iφ j (t) } j,t .

Figure 4.17 shows an example. The signal is a sum of sinusoids. The relative difference between the modulus is 0.3%, but the difference between the initial and reconstructed signals is more than a hundred times larger; it is 46%. The right subfigure shows the difference between the phases of the two wavelet transforms. It indeed varies slowly, in both time and frequency (actually, it is almost constant along the frequency axis), and a bit faster at the extremities, where the wavelet transform is closer to zero.

Influence of the parameters

In this paragraph, we analyze the importance of the two main parameters of the algorithm: the choice of the wavelets (paragraph 4.4.4) and the number of iterations allowed per local optimization step (paragraph 4.4.4).

Choice of the wavelets

Two properties of the wavelets are especially important: the exponential decay of the wavelets in the Fourier domain (so that the Q j 's (4.2) are correctly computed) and the amount of overlap between two neighboring wavelets (if the overlap is too small, then f ψ J , ..., f ψ j+1 contain not much information about f ψ j and the multiscale approach is less efficient).

We compare the reconstruction results for four families of wavelets. Morlet wavelets have a fast decay and consecutive wavelets overlap well. This does not depend upon the dilation factor so the reconstruction performances are similar for the two Morlet families (figures 4.19a and 4.19b).

Laplacian wavelets are similar, but the overlap between consecutive wavelets is not as good. So Laplacian wavelets globally have the same behavior as Morlet wavelets but require significantly more computational effort to reach the same precision. Figures 4.19a and 4.19b have been generated with a maximal number of iterations per optimization step equal to 30000 (instead of 10000) and the reconstruction error is still larger. The decay of Gammatone wavelets is polynomial instead of exponential. The products Q j cannot be efficiently estimated and our method performs significantly worse. In the case of lines extracted from images (4.19a), the reconstruction error stagnates at 0.1%, even when the noise is of the order of 0.01%. For audio signals (4.19b), it is around 1% for any amount of noise.

Number of iterations in the optimization step

The maximal number of iterations allowed per local optimization step (paragraph 4.1.3) can have a huge impact on the quality of the reconstruction.

Figure 4.20 represents, for an audio signal, the reconstruction error as a function of this number of iterations. As the objective functions are not convex, there are no guarantees on the speed of the decay when the number of iterations increases. It can be slow and even nonmonotonic. Nevertheless, it clearly globally decays.

The execution time is roughly proportional to the number of iterations. It is thus important to adapt this number to the desired application, so as to reach the necessary precision level without making the algorithm excessively slow. This problem has at most 2 K-1 solutions, up to a global phase, and there exist a simple algorithm which, from m and N , returns the list of all possible solutions.

Proof. We define: P

(g)(X) = ĝ[1] + ĝ[2]X + ... + ĝ[K]X K-1
We show that the constraint |g| = m amounts to knowing P (g)(X)P (g)(1/X). This is in turn equivalent to knowing the roots of P (g) (and thus knowing g) up to inversion with respect to the unit circle. There are in general K -1 roots, and each one can be inverted. This gives 2 K-1 solutions.

We set:

Q(g)(X) = P (g)(1/X) = ĝ[K]X -(K-1) + ĝ[K -1]X -(k-2) + ... + ĝ[1] Equation |g| 2 = m 2 is equivalent to |g| 2 = m 2 , that is 1 N ĝ ĝ = m 2 . For each k ∈ {-(K - 1), ..., K -1}: ĝ ĝ[k] = s ĝ[k -s]ĝ[-s]
This number is the coefficient of order k of P (g)(X)Q(g)(X), so |g| = m if and only if:

P (g)(X)Q(g)(X) = N K-1 k=-(K-1) m 2 [k]X k (4.11)
Let us denote by r 1 , ..., r K-1 the roots of P (g)(X), so that:

P (g)(X) = ĝ[K](X -r 1 )...(X -r K-1 ) Q(g)(X) = ĝ[K](1/X -r 1 )...(1/X -r K-1 )
From (4.11), the equality |g| = m holds if and only if ĝ[K], r 1 , ..., r K-1 satisfy:

|ĝ[K]| 2 K-1 j=1 (X -r j )(1/X -r j ) = N K-1 k=-(K-1) m 2 [k]X k (4.12)
If we denote by s 1 , 1/s 1 , ..., s K-1 , 1/s K-1 the roots of the polynomial function

K-1 k=-(K-1) m 2 [k]X k ,
then the only possible choices for r 1 , ..., r K-1 are, up to permutation:

r 1 = s 1 or 1/s 1 r 2 = s 2 or 1/s 2 . . .
So there are 2 K-1 possibilities. Once the r j have been chosen, ĝ[K] is uniquely determined by (4.12), up to multiplication by a unitary complex.

From r 1 , ..., r K-1 , ĝ[K], P (g) is uniquely determined and so is g. The algorithm is summarized in 6.

Algorithm 6 reconstruction by exhaustive search for a small problem Input: K,m 1: Compute the roots of

K-1 k=-(K-1) m 2 [k]X k 2:
Group them by pairs (s 1 , 1/s 1 ), ..., (s K-1 , 1/s K-1 ) 3: List the 2 K-1 elements (r 1 , ..., r K-1 ) of {s 1 , 1/s 1 } × ... × {s K-1 , 1/s K-1 } 4: for all the elements do 5:

Compute the corresponding ĝ[K] by (4.12)

6:

Compute the coefficients of P (g)(X) = ĝ[K](X -r 1 )...(X -r K-1 )

7:

Apply an IFFT to the coefficients to obtain g 8: end for Output: the list of 2 K-1 possible values for g 4.5.2 Proof of Lemma 4.2 Lemma (4.2). For any f satisfying the analycity condition (4.1):

∀z ∈ C P (|f ψ j | 2 )(rz) = P ((f ψ low j )(f ψ high j ))(z) and P (g 2 j )(rz) = P (Q j )(z)
Proof. Recall that, by definition, for any h ∈ C N :

∀z ∈ C P (h)(z) = N 2 k= N 2 -N +1 ĥ[k]z k
So for any two signals h, H, the condition P (h)(rz) = P (H)(z), ∀z ∈ C is equivalent to:

∀k = N 2 -N + 1, ..., N 2 ĥ[k]r k = Ĥ[k] (4.13)
Applied to g 2 j and Q j , this property yields the equality P (g 2 j )(rz) = P (Q j )(z), ∀z ∈ C: by the definition of Q j in (4.2), Equation (4.13) is clearly satisfied.

Let us now show that:

P (|f ψ j | 2 )(rz) = P ((f ψ low j )(f ψ high j ))(z), ∀z ∈ C
It suffices to prove that (4.13) holds, that is:

∀k ∈ N 2 -N + 1, ..., N 2 , |f ψ j | 2 [k]r k = (f ψ low j )(f ψ high j )[k]
Indeed, because the analycity condition (4.1) holds, we have for all k:

|f ψ j | 2 [k] = 1 N f ψ j f ψ j [k] = 1 N N/2 l=1 f [l] ψj [l] f [l -k] ψj [l -k] = r -k N N/2 l=1 f [l] ψlow j [l] f [l -k] ψhigh j [l -k] = r -k N f ψ low j f ψ high j [k] = r -k (f ψ low j )(f ψ high j ) [k]
Chapter 5

Exponential decay of scattering coefficients

In Chapters 3 and 4, we have studied, from the point of view of phase retrieval, the wavelet transform modulus. Our motivation was to understand the behavior of this operator, commonly used in data analysis as a convenient representation of signals, especially audio signals. However, if the wavelet transform modulus is an ubiquitous tool in signal processing, it can be insufficient for some tasks. It is indeed not invariant enough to perceptually negligible modifications of the input signal. It can then be used as a building block to construct more sophisticated representations, with more invariance properties.

In this chapter, we consider such a representation, the scattering transform. Introduced in [START_REF] Mallat | Group invariant scattering[END_REF], the scattering transform has the property to be stable to translations and small deformations of the input signal. It gives excellent results in tasks which are naturally invariant to this transformations, like classification of audio signals and images [Bruna and Mallat, 2013a;[START_REF] Andén | Multiscale scattering for audio classification[END_REF].

The scattering is defined as a cascade of wavelet transform modulus. At each step of the cascade, the components of the wavelet transform are averaged, to produce so-called scattering coefficients. Each scattering coefficient has an order, which is the number of wavelet transform modulus that have been necessary to compute it.

Our long-term goal is to understand which properties of input signals are characterized by their scattering transform, in particular which regularity properties.

In this chapter, we focus on the relation between the order of scattering coefficients and the frequency band that they describe in the input signal. Theorem 5.2 indeed states that scattering coefficients of order n do almost not contain information on the frequential band centered at 0, of bandwidth proportional ra n , for some constants r > 0, a > 1.

This indicates that there exists in scattering a phenomenon of propagation from high fre-quencies towards low frequencies. Intuitively, the information initially contained in a frequency band of width A is moved, after one application of the wavelet transform modulus, to a frequency band of width Aa -1 , for some a > 1. After a second application of the wavelet transform modulus, it is in a band of width Aa -2 . And so on, until it is in the frequency band covered by the averaging operator, and disappears, absorbed in a scattering coefficient.

In Section 5.1, we precisely define the scattering transform and some known results about it. We also describe in more detail the propagation phenomenon. In Section 5.2, we state the theorem. In Section 5.3, we give the principle of the proof. Section 5.4 adapts the theorem to the scattering transform on stationary processes. Finally, Section 5.5 proves the lemmas necessary for the proof of the main theorem.

The scattering transform

As in the previous chapters, once a wavelet ψ ∈ L 1 ∩ L 2 (R) (such that R ψ = 0) has been chosen, we define a family of wavelets (ψ j ) j∈Z by:

∀j ∈ Z, t ∈ R ψ j (t) = 2 -j ψ(2 -j t) ⇐⇒ ∀j ∈ Z, ω ∈ R ψj (ω) = ψ(2 j ω)

Definition

We follow the definition of [START_REF] Mallat | Group invariant scattering[END_REF].

The scattering transform consists in a cascade of modulus of wavelet transforms. After each application of the modulus, the resulting functions are locally averaged. The set of averages constitutes the scattering transform.

The averaging is performed with a real-valued positive function φ ∈ L 1 ∩ L 2 (R) such that φ(0) = 1. We define:

∀J ∈ Z, t ∈ R φ J (t) = 2 -J φ(2 -J t)
The convolution with φ J represents an average on an interval of characteristic size 2 J .

We now formally define the cascade of modulus of wavelet transforms. For any function f ∈ L 2 (R), we set:

U [ø]f = f
and iteratively define, for any n-uplet (j 1 , ..., j n ) ∈ Z n , with n ≥ 1: For any J ∈ Z, we set P J = {(j 1 , ..., j n ), n ∈ N, j 1 , ..., j n ∈ {-∞, ..., J}}; we refer to the elements of P J as paths. We denote the length (that is, the number of elements) of a path p by |p|.

U [(j 1 , ..., j n )]f = U [(j 1 , ..., j n-1 )]f ψ jn f |f ψ J-2 | |f ψ J-1 | |f ψ J | ||f ψ J | ψ J-1 | ||f ψ J | ψ J | f φ J ... |f ψ J-2 | φ J |f ψ J-1 | φ J |f ψ J | φ J ... ||f ψ J-1 | ψ J | φ J ... ||f ψ J | ψ J-1 | φ J ||f ψ J | ψ J | φ J ...
For any p ∈ P J , we define:

S J [p]f = U [p]f φ J
The scattering coefficients associated to f at scale J are the set {S J [p]f } p∈P J .

The computation of the scattering coefficients is schematized in Figure 5.1.

Norm preservation and energy propagation

When the wavelets are suitably chosen, the scattering transform preserves the norm [Mallat, 2012, Theorem 2.6]:

p∈P J ||S J [p]f || 2 2 = ||f || 2 2 (5.1)
In this paragraph, we briefly describe this result. To explain it, we introduce the informal idea that the repeated application of the modulus of the wavelet transform moves the energy contained in the high frequencies of the initial signal towards the low frequencies.

For the moment, we assume the wavelets to be analytical:

ψ(ω) = 0 if ω < 0
and, together with the low-pass filter φ, to satisfy a Littlewood-Paley condition:

∀ω ≥ 0 | φ(ω)| 2 + 1 2 j≤0 | ψj (ω)| 2 = 1
This condition ensures that, for any scale J ∈ Z, the wavelet transform with low-pass filter f → {f φ J , {f ψ j } j≤J } is unitary over the set of real-valued functions:

∀f ∈ L 2 (R, R) ||f φ J || 2 2 + j≤J ||f ψ j || 2 2 = ||f || 2 2
Consequently, the application W J : f → {f φ J , {|f ψ j |} j≤J } preserves the norm. As the scattering transform is computed by recursively applying this operator to the input function, we can prove by iteration over n that, for any length n ≥ 0:

∀f ∈ L 2 (R, R) p∈P J ,|p|<n ||S J [p]f || 2 2 + p∈P J ,|p|=n ||U [p]f || 2 2 = ||f || 2 2
If we can prove that p∈P J ,|p|=n ||U [p]f || 2 2 goes to zero when n goes to infinity, then we can prove Equation (5.1). Under an additional condition on the wavelets, this is done in [Mallat, 2012, Theorem 2.6]: Theorem 5.1. A family (φ, {ψ j } j∈Z ) is said to be admissible if there exists η ∈ R and a positive real-valued function ρ ∈ L 2 (R) such that:

∀ω ∈ R, |ρ(ω)| ≤ | φ(2ω)| and ρ(0) = 1
and that the function:

Ψ(ω) = |ρ(ω -η)| 2 - +∞ k=1 k(1 -|ρ(2 -k (ω -η))| 2 ) satisfies: inf 1≤ω≤2 +∞ j=-∞ Ψ(2 j ω)| ψj (ω)| 2 > 0 (5.2)
If (φ, {ψ j } j∈Z ) is admissible, then, for any real-valued function f ∈ L 2 (R) and any scale J ∈ Z:

p∈P J ,|p|=n ||U [p]f || 2 2 → 0 when n → +∞ (5.3)
which implies the norm preservation:

p∈P J ||S J [p]f || 2 2 = ||f || 2 2
Intuitively, the property (5.3) holds because the iterative application of the modulus of the wavelet transform moves the energy carried by the high frequencies of the signal towards the low frequencies. At each step of the scattering transform, the energy in the lowest frequency bands is output by convolution with φ J . The remaining part is shifted towards the lower frequencies by a new application of the modulus of the wavelet transform and so on.

For the displacement of the energy towards the low frequencies, the modulus is essential: for each signal f , f ψ j is a function whose energy is concentrated in a frequency band of characteristic size 2 -j , with a mean frequency also of the order of 2 -j . After application of the modulus, |f ψ j | tends to have its energy concentrated in a frequency band of characteristic size still equal to 2 -j , but now centered around zero. So the frequencies that it contains are globally lower than the frequencies of f ψ j . An example of this phenomenon is displayed on Figure 5.2.

According to this simplistic reasoning, the modulus of the wavelet transform approximately moves the energy contained in a frequency band around 2 j to the frequency band [-2 j-1 , 2 j-1 ]. By iteratively applying this argument, we expect the energy to arrive in the frequency band [-2 J , 2 J ] (and thus disappear) after a number of scattering steps proportional to j. The theorem of the next section will formalize this idea.

Theorem statement

The theorem of this section relies on similar ideas to the ones of Theorem 5.1 but improves it in two aspects:

• It gives a precise bound on p∈P J ,|p|=n ||U [p]f || 2 2
, instead of simply ensuring that this quantity goes to zero. This bound formalizes the idea described in the end of the previous paragraph, that the energy of the signal carried by the frequencies around 2 j disappears after a number of scattering step proportional to j.

• It holds for a much more general class of wavelets that Theorem 5.1. In particular, it does not require the wavelets to be analytical. It notably applies to Morlet wavelets, and also to compactly-supported wavelets like Selesnick wavelets [START_REF] Selesnick | Hilbert transform pairs of wavelet bases[END_REF], two important choices for practical applications. The wavelets have to satisfy three simple conditions, which will be commented after the statement of the theorem.

For any a > 0, we denote by χ a the Gaussian function:

∀t ∈ R χ a (t) = √ πa exp -(πa) 2 t 2
whose Fourier transform satisfies:

∀ω ∈ R χa (ω) = exp(-(ω/a) 2 )
Theorem 5.2. Let (ψ j ) j∈Z be a family of wavelets. We assume the following Littlewood-Paley inequality to be satisfied:

∀ω ∈ R 1 2 j∈Z | ψj (ω)| 2 + | ψj (-ω)| 2 ≤ 1 (5.4)
We also assume that:

∀j ∈ Z, ω > 0 | ψj (-ω)| ≤ | ψj (ω)| (5.5)
and, for any ω, the inequality is strict for at least one value of j.

Finally, we assume that, for some > 0:

ψ(ω) = O(|ω| 1+ ) (5.6)
when ω → 0.

Then, for any J ∈ Z, there exist r > 0, a > 1 such that, for all n ≥ 2 and f ∈ L 2 (R, R):

p∈P J ,|p|=n ||U [p]f || 2 2 ≤ ||f || 2 2 -||f χ ra n || 2 2 = R | f (ω)| 2 1 -| χra n (ω)| 2 dω (5.7)
Before turning to the proof of this theorem, let us briefly comment the conditions (5.4), (5.5) and (5.6).

The Littlewood-Paley inequality (5.4) is equivalent to the fact that the wavelet transform is contractive over L 2 (R, R). Without a condition of this kind, the wavelet transform can amplify some frequencies of the initial signal, and the energy contained in the long paths of length n will not necessarily decrease when n increases.

The condition (5.5) describes the fact that, although they do not need to be analytical, the wavelets must give more weight to the positive frequencies than to the negative ones. If this condition is not met, the phenomenon of energy shift towards the low frequencies may not happen. This is in particular the case if the wavelets are real.

The condition (5.6) states that the wavelets have a bit more than one zero momentum. It is necessary for our proof, but it is not clear whether the theorem stays true without it or not.

Principle of the proof

The proof proceeds by iteration over n.

In this paragraph, we show that, if a is close enough to 1, the fact that the property holds for n implies that it also holds for n + 1. The initialization, for n = 2, relies on the same principle, but is more technical; it is done in the paragraph 5.5.3.

Lemma 5.3. For any j ∈ Z, x ∈ R * + , δ j ∈ R: || |f ψ j | χ x || 2 2 ≥ ||f ψ j (χ x e 2πiδ j . )|| 2 2 = R | f (ω)| 2 | ψj (ω)| 2 | χx (ω -δ j )| 2 dω
This lemma is already present in [START_REF] Mallat | Group invariant scattering[END_REF]. For sake of completeness, we prove it again in the paragraph 5.5.1. If the property (5.7) holds for n ≥ 2, then:

p∈P J ,|p|=n+1 ||U [p]f || 2 2 = j≤J   p∈P J ,|p|=n ||U [j, p]f || 2 2   = j≤J   p∈P J ,|p|=n ||U [p] |f ψ j | || 2 2   ≤ j≤J || |f ψ j | || 2 2 -|| |f ψ j | χ ra n || 2 2
(5.8) By Lemma 5.3, for any choice of (δ j ) j≤J :

p∈P J ,|p|=n+1 ||U [p]f || 2 2 ≤ j≤J ||f ψ j || 2 2 -||f ψ j (χ ra n e 2πiδ j )|| 2 2 = R | f (ω)| 2 j≤J | ψj (ω)| 2 (1 -| χra n (ω -δ j )| 2 ) dω
We symmetrize the expression, by taking into account the fact that f is real, so

| f (ω)| = | f (-ω)|: p∈P J ,|p|=n+1 ||U [p]f || 2 2 ≤ R | f (ω)| 2 × 1 2 j≤J | ψj (ω)| 2 (1 -| χra n (ω -δ j )| 2 ) +| ψj (-ω)| 2 (1 -| χra n (-ω -δ j )| 2 ) dω
To conclude, it is thus sufficient to show that, if the δ j 's are well-chosen, then:

1 2 j≤J | ψj (ω)| 2 (1 -| χra n (ω -δ j )| 2 ) + | ψj (-ω)| 2 (1 -| χra n (-ω -δ j )| 2 ) ≤ 1 -| χar n+1 (ω)| 2 (∀ω ∈ R)
which is a direct consequence of the next lemma, proven in the paragraph 5.5.2.

Lemma 5.4. For any x > 0, if a > 1 is close enough to 1, then there exist (δ j ) j≤J such that:

∀ω ∈ R 1 2 j∈Z | ψj (ω)| 2 (1 -| χx (ω -δ j )| 2 ) + j∈Z | ψj (-ω)| 2 (1 -| χx (-ω -δ j )| 2 ) ≤ 1 -| χax (ω)| 2
The proof is summarized in Figure 5.3.

Inductive hypothesis: the energy contained in the paths of length n + 1 beginning by j is at most the energy of f ψ j , multiplied by a high-pass filter (Equation (5.8)).

| f | 2 | ψj | 2 1 -| χar n | 2
Because of the modulus around f ψ j , the high-pass filter can be arbitrarily shifted in frequency.

| f | 2 1 2 | ψj | 2 (1 -| χar n (. -δ j )| 2 ) + | ψj (-.)| 2 (1 -| χar n (-. -δ j )| 2 )
Summing over j gives: In what follows, X is a real-valued stationary stochastic process, with continuous and integrable autocovariance R X .

| f | 2 j≤J 1 2 | ψj | 2 (1 -| χar n (. -δ j )| 2 ) + | ψj (-.)| 2 (1 -| χar n (-. -δ j )| 2 ) ≤ | f | 2 × 1 -| χar n+1 | 2 (shown in red)
The scattering transform of X is defined in the same way as for elements of L 2 (R):

U [ø]X = X ∀n ≥ 1, (j 1 , ..., j n ) ∈ Z n U [(j 1 , ..., j n )]X = |U [(j 1 , ..., j n-1 )]X ψ jn | ∀p ∈ P J S J [p]f = U [p]f φ J
The results proven for the deterministic wavelet transform tend to be also valid for stationary processes, if one replaces the squared L 2 -norms by the expectations of the squared modulus. In particular, Theorem 5.2 can be adapted to the case of stationary processes.

Theorem 5.5. Let (ψ j ) j∈Z be a family of wavelets, satisfying the same conditions (5.4), (5.5) and (5.6) as in Theorem 5.2.

Then, for any J ∈ Z, there exist r > 0, a > 1 such that, for all n ≥ 2 and f ∈ L 2 (R, R):

p∈P J ,|p|=n E(|U [p]X| 2 2 ) ≤ E(|X| 2 ) -E(|X χ ra n | 2 ) = R RX (ω)(1 -| χra n (ω)| 2 )dω
The proof of this theorem is exactly the same as the one of Theorem 5.2. The only lines to be modified are the ones where we use the fact that the Fourier transform is an isometry of L 2 (R). We use instead the property according to which, for any stationary process with continuous and integrable autocovariance R Y , and for any h ∈ L 1 ∩ L 2 (R): This implies the inequality. The equality is a consequence of the unitarity of the Fourier transform.

E(|Y h| 2 ) = |E(Y h)| 2 + R | ĥ(ω)| 2 RY (ω)dω = E(Y ) × R h (
5.5.2 Proof of Lemma 5.4

Lemma (5.4). For any x > 0, if a > 1 is close enough to 1, then there exist (δ j ) j≤J such that:

∀ω ∈ R 1 2 j∈Z | ψj (ω)| 2 (1 -| χx (ω -δ j )| 2 ) + j∈Z | ψj (-ω)| 2 (1 -| χx (-ω -δ j )| 2 ) ≤ 1 -| χax (ω)| 2
Proof. We are going to look for δ j 's of the form δ j = δ2 -j . δ 2 2 -2j

x 2

)

≤ S(ω) 1 -exp 2 - ω 2 x 2 + j∈Z 1 2 | ψj (ω)| 2 S(ω) 2δω2 -j x 2 - δ 2 2 -2j x 2 + j∈Z 1 2 | ψj (-ω)| 2 S(ω) - 2δω2 -j x 2 - δ 2 2 -2j x 2 = S(ω) 1 -exp 2 - ω 2 x 2 + 2δω x 2 F 1 (ω) - δ 2
x 2 F 2 (ω) (5.9) if we set:

F 1 (ω) = j∈Z 1 2 | ψj (ω)| 2 -| ψj (-ω)| 2 S(ω) 2 -j F 2 (ω) = j∈Z 1 2 | ψj (ω)| 2 + | ψj (-ω)| 2 S(ω) 2 -2j
The functions F 1 and F 2 are both continuous. Indeed, for any j, ψj is continuous, because ψ ∈ L 1 (R). The function S is also continuous, and lower-bounded by a strictly positive constant. Finally, the sums converge uniformly on every compact subset of R, because of the assumption (5.6). This implies the continuity. Because of the hypothesis (5.5), F 1 (ω) > 0 when ω > 0. Moreover, for any ω > 0, F 1 (2ω) = 2F 1 (ω). By a compacity argument, there exist c > 0 such that: ∀ω > 0 F 1 (ω) ≥ cω which, as F 1 is odd, implies: ∀ω ∈ R ωF 1 (ω) ≥ cω 2 (5.10)

Similarly, there exist C > 0 such that: 5.11) By combining (5.10) and ( 5.11) with (5.9), we get that, for all ω = 0:

∀ω ∈ R F 2 (ω) ≤ Cω 2 (
1 2 j∈Z | ψj (ω)| 2 (1 -| χx (ω -δ2 -j )| 2 ) + j∈Z | ψj (-ω)| 2 (1 -| χx (-ω -δ2 -j )| 2 ) ≤ S(ω) 1 -exp -2 ω 2 x 2 (1 -2cδ + Cδ 2 )
If we take δ = c/C, this yields, for any a such that 1 < a ≤ ≤ S(ω) 1 -exp -2 ω 2

x 2 1 -

c 2 C ≤ S(ω) 1 -exp -2 ω 2 (ax) 2 ≤ 1 -exp -2 ω 2 (ax) 2 = 1 -| χax (ω)| 2
The bound S(ω) ≤ 1 comes from the Littlewood-Paley inequality (5.4).

The inequalities are also true for ω = 0 because all terms are then equal to zero.

Initialization

In this paragraph, we prove that Theorem 5.2 holds for n = 2. More precisely, we prove that, for any a > 1, there exist r > 0 such that Equation (5.7) is valid for n = 2.

Proof. For any real-valued function g ∈ L 2 (R): If the following inequality held, for some r > 0, a > 1:

1 2 j≤J | ψj (ω)| 2 + | ψj (-ω)| 2 ≤ 1 -| χra (ω)| 2
then Theorem 5.2 would be valid for n = 1 and it could be proven for n = 2 in the exact same way as in Section 5.3. Unfortunately, this inequality is not necessarily valid (in particular, it is never the case if the wavelet transform is unitary and the wavelets are band-limited).

The next lemma (proven at the end of the paragraph) nevertheless shows that the inequality is satisfied, if one allows the Gaussian function to be replaced by a more general function.

Lemma 5.6. There exist a real-valued positive function φ ∈ L 1 ∩ L 2 (R) such that:

| φ(ω)| 2 = 1 -O(ω 2 )
when ω → 0 (5.13) and:

∀ω ∈ R 1 2 j≤J | ψj (ω)| 2 + | ψj (-ω)| 2 ≤ 1 -| φ(ω)| 2 (5.14)
The rest of the proof consists in showing how to adapt Section 5.3, when the Gaussian function has been replaced by the φ of the lemma.

Because of (5.12), with φ defined as in Lemma 5.6, we have, for any real-valued function g ∈ L 2 (R): (5.15) Lemma 5.3 is still true when χ x is replaced by φ, because the only property of χ x which is really needed is its positivity. So for any δ ∈ R:

∀j ∈ Z || |f ψ j | φ|| 2 2 ≥ R | f (ω)| 2 | ψj (ω)| 2 | φ(ω -δ)| 2 dω
In Section 5.3, we used this same inequality, with a different δ for each value of j. Here, we do not need δ to vary as a function of j; however, we will need to consider different values of δ and to average the inequalities over these different values.

If we combine the last inequality with (5.15), we obtain:

∀δ ∈ R p∈P J ,|p|=2 ||U [p]f || 2 2 ≤ R | f (ω)| 2 (1 -| φ(ω -δ)| 2 ) j≤J | ψj (ω)| 2 dω
have a clear spatial or temporal structure [START_REF] Sermanet | Pedestrian detection with unsupervised multi-stage feature learning[END_REF][START_REF] Chen | Unsupervised deep haar scattering on graphs[END_REF]. It could possibly be further improved.

In this chapter, we focus on the first body of questions, and describe a few properties of the initial process that can be retrieved from the scattering coefficients. In finite dimension, we prove that the scattering operator preserves the norm when the W n are unitary (Theorem 6.2). We study in more detail the one-dimensional case, where all the X n take their values in R. We show that the scattering coefficients characterize the tail of the distribution of X 0 (Theorem 6.4), but not necessarily the distribution in the low values.

In Section 6.1, we define the generalized scattering, and explain its link with the regular scattering. In Section 6.2, we prove the energy preservation theorem. Section 6.3 is devoted to the one-dimensional case. We conclude with numerical experiments in Section 6.4.

The first two sections of this chapter come from [START_REF] Mallat | Deep learning by scattering[END_REF]].

Definition of the generalized scattering

In this paragraph, we explain the generalization of the regular scattering as defined in [START_REF] Mallat | Group invariant scattering[END_REF] which leads to the definition that we are going to study.

We consider a finite-dimensional version of the regular scattering, where signals are of finite dimension N and we use only a finite number J of wavelets. This is the setting of all concrete applications.

The scattering of a stationary process X 0 is written as the following sequence of computations:

X 0 ∈ R N ∀n ≥ 0 X n+1 = |W n X n | ∈ (R N ) J n+1
where W n is the wavelet transform operator:

W n : (g s ) 1≤s≤J n ∈ (R N ) J n → (g s ψ j ) 1≤s≤J n ,1≤j≤J ∈ (C N ) J n+1
and the scattering coefficients are the A n X n , n ∈ N, where A n is the averaging operator:

A n : (g s ) 1≤s≤J n ∈ (R N ) J n → (g s φ J ) 1≤s≤J n ∈ (R N ) J n
When J is large enough, g s φ J is approximately the spatial mean of g s :

∀s, ∀k = 1, ..., N (g s φ J ) k ≈ 1 N N l=1 (g s ) l def = mean(g s )

Proof of Lemma 6.3. We deduce the convergence E(||X n ||) → 0 from the weaker convergence ||E(X n )|| → 0, which comes from Equation (6.2.2). The proof relies on a compacity argument, and strongly uses the fact that X 0 takes its values in a finite-dimensional space.

For any z ∈ R N 0 and a > 0, we denote by B(z, a) the set {z ∈ R N 0 s.t. ||z -z|| < a}.

Let us fix M > E(||X 0 || 2 ), and > 0. Let z 1 , ..., z S be a finite number of elements of R N 0 such that:

B(0, M ) ⊂ S s=1 B(z s , )
We iteratively define:

φ 0 : Id : R N 0 → R N 0 ∀n ≥ 0, φ n+1 : x ∈ R N 0 → |W n (φ n (x) -E(X n ))| ∈ R N n+1
so that, for any n ∈ N, X n = φ n (X 0 ). These applications are all 1-Lipschitz: they are compositions of 1-Lipschitz operators. So if we denote by α : R N 0 → {z 1 , ..., z s } any measurable function such that ||z -α(z)|| < for any z ∈ B(0, M ), we have: In this thesis, we have studied the modulus of the wavelet transform. We have seen that this operator produces a representation of audio signals which is both discriminative (two signals that are different for a human person have different representations) and stable to transformations which a person can not hear (multiplication by a slow-varying phase in the time-frequency domain). In chapters 5 and 6, we have moreover seen how it could be used to construct deep representations.

E(||X n ||) = E(||X n ||1 ||X 0 ||<M ) + E(||X n ||1 ||X 0 ||≥M ) = E(||φ n (X 0 )||1 ||X 0 ||<M ) + E(||X n ||1 ||X 0 ||≥M ) ≤ E ||φ n (α(X 0 ))|| + ||φ n (α(X 0 )) -φ n (X 0 )|| 1 ||X 0 ||<M + E(||X n ||1 ||X 0 ||≥M ) ≤ E ||φ n (
Thanks to these properties, the modulus of the wavelet transform is a crucial tool for the analysis of audio signals. In audio processing, almost all representations used to analyze or classify sounds rely on the modulus of a time-frequency transform (spectrogram or scalogram, often under the form of mel-frequency cepstral coefficients). In recent works, the modulus of the wavelet transform is sometimes followed by sophisticated additional transformations [START_REF] Hinton | Deep neural networks for acoustic modeling in speech recognition[END_REF]. However, it is always the starting point of the representation.

The goal of this conclusion is to explain why the situation is somewhat similar in image processing, although it may be less obvious, and to describe future directions of research.

In image processing, time-frequency representations have also been the cause of important successes, with in particular descriptors like SIFT [START_REF] Lowe | Object recognition from local scale-invariant features[END_REF] or HOG [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF]. However, a recent and very successful trend in image processing consists in learning deep representations from end to end, using raw pixels as inputs [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF][START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF][START_REF] Szegedy | Going deeper with convolutions[END_REF]. In this case, no modulus of wavelet transform is explicitly computed.

Nevertheless, it is known that part of filters in the first layer of a neural network spontaneously tend to resemble Gabor filters, responding to particular orientations and scales [START_REF] Zeiler | Visualizing and understanding convolutional networks[END_REF]. So they are the same filters as in a wavelet transform, and they are also followed by a non-linearity, even if it is not exactly a modulus. Indeed, we are going to see that, at least in AlexNet [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF], the first half of the first convolutional layer is essentially the modulus of a wavelet transform. This observation is coherent with the results of [START_REF] Yosinski | How transferable are features in deep neural networks[END_REF], suggesting that the first layers of a neural network are relatively universal: they do not depend much on the dataset on which the representation has been learned. It is also in line with [START_REF] Perronnin | Fisher vectors meet neural networks: a hybrid classification architecture[END_REF], who show that replacing the first layers of a neural network by more traditional descriptors (Fisher vectors) simplifies the learning, while retaining a high classification accuracy.

In the first paragraph of this conclusion, we compare the filters of the first layer of AlexNet with the ones of a wavelet transform. In the second paragraph, we discuss the non-linearity. We conclude with preliminary remarks on the second layer.

The images are generated with Caffe [START_REF] Jia | Caffe: convolutional architecture for fast feature embedding[END_REF]. We use the version of AlexNet implemented in Caffe, which slightly differs from the original one.

First convolutional layer

AlexNet contains eight layers. The first five are convolutional, and the last three fullyconnected. At the first layer, there are 96 filters, of size 3 × 11 × 11. The first dimension corresponds to the three color channels of the input image, while the second and third dimensions are spatial. The 96 filters are displayed in color on Figure 7.1.

In the implementation of AlexNet, due to memory constraints, the filters of each convolutional layer are split in two halves, stored on different GPUs. There is no communication between the two GPUs, except in the third convolutional layer and in the fully-connected ones. Even if this is not imposed by the architecture, the first GPU tends to learn black-and-white filters, while the second one learns colored filters.

Here, we focus our analysis on the black-and-white filters, that is, the first 48 ones. They are displayed on Figure 7.2 (a). Except for numbers 11, 25, 43, 46 and 47, these filters have zero-mean and present oscillating patterns, with distinct orientations. They are similar to filters of a two-dimensional wavelet transform.

As shown in Figure 7.2 (b), their Fourier transforms are very well-localized in frequency. The distribution of characteristic frequencies in the plane is shown on Figure 7.3. It is relatively uniform across the different orientations. However, it is logarithmic in scale: the three circular rings of figure 7.3 (obtained one from each other by dilations of factor 2) contain approximately the same number of frequencies. This is the same distribution as in the case of a wavelet transform.

Figure 7.2 (c) shows, for each of the 48 filters, its closest approximation in a frame of real Gabor wavelets. Except for filters 11, 25, 43, 46 and 47, the approximation is good.

Non-linearity

In AlexNet (as in most deep networks), the filters are real-valued. This is a priori an important difference with the complex wavelet transforms used throughout this thesis. Moreover, the non-linearity used in AlexNet is a Rectified Linear Unit (ReLU: x → max(0, x)), followed by a max-pooling; it is not a modulus.

However, depending on the choice some parameters, the combination "real wavelet transform + ReLU + max-pooling" can produce coefficients very similar to the ones obtained with a complex wavelet transform followed by a modulus. Indeed, if ψ is a complex-valued wavelet, sufficiently localized in frequency, with characteristic frequency ν ∈ R 2 , then, for each real-valued function f , the convolved function f ψ is of the form:

f ψ(x) ≈ f 0 (x)e i ν,x (x ∈ R 2 )
where f 0 is low-frequency. which is exactly Equation (7.1).

This reasoning seems to apply for AlexNet. Figure 7.4 (a) displays, for the input image shown in (c), the coefficients returned by AlexNet after the first max-pooling layer (for the first 48 channels). In subfigure (b) are displayed the same coefficients, where the real filters have been replaced by complex ones, and the combination of ReLU and max-pooling has been replaced by a modulus. The coefficients in (a) and (b) are not exactly identical, but all the same very similar. In particular, in each case, the non-negligible coefficients are located in the same channels, at the same spatial position.

Second layer

At the second layer, there are 256 filters. Again, we focus on the first half. Each filter has dimensions 48 × 5 × 5. The first dimension corresponds to the indexes of the 48 first filters of the first layer. As first-layer filters are localized in the frequency domain, it can be seen a frequential index. The second and third dimensions are spatial variables.

As for the first layer, we would like to describe the transformation performed by these filters in terms of a simple operator like the wavelet transform. This question is still unsolved for us; in this last paragraph, we limit ourselves to preliminary considerations. The key point is to understand the interaction between frequential and spatial variables.

To partially overcome the difficulty raised by the three-dimensionality of filters, a possible approach is to use the fact that most filters at this layer can be written as a linear combination of the rotation group) [START_REF] Sifre | Rotation, scaling and deformation invariant scattering for texture discrimination[END_REF][START_REF] Oyallon | Deep roto-translation scattering for object classification[END_REF]. There are also more complicated shapes. Future work will consist in precisely analyzing these second-layer filters, and see to what extent they can be replaced by operators with simple analytic expressions. We can then determine what information it brings on the geometric properties of images that deep networks implicitly use, and see if it suggests possible ways to improve the first layers of deep representations.

Figure 1

 1 Figure 1.1: Example of a wavelet family. (a) (b) (c) Functions ψ 1 , ψ 0 , ψ -1 ; only the real part is displayed (d) (e) (f) Functions ψ1 , ψ0 , ψ-1

  of modulus of wavelet transform. On the left: the same function f as previously. On the right, from top to bottom: |f ψ 1 |, |f ψ 0 |, |f ψ -1 |.

Figure 1 . 2 :

 12 Figure 1.2: Wavelet transform and modulus of the wavelet transform

Figure 1 . 3 :

 13 Figure 1.3: Schematic illustration of the scattering transform

Figure 2 .

 2 Figure 2.1: Schematic representation of the sets involved in equations (2.13) and (2.14): the cone of positive hermitian matrices H + n (in light grey), its intersection with the affine subspace H b , and F ∩ H + n , which is a face of H + n .

Figure 2 . 2 :

 22 Figure 2.2: Real parts of sample test signals. (a) Gaussian white noise. (b) Sum of 6 sinuoids of random frequencies and amplitudes. (c) Scan-line of an image.

Figure 2 . 3 :

 23 Figure 2.3: Comparison of PhaseLift and Weak PhaseLift performance, for 64-sized signals, as a function of the number of measurements. Reconstruction rate, after Gerchberg-Saxton iterations (left) and proportion of rank one solutions (right).

Figure 2 .

 2 Figure 2.4 displays the performance of PhaseCut in the wavelet transform case. It shows that PhaseCut is stable up to around 5 -10% of noise. Indeed, the reconstructed x usually satisfies (|Ax|, |Ax|) = |Ax| -|Ax| 2 ≤ b noise 2 , which is the best we can hope for. Wavelet transform is a case where the underlying phase retrieval problem may present instabilities, therefore the reconstruction error (x, x) is sometimes much larger than(|Ax|, |Ax|). This remark applies especially to sums of sinusoids, which represent the most unstable case.When all coefficients of Ax have approximately the same amplitude, PhaseLift and PhaseCut produce similar results, but when Ax is sparse, PhaseLift appears less stable. We gave a qualitative explanation of this behavior at the end of Paragraph 2.3.4 which seems to be confirmed by the results in Figure2.4. Indeed, the performance of PhaseLift and PhaseCut are equivalent in the case of Gaussian random filters (where measurements are never sparse), they are a bit worse

Figure 2 . 4 :

 24 Figure 2.4: Mean reconstruction errors versus amount of noise for PhaseLift and PhaseCut, both in decimal logarithmic scale, for three types of signals: Gaussian white noises, sums of sinusoids and scan-lines of images. Both algorithms were followed by a few hundred Gerchberg-Saxton iterations.

Figure 3 Figure 3 . 2 :

 332 Figure 3.1: Cauchy wavelets of order p = 5 for j = 2, 1, 0, -1, a = 2

  Re (e iφ g + ) Remark 3.4. Although Corollary 3.2 holds for only two wavelets and does not require |f ψ s | = |g ψ s | for each s ∈ Z, the reconstruction of f from only two components, |f ψ j | and |f ψ k |, is very unstable in practice. Indeed, ψj and ψk are concentrated around characteristic frequencies of order 2 -j and 2 -k . Thus, from f ψ j and f ψ k (and even more so from |f ψ j | and |f ψ k |), reconstructing the frequencies of f which are not close to 2 -j or 2 -k is numerically impossible. It is an ill-conditioned deconvolution problem.

  Let α > 0 be fixed. Let F, G : H → C be holomorphic functions such that, for some M > 0: R |F (x + iy)| 2 dx < M and R |G(x + iy)| 2 dx < M ∀y > 0 (3.8)

  y→0 + |F (. + iy)| = lim y→0 + |G(. + iy)| and |F (. + iα)| = |G(. + iα)|, we have D F = D G and DF = DG .We show that it implies a relation between the B's, that is, a relation between the zeros of F and G. From this relation, we will be able to prove that F and G have the same zeros and that, up to a global phase, they are equal.

Figure 3 . 3 :

 33 Figure 3.3: (a) Wavelet transform of f 1(b) Wavelet transform of f 2 (c) Wavelet transform of f 1 + f 2 (solid blue) and f 1 -f 2 (dashed red) (d)Modulus of the wavelet transforms of f 1 + f and f 1 -f 2 ; the two modulus are almost equal In each column, each graph corresponds to a specific frequency; the highest frequency is on top and the lowest one at bottom. For complex functions, only the real part is displayed.

  j+1

Figure 3 . 4 :

 34 Figure 3.4: examples of signals: (a) realization of a Gaussian process (b) sum of sinusoids (c) piecewise regular

Figure 3 . 5 :Figure 3 . 6 :

 3536 Figure 3.5: Reconstruction results for the three considered classes of signals. Left column: our algorithm. Right column: alternate projections (Gerchberg-Saxton)

  ,b and continuous on B a,b . It is upper-bounded and takes negative values on ∂B a,b . Moreover, L(z) → -∞ when Re (z) → ±∞. From the maximum principle, this function must be negative on B a,b .

Figure 3 . 7 :

 37 Figure 3.7: Positions of the points used in the definition of f

Figure 4 .

 4 Figure 4.1: Example of wavelets; the figure shows ψ j for J -5 ≤ j ≤ J in the Fourier domain. Only the real part is displayed.

Figure 4 . 2 :

 42 Figure 4.2: ψ J , ..., ψ j+1 , ψ j (in the Fourier domain), along with ψ low j and ψ high j

  Compute f 10: Refine f with Gerchberg-Saxton Output: f

4 Figure 4 . 3 :

 443 Figure 4.3: Four window signals, whose supports cover the interval in which errors have been detected

Figure 4 . 4 :

 44 Figure 4.4: For an audio signal, the reconstructed value of f ψ low 7 at the scale 7 of the multiscale algorithm, in modulus (dashed line); the solid line represents the ground true. (a) Before the error correction step (b) After the error correction step

Figure 4 . 5 :

 45 Figure 4.5: Mean reconstruction error (4.9) as a function of the noise, for an audio signal representing a human voice. (a) Maximal number of iterations per local optimization step equal to 200 (b) Maximal number equal to 2000.

Figure 4 . 6 ,

 46 we plot the mean reconstruction error obtained with

Figure 4 . 6 :

 46 Figure 4.6: Mean reconstruction error, as a function of the size of the signal; the solid blue line corresponds to the multiscale algorithm and the dashed red one to the non-multiscale one.

  We describe our experimental setting inParagraph 4.4.1. In Paragraph 4.4.2, we give detailed numerical results for various types of signals. In Paragraph 4.4.3, we use our algorithm to investigate the stability to noise of the underlying phase retrieval problem. Finally, in Paragraph 4.4.4, we study the influence of various parameters on the quality of the reconstruction. Code and audio examples are available at: http://www.di.ens.fr/ ~waldspurger/wavelets_phase_retrieval.html

Figure 4 . 7 :Figure 4 . 8 :Figure 4 . 9 :

 474849 Figure 4.7: Realization of a Gaussian process (left) and modulus of its wavelet transform (right)

Figure 4 .

 4 Figure 4.10: Mean reconstruction error as a function of the noise, for lines extracted from images, of size N = 256 or 10000

Figure 4 .Figure 4 .Figure 4 .

 444 Figure 4.11: Random sum of sinusoids, multiplied by a window function (left) and modulus of its wavelet transform (right)

Figure 4 .

 4 Figure 4.14: mean reconstruction error as a function of the noise, for the audio signal "Rimsky-Korsakov"

Figure 4 .

 4 Figure 4.15: mean reconstruction error as a function of the noise, for the audio signal "I'm sorry"

  error on the modulus = j || |f ψ j | -|f rec ψ j | || 2

Figure 4 .

 4 Figure 4.16: error on the signal (4.10) as a function of the error on the modulus of the wavelet transform (4.9), for several reconstruction trials; the red line y = x is here to serve as a reference (a) Gaussian signals (b) lines from images (c) sums of sinusoids (d) audio signal "I'm sorry"

Figure 4 .

 4 Figure 4.18: Four wavelet families. (a) Morlet (b) Morlet with dilation factor 2 1/8 (c) Laplacian (d) Gammatone

Figure 4 .

 4 Figure 4.19: Mean reconstruction error as a function of the noise for the four wavelet families displayed in 4.18. (a) Lines from images (b) Audio signal "I'm sorry"

Figure 4 .

 4 Figure 4.19 displays the mean reconstruction error as a function of the noise, for two classes of signals: lines randomly extracted from natural images and audio signals.Morlet wavelets have a fast decay and consecutive wavelets overlap well. This does not depend upon the dilation factor so the reconstruction performances are similar for the two Morlet families(figures 4.19a and 4.19b).Laplacian wavelets are similar, but the overlap between consecutive wavelets is not as good.

Figure 4 .

 4 Figure 4.20: for the audio signal "I'm sorry", reconstruction error as a function of the maximal number of iterations (a) with 0.01% of noise (b) with 0.6% of noise

4 )

 4 Let m ∈ R N and K ∈ N * be fixed. We consider the problem:Find g ∈ C N s.t. |g| = mand Supp(ĝ) ⊂ {1, ..., K}

Figure 5 . 1 :

 51 Figure 5.1: Schematic illustration of the scattering transform: the tree on the right represents the cascade of modulus of wavelet transforms; the output scattering coefficients are on the left.

Figure 5 . 2 :

 52 Figure 5.2: Illustration of the shift towards low frequencies due to the modulus. (a) f ψ j(real part) (b) f ψ j (in modulus) (c) |f ψ j | (d) |f ψ j |.Observe that f ψ j and |f ψ j | are localized on frequency bands of the same width, but that, for |f ψ j |, the band is centered around 0, and thus globally lower in frequency.

2 | 5 | f | 2 |

 252 ψj | 2 1 -| χar n (. -δ j )| 2 ψj | 2 (1 -| χar n (. -δ j )| 2 )We consider real signals so we can symmetrize the filter.

Figure 5 . 3 :

 53 Figure 5.3: Schematic summary of the proof of Theorem 5.2 148

  For any j ∈ Z, x ∈ R * + , δ j ∈ R:|| |f ψ j | χ x || 2 2 ≥ ||f ψ j (χ x e 2πiδ j . )|| 2 2 = R | f (ω)| 2 | ψj (ω)| 2 | χx (ω -δ j )| 2 dωProof. For any t, because χ x is a positive function:|f ψ j | χ x (t) = |(f ψ j )e -2πiδ j . | χ x (t) = |(f ψ j )e -2πiδ j . | |χ x |(t) ≥ |((f ψ j )e -2πiδ j . ) χ x (t)| = |e -2πiδ j t f ψ j (χ x e 2πiδ j . ) (t)|= |f ψ j (χ x e 2πiδ j . )(t)|

1 2 j∈Z| 2 2 -ω 2 x 2

 2222 ψj (ω)| 2 (1 -| χx (ω -δ2 -j )| 2 ) + j∈Z | ψj (-ω)| 2 (1 -| χx (-ω -δ2 -j )| 2 ) This function is never 0 if ω = 0: if ω > 0, there exist j such that | ψj (ω)| > | ψj (-ω)| ≥ 0 so S(ω) > 0; as S is even, we also have S(ω) > 0 if ω < 0.The function y → 1 -e +y is concave so, for any ω =

|

  ψj (ω)| 2 (1 -| χx (ω -δ2 -j )| 2 ) + j∈Z | ψj (-ω)| 2 (1 -| χx (-ω -δ2 -j )| 2 )

|

  ψj (ω)| 2 + | ψj (-ω)| 2 dω(5.12)

j 1 2 ≤

 12 ≤J j 2 ≤J || ||f ψ j 1 | ψ j 2 | || 2 j≤J ||f ψ j || 2 2 -|| |f ψ j | φ|| 2 2

Figure 6 . 4 :

 64 Figure 6.2: (a) Laplace probability density function p(x) = e -x 1 x≥0 (b) Function 2f , with f defined as in Equation (6.4), and the pairs (S n , E(X n )) for n = 0, ..., 69 (c) Pareto-like probability density function p(x) = (8/7)(1 + |x -1|) -3 1 x≥0 (d) Function 2f , and the pairs (S n , E(X n )) for n = 0, ..., 10

Figure 7 . 1 :Figure 7 . 3 :

 7173 Figure 7.1: The 92 filters of the first layer of AlexNet. Each filter has size 3 × 11 × 11.

  the complex wavelet ψ by its real part Re (ψ) = (ψ + ψ)/2, then, because f is real-valued:f Re (ψ)(x) = Re (f ψ)(x) = Re (f 0 (x)e i ν,x )As f 0 is a low-frequency function, it can be considered approximately constant in the neighborhood of each point x 0 : f 0 (x) ≈ |f 0 (x 0 )|e iφ 0 , where φ 0 is a phase depending on x 0 . So:f Re (ψ)(x) ≈ Re (|f 0 (x 0 )|e i(φ 0 + ν,x ) ) = |f 0 (x 0 )| cos(φ 0 + ν, x )Combining with ReLU and max-pooling yields:max-pool(ReLU(f Re (ψ)))(x) ≈ max ||y-x|| 1 ≤M max 0, |f 0 (x 0 )| cos(φ 0 + ν, y ) = |f 0 (x 0 )| max ||y-x|| 1 ≤M max 0, cos(φ 0 + ν, y )where M is the spatial extent of the max-pooling. If 2M is larger than the period of the cosine function, 2π/||ν|| 2 , then the cosine always reaches 1 on the set {||y -x|| 1 ≤ M }, whatever the value of x is. So: max-pool(ReLU(f Re (ψ)))(x) ≈ |f 0 (x 0 )| (7.2)

Figure 7 . 6 :

 76 Figure 7.6: Frequential components for the filters of the second layer of AlexNet. Each component is pictured as a function from the frequency plane R 2 to R.

Table 2 .

 2 1: Percentage of perfect reconstruction from |Ax|, over 300 test signals, for the three different operators A (columns) and the three algorithms (rows).

		Fourier Random Filters Wavelets
	Gerchberg-Saxton	5%	49%	0%
	PhaseLift with reweighting	3%	100%	62%
	PhaseCut	4%	100%	100%
		Fourier Random Filters Wavelets
	Gerchberg-Saxton	0.9	1.2	1.3
	PhaseLift with reweighting	0.8	exact	0.5
	PhaseCut	0.8	exact	exact

Table 2 .

 2

2: Average relative signal reconstruction error (x, x) over all test signals that are not perfectly reconstructed, for each operator A and each algorithm.

Table 2 .

 2 . The resulting operator A is the discrete Fourier transform 3: Average relative error (|Ax|, |Ax|) of coefficient amplitudes, over all test signals that are not perfectly reconstructed, for each operator A and each algorithm.

		Fourier Random Filters Wavelets
	Gerchberg-Saxton	9.10 -4	0.2	0.3
	PhaseLift with reweighting 5.10 -4	exact	8.10 -2
	PhaseCut	6.10 -4	exact	exact

of x multiplied by each filter h j of size p

  Phase retrieval problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Theoretical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 A general phase retrieval algorithm, PhaseCut (chapter 2) . . . . . . . . 1.2 Phase retrieval for the wavelet transform . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Wavelet transform modulus . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Interest of the phase retrieval problem . . . . . . . . . . . . . . . . . . . 1.2.3 Phase retrieval for the Cauchy wavelet transform (chapter 3) . . . . . . . 1.2.4 Phase retrieval for wavelet transforms: a non-convex algorithm (chapter 4) 1.3 Scattering transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Definition of scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Success and open question . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Exponential decay of scattering coefficients (chapter 5) . . . . . . . . . . 1.3.4 Generalized scattering (chapter 6) . . . . . . . . . . . . . . . . . . . . . . 2 A general phase retrieval algorithm, PhaseCut Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Phase recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Greedy optimization in the signal . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Splitting phase and amplitude variables . . . . . . . . . . . . . . . . . . . 2.1.3 Greedy optimization in phase . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Complex MaxCut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Interior point methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 First-order methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Block coordinate descent . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Initialization & randomization . . . . . . . . . . . . . . . . . . . . . . . .

	Contents
	1 Introduction
	1.1

"natural" in the sense that it appears in other fields than phase recovery

which is true by hypothesis.

A few examples are available at http://www.di.ens.fr/ ~waldspurger/wavelets_phase_retrieval.html.

E(X n 2s+2 -3 )

Remerciements Acknowledgments

The authors are grateful to Richard Baraniuk, Emmanuel Candès, Rodolphe Jenatton, Amit Singer and Vlad Voroninski for very constructive comments. In particular, Vlad Voroninski showed in [Voroninski, 2012] that the argument in the first version of this text, proving that PhaseCutMod is tight when PhaseLift is, could be reversed under mild technical conditions and pointed out an error in our handling of sparsity constraints. AA would like to acknowledge support from a starting grant from the European Research Council (project SIPA), and SM acknowledges support from ANR grant BLAN 012601.

Second part: we define, for any z ∈ C such that -2a n l -1 + a n l < Im z < w l : H(z) = F (1) (z + iw l )F (1) (z + i(2a n l -1 -a n l )) -F (2) (z + iw l )F (2) (z + i(2a n l -1 -a n l ))

We write:

From the first part:

Moreover, if we set y l = w l -2a n l -1 + a n l :

H(x + iy l ) = F (1) (x + i(2a n l -1 -a n l ))F (1) (x + iw l ) -F (2) (x + i(2a n l -1 -a n l ))F (2)

Thus, we also have:

We apply Lemma 3.26 with a = 0, b = y l . For Im z = (a -1) 2 a n l -2 and |Re z| ≤ (M -(l + 1)µ)a j+K : .

From the definition of w l , one may check that (w l ) is a decreasing sequence which converges to 2a j+K a+1 when l goes to ∞. So, for any l ≥ 0:

This solves the problem.

The number of elements in the lists is exponential in the size of the supports of ψj and ψj-1 , so this algorithm has a prohibitive complexity when the supports become large. Otherwise, our numerical experiments show that it works well.

Error correction

When the modulus are noisy, there can be errors during the phase propagation step. The local optimization generally corrects them, if run for a sufficient amount of time, but, for the case where some errors are left, we add, at each scale, a multigrid error correction step. This step does not totally remove the errors but greatly reduces their amplitude.

Principle

First, we determine the values of n for which f ψ low j [n] and f ψ high j+1 [n] seems to have been incorrectly reconstructed. We use the fact that f ψ low j and f ψ high j+1 must satisfy:

The points where this equality does not hold provide a good estimation of the places where the values of f ψ low j and f ψ high j+1 are erroneous. We then construct a set of smooth "windows" w 1 , ..., w S , whose supports cover the interval on which errors have been found (see figure 4.3), such that each window has a small support. For each s, we reconstruct (f ψ low j ).w s and (f ψ high j+1 ).w s , by expressing these functions as the solutions to phase retrieval problems of small size, which we can solve by the exhaustive search method described in Paragraph 4.2.2.

As w s is smooth, the multiplication by w s approximately commutes with the convolution by ψ j , ψ j+1 :

The wavelets ψ j and ψ j+1 have a small support in the Fourier domain, if we truncate them to the support of w s , so we can solve this problem by exhaustive search, and reconstruct (f.w s ) ψ j and (f.w s ) ψ j+1 .

From (f.w s ) ψ j and (f.w s ) ψ j+1 , we reconstruct (f ψ low j ).w s ≈ (f.w s ) ψ low j and (f ψ high j+1 ).w s ≈ (f.w s ) ψ high j+1 by deconvolution.

As it holds for any δ ∈ R, we have, for any positive c ∈ L 1 (R) whose integral over R is 1:

We limit ourselves to the case where c is even. As | φ| 2 is also even, the last inequality yields, by using the fact that f is real:

The conclusion comes from a last lemma, proven at the end of the paragraph.

Lemma 5.7. If we take c(δ) = 1 √ π exp(-δ 2 ), then there exists x > 0 such that:

We now give the proofs of Lemmas 5.6 and 5.7.

Proof of Lemma 5.6. Let γ : R → R be any even, rapidly decreasing and band-limited function such that R γ 2 = 1. We set φ 0 = γ 2 . This is also an even, rapidly decreasing and band-limited function.

Because R γ 2 = 1, we have φ0 (0) = 1. As φ 0 is even, φ 0 (0) = 0. But the second derivative is strictly negative: φ 0 (0) = -(2π) 2 R t 2 φ 0 (t)dt < 0. We deduce from these relations that there exists α > 0 such that:

when ω → 0 (5.16)

Let us show that, for M large enough, φ : t → M -1 φ 0 (M -1 t) satisfies the desired properties. By construction, it is a real-valued positive function. It is rapidly decreasing, so φ ∈ L 1 ∩ L 2 (R).

The property (5.13) holds; only the inequality (5.14) is left to prove. Because of Equation (5.16) and because φ0 is compactly-supported, there exists α > 0 such that for all ω ∈ Supp( φ0 ):

By the assumption (5.6), | ψ(ω)| = O(|ω| 1+ ) when ω → 0, for some > 0. This implies that:

Because this sum is moreover bounded by 1 on all R, there exists A > 0 such that:

If M ≥ A/α, then, on the support of φ:

Outside the support of φ, the inequality is also true, because of the Littlewood-Paley condition (5.4). So Equation (5.14) holds.

Proof of Lemma 5.7. Let us define:

We are going to prove that F has the following three properties:

These three properties imply that F is bounded by 1 -| χx | 2 on all R, for x small enough. This assertion relies on a compacity argument; as it is relatively straightforward, we do not prove it. The first property is an immediate consequence of the Littlewood-Paley inequality (5.4) and of the fact that | φ| 2 c > 0 over R.

The second one is a consequence of a fact that has been explained in the proof of Lemma 5.6:

when ω → 0

For the last one, we remark that, for any ω ∈ R such that ω ≥ 1:

As all the functions are even, this is also true for ω ≤ -1. So when |ω| is large enough:

This proves the third property and concludes.

Chapter 6

Generalized scattering

In this last chapter, we define a generalization of the scattering transform for stationary processes, by keeping the structure in cascade described in Section 5.1 but replacing the wavelet transform by arbitrary linear functions.

The resulting operator takes as input a random process X 0 and, from it, defines a sequence of processes by the iterative application of the following operation:

where the W n are arbitrary linear operators and |.| denotes the pointwise modulus. The scattering coefficients are, by definition, the E(W n X) for n ∈ N.

The interest of this generalized scattering is to offer a mathematical framework for the theoretical analysis of deep neural networks. Indeed, it models the architecture of a neural network, with the modulus used as non-linearity.

The first questions that arise are again: for a given choice of W n , what information does the generalized scattering preserve about the input process? What do scattering coefficients say about the regularity of the input process? Which processes have the same scattering coefficients? For neural networks, these questions have been empirically studied in [START_REF] Simonyan | Deep inside convolutionnal networks: visualising image classification models and saliency maps[END_REF][START_REF] Mahendran | Understanding deep image representations by inverting them[END_REF][START_REF] Dosovitskiy | Inverting convolutional networks with convolutional networks[END_REF].

In the future, we would also be interested in studying the learning of the linear operators W n . Indeed, while, in the regular scattering, the W n are fixed (they represent the wavelet transform), they can here be learned from the data. We are especially interested in unsupervised learning: how to adapt the W n to the law of the input random process, so that the resulting scattering operator yields a relevant representation of realizations of this process? The question of unsupervised learning has attracted a great deal of attention in the last ten years [START_REF] Bengio | Representation learning: a review and new perspectives[END_REF]. On large-scale datasets, it tends to be supplanted by purely supervised learning [START_REF] Russakovsky | Imagenet large scale visual recognition challenge[END_REF], but it is competitive when labeled data is less abundant or does not If X is a sufficiently ergodic stationary process, then the spatial mean is an approximation of the expectation: ∀k = 1, ..., N mean(g s ) ≈ E((g s ) k )

So the scattering coefficients are approximately the set of E(X n ), n ∈ N.

If the wavelets have been well-chosen, the wavelet transform has the set of constant signals as kernel, and it is unitary on the set of signals with mean zero. So:

And we arrive at Definition (6.1).

We summarize this definition. Let an increasing sequence (N n ) n∈N of integers be fixed. For each n, let W n : R Nn → R N n+1 be a linear unitary operator. Starting from a random process X 0 , taking its values in R N 0 , we iteratively define:

The scattering coefficients of X 0 are defined as {E(X n )} n∈N .

6.2 Energy preservation

One-dimensional case

We shall prove an energy preservation result. To simplify the exposition of the proof, we begin with the one-dimensional case: X 0 is a real-valued random variable, with finite secondorder moment.

In dimension 1, there are only two unitary operators: Id or -Id. The modulus removes the sign, so the generalized scattering reduces to:

From the definition, we immediately see that:

So, for any N ≥ 0:

Our first result is that this equality goes to the limit N → +∞: the l 2 -norm of the sequence of scattering coefficients (E(X n )) n∈N is the norm of X 0 .

Theorem 6.1. Let X 0 be such that E(||X 0 || 2 ) < +∞. Then:

Proof. From Equation (6.3), it is enough to show that E(X 2 n ) goes to zero when n goes to infinity. This equation also implies that E(X n ) → 0 when n → +∞.

For any n ≥ 1, X n is positive. So if the X n 's were uniformly bounded by a constant M > 0, we would have, when n → +∞:

Here, the processes X n may not be uniformly bounded. However, they are uniformly bounded on an event of arbitrarily small probability.

Indeed, let us fix any M > E(X 2 0 ). For any n ≥ 1, X n ≥ 0 so:

which implies (using Equation ( 6.3) for the second inequality):

We thus have the inclusion, for any n ≥ 1:

and:

Letting M go to infinity proves:

Generalization to higher dimensions

We now remove the assumption of dimension one and return to the generalized scattering as defined in Equation ( 6.2):

Then:

Proof. The proof follows the one of Theorem 6.1. We first remark that, for any n ∈ N:

which implies a formula analog to Equation ( 6.3):

To prove the theorem, it suffices to show that E(||X n || 2 ) → 0 when n → +∞. We temporarily admit a lemma:

From this lemma, we can deduce the result. As the reasoning is the same as in the onedimensional case, we only review its main steps.

We start by fixing M > E(||X 0 || 2 ). For any n ≥ 1:

from which we deduce:

Letting n then M go to infinity yields:

This is what was needed; only Lemma 6.3 is left to show.

Moreover, for any n ≥ 1, using the positivity of the coordinates of X n :

Letting M and go respectively to +∞ and 0 implies:

Characterization of the distribution tail

In this section, we focus on the one-dimensional case:

We assume that the process X 0 takes only positive values and is not almost surely bounded. We show that the sequence of generalized scattering coefficients (E(X n )) n∈N precisely describes the distribution tail of X 0 . We will not tackle the subject under this point of view, but it is somewhat related to the problem of estimating a probability density function from samples of the corresponding distribution [START_REF] Scott | Multivariate Density Estimation: Theory, Practice, and Visualization[END_REF][START_REF] Antoniadis | Wavelets in statistics: a review[END_REF].

We define: This decreasing function fully characterizes the distribution of X 0 . The next theorem shows that the knowledge of (E(X n )) n∈N gives an equivalent of this function in +∞. So the scattering coefficients precisely describe the distribution of X 0 in the high values, although they do not uniquely determine the law of X 0 . Theorem 6.4. When n → +∞, if X 0 takes positive values and is not almost surely bounded:

The sequence (E(X n )) n∈N thus provides an equivalent of f at infinity, because n-1 s=0 E(X s ) goes to infinity with n (it is proven at the beginning of Paragraph 6.3.4).

Proof

To simplify the notations, we set:

We iteratively define:

By definition, we have X n = φ n (X 0 ), and in particular E(X n ) = E(φ n (X 0 )). Figure 6.1 illustrates this definition. For n large enough, we see that φ n has two distinct parts: on [0; S n ], it is an oscillating function with small values and, on [S n ; +∞[, we have φ n (x) = x-S n (this is proven by iteration over n). The value of E(X n ) is the sum of the expectations of these two parts:

) Informally, at infinity, these two terms balance each other, so when n goes to +∞:

To rigorously prove this equivalence, we begin with upper and lower bounds for E(X n ). We define:

∀n ≥ 0

The next lemma is proven in Paragraph 6.3.2.

Lemma 6.5. For any n ≥ 1:

To establish Theorem 6.4, we have to bound the error term 2 max(0, m n-1 -E(X n-1 )).

We first list a few useful properties, which can be directly deduced from the definitions. Their detailed proof is in Paragraph 6.3.3. Lemma 6.6. For all n ≥ 0:

and:

-1 (6.9)

and:

In any case, the above properties imply:

These properties do not directly show that the error term 2 max(0, m n-1 -E(X n-1 )) of Lemma 6.5 is small. However, they allow to show that there are infinitely large values for which it is small. Lemma 6.7. Let c ∈]0; 1/2[ be fixed. There exists an extraction φ : N → N such that:

The proof of this lemma is in Paragraph 6.3.4. We immediately rewrite it under a more practical form: Corollary 6.8. Let α > 2 be fixed. There exists an extraction φ : N → N such that:

Proof of Corollary 6.8. If we use the extraction of Lemma 6.7 for a value of c such that 2 1-c ≤ α, then, for any n ≥ 1, from Lemma 6.5:

Using again the properties of Lemma 6.6, we can show that, for n large enough, if n satisfies the equation of Corollary 6.8, then so does n + 2. It is proven in Paragraph 6.3.5. Lemma 6.9. Let α ∈]2; 5/2[ be fixed. For any n large enough, if:

Combining this lemma and Corollary 6.8, we obtain that, for all α > 2:

) and m 2n+2 = E(X 2n+1 ) for all n large enough As α can take any value larger than 2 and, by Lemma 6.5, E(X n ) ≥ 2f (S n ) for all n:

) and m 2n+1 = E(X 2n ) when n goes to infinity or E(X 2n+1 ) ∼ 2f (S 2n+1 ) and m 2n+2 = E(X 2n+1 ) when n goes to infinity

Let us for example assume that we are in the first case: E(X 2n ) ∼ 2f (S 2n ). A final lemma, proven in Paragraph 6.3.6, concludes.

Lemma 6.10.

) and m 2n+1 = E(X 2n ) when n goes to infinity, then E(X 2n+1 ) ∼ 2f (S 2n+1 ) when n goes to infinity 6.3.2 Proof of Lemma 6.5

Proof of Lemma 6.5. For any n ≥ 1, when x ≥ 0:

From the first equality of the proof, this means:

Replacing x by X 0 and taking the expectation gives:

)) (6.5)

Proof of Lemma 6.6

Proof of Lemma 6.6. Equation (6.6): for any x ∈ [0; S n ]:

For any x ∈ [S n ; S n+1 ]:

Equation ( 6.7): for any n, X n only takes positive values.

Equation ( 6.8): we remark that, if E(X n ) ≥ E(X n+1 ), then:

Indeed, the functions on both sides of the inequality are affine on the following intervals:

On the last interval, both functions have a slope equal to 1. So to prove the inequality, it suffices to verify that it holds at the junction points:

which can be done by a simple computation. Replacing x by X n in (6.12), then taking the expectation yields:

Equation ( 6.9): this proof is similar to the previous one.

If E(X n ) ≤ E(X n+1 ), then:

where we have set:

By the same reasoning as in the proof of Equation ( 6.8), this inequality is true because the two functions involved have a slope equal to 1 at infinity, and because the inequality holds in 0, E(X n ) and E(X n ) + E(X n+1 ).

If we replace x by X n and take the expectation:

The proof of the other inequality is easier. By the triangular inequality:

If we replace x by X n and take the expectation:

-1 (6.9) Equation ( 6.10): if E(X n ) ≤ E(X n+1 ), then, by the previous equation, E(X n+2 ) ≤ E(X n+1 ).

Moreover:

It allows us to show the following inequality:

where:

By the same argument as in the proofs of Equations ( 6.8) and ( 6.9), it suffices to remark that the functions on both sides of the inequality have a slope equal to 1 at infinity and to check the inequality at points:

which can be done by a simple computation. Replacing x by X n and computing the expectation gives:

), then, by Equation (6.9):

), then, by Equation (6.8):

So in any case:

)) (6.11)

Proof of Lemma 6.7

Proof of Lemma 6.7. We need to show that there exist arbitrarily large values of n for which the following equation holds:

By contradiction, we assume that there exists N ∈ N such that the equation is never satisfied when n ≥ N . It means that, for all n ≥ N :

We will show that:

This is absurd because then, S n = n-1 k=0 E(X k ) converges to a finite limit L. By continuity, f (S n ) → f (L) so, from Lemma 6.5, E(X n ) ≥ 2f (S n ) ≥ 2f (L) for all n ≥ 1. So E(X n ) is lower bounded by a strictly positive constant and Equation (6.15) does not hold.

We distinguish the values of n, depending on which of the following two properties they satisfy:

The following assertions are true; they are proven at the end of the proof.

(1) If n ≥ N and P 1 (n), P 1 (n + 1), then:

(2) If n ≤ n and P 1 (n), P 2 (n + 1), ..., P 2 (n ), then:

These properties imply that there exists C > 0 such that, for any n, n , n with N ≤ n < n < n :

If P 1 (n), ..., P 1 (n -1), P 2 (n ), ..., P 2 (n -1) then

Indeed, because of Property (1), for any k ≥ 0:

For the last inequality, we have used Equation (6.7) of Lemma 6.6.

Because of Property ( 2):

From relations (6.17

So in any case, from Equation (6.7) of Lemma 6.6:

and:

which proves Equation (6.16).

If P 1 (n) is satisfied for all n large enough, then, by Property (1), (E(X n )) n∈N decays geometrically and n≥N E(X n ) < +∞. On the other hand, if P 2 (n) is satisfied for all n large enough, then, by Property ( 2), we also have n≥N E(X n ) < +∞. We can thus assume that there exist arbitrarily large values of n such that P 1 (n) holds, and arbitrarily large values such that P 2 (n) holds.

Even if we have to replace N by a larger integer, we can assume that P 1 (N ) holds.

We define the sequence of integers (n s ) s∈N * such that n 1 = N and:

P 1 (n 1 ) P 1 (n 1 + 1) . . . P 1 (n 2 -1) P 2 (n 2 ) . . . P 2 (n 3 -1) P 1 (n 3 ) . . .

From Equation ( 6.16):

So the proof is over if we show that s≥0 E(X n 2s+1 ) < +∞. This is a consequence of the following equation:

The proof of this equation is done at the end of the paragraph.

Proof of the properties ( 1) and ( 2).

Property (1): we begin with the case where P 1 (n + 2) holds.

In this case, by definition of P 1 (n + 2), m n+2 = E(X n+1 ) so, by Equation ( 6.14):

As E(X n+1 ) is positive, this equation in particular implies E(X n-1 ) ≥ E(X n ) so, by Equation (6.8) Lemma 6.6:

Combining the last two equations:

Computing the maximum of the right side when E(X n ) varies in [0; E(X n-1 )] yields:

Using the inequality c < 1/2 and the fact that, for any positive x,

Let us now handle the case where P 1 (n+2) does not hold. From Equation (6.6) of Lemma 6.6,it

), then, from Equation (6.9) of Lemma 6.6:

-1

As we have seen that E(X n+1 ) < E(X n )/2:

Property ( 2): because P 1 (n), P 2 (n + 1), ..., P 2 (n ),

The result follows from the fact that m n = max(E(X n -1 ), m n -1 -E(X n -1 )) is always positive.

Proof of Equation (6.18). We divide the proof in three cases:

Indeed, we use Property (2) for n = n 2s+2 -1 and n = n 2s+3 -1:

In particular:

If, on the other hand, E(X n 2s+3 -3 ) > E(X n 2s+3 -2 ), then, by Equation ( 6.11):

Then Equation (6.19), combined with the fact that E(X n 2s+3 -1 ) ≤ 2E(X n 2s+3 -2 ) yields:

This upper bound is increasing in E(X n 2s+2 -1 ) over [0; E(X n 2s+2 -2 )/2], so:

In either case, we have, as announced:

If n 2s+2 -n 2s+1 is odd, Property (1) implies:

Combining the previous two equations:

), then, again by Property (1):

so we reach again the same conclusion:

Because P 1 (n 2s+2 -1) and P 2 (n 2s+2 ) hold, and because

As E(X n 2s+2 -2 ) > E(X n s+2 -3 ), Equation (6.10) of Lemma 6.6 implies that:

This upper bound is increasing in E(X n 2s+2 -1 ) on [0; E(X n 2s+2 -2 )/2]: it can be seen by computing the derivative and using the fact that:

So, in the upper bound, we can replace E(X n 2s+2 -1 ) by E(X n 2s+2 -2 )/2 and get:

When E(X n 2s+2 -2 ) varies, the upper bound attains its maximum in E(X n 2s+2 -2 ) = 5 4 E(X n 2s+2 -3 ), which yields:

By Property (1), as n 2s+2 -n 2s+1 is even:

). For any n = n 2s+2 , ..., n 2s+3 -1, P 2 (n) holds so, as m n = max(E(X n-1 ), m n -E(X n-1 )):

By iteration:

In particular:

From Equation (6.11) of Lemma 6.6:

Now we show that:

Indeed, by the same reasoning as above, as P 1 (n 2s+2 -1) and P 2 (n 2s+2 ) hold:

But, as E(X n 2s+2 -2 ) > E(X n 2s+2 -3 ), from this equation and Equation (6.9) of Lemma 6.6:

Combining this with (6.20) yields:

From Property (1), as n 2s+2 -n 2s+1 is even:

6.3.5 Proof of Lemma 6.9

Proof of Lemma 6.9. We first remark that, if E(X n ) ≤ αf (S n ), then:

The derivative of f goes to zero at infinity, so:

which implies Equation (6.21).

We fix x ∈]0; 1/2 -1/α[ and divide the proof in three cases.

First case:

Then, from Equation (6.8) of Lemma 6.6:

so, by Equation ( 6.21), if n is large enough:

We must also prove that m n+3 = E(X n+2 ). By Lemma 6.5:

For n large enough, from Equation ( 6.21) and the fact that α < 4, we conclude that:

Indeed, by assumption,

As m n+3 = max(E(X n+2 ), m n+2 -E(X n+2 )), we deduce from the last equation that:

From Lemma 6.5:

We have chosen x < 1 2 -1 α , so that 2(1 + αx) < α. From Equation ( 6.21), this implies, for n large enough:

The proof that m n+3 = E(X n+2 ) is identical to the one done in the first case.

Third case: E(X n+1 ) > E(X n ) From Lemma 6.5:

From Equation (6.9) of Lemma 6.6:

By solving a polynomial equation of degree 2, we see that it implies:

Combining this with Equation ( 6.21) and the fact that E(X n+2 ) = 2f (S n+2 ) gives, for all n large enough:

Proof of Lemma 6.10

Proof of Lemma 6.10. We assume that E(X 2n ) ∼ 2f (S 2n ) when n goes to +∞ and that, for all n large enough:

As in the proof of Lemma 6.9, we see that E(X 2n ) ∼ 2f (S 2n ) implies:

From Equation (6.7) of Lemma 6.6, E(X 2n+1 ) ≤ 2E(X 2n ) for any n ≥ 1. Combining this with E(X 2n ) ∼ 2f (S 2n ) ∼ 2f (S 2n+1 ) yields:

We define:

and show that M = 2. As, from Lemma 6.5,

We first find an upper bound for E(X 2n+1 )/f (S 2n+1 ) as a function of E(X 2n-1 )/f (S 2n-1 ). We distinguish two cases.

First case: E(X 2n-1 ) > E(X 2n ) Then, from Equation (6.8) of Lemma 6.6:

The upper bound is increasing in E(X 2n-1 ). By definition of M , for all M > M , E(X 2n-1 ) < M f (S 2n-1 ) when n is large enough. So, for n large enough:

As this inequality holds for any M > M , we obtain:

By assumption, for n large enough, m 2n-1 = E(X 2(n-1) ). As E(X 2(n-1) ) ∼ 2f (S 2(n-1) ) ∼ 2f (S 2n-1 ), the following inequality holds for all n large enough:

As m 2n = max(E(X 2n-1 ), m 2n-1 -E(X 2n-1 )), we must have m 2n = E(X 2n-1 ). So m 2n -E(X 2n ) = E(X 2n-1 ) -E(X 2n ) ≤ 0 and, by Lemma 6.5: (6.23) We can now conclude. If we combine Equations (6.22) and ( 6.23), we obtain:

A simple computation shows that, if M > 2, then this upper bound is strictly smaller than M , which is a contradiction with the definition of M . So M = 2.

Numerical illustrations 6.4.1 Characterization of the distribution tail

We perform numerical experiments for two different choices of X 0 . The first process X 0 follows a Laplace law, with a probability density function equal to:

The second process follows a variant of a Pareto law:

For each of these two choices, we plot the density function p on Figure 6.2 (graphs (a) and (c)). We also plot the first pairs (E(X n ), S n ), with S n defined as in the last section:

These pairs correspond to the red crosses of graphs (b) and (d). They are displayed along with the function 2f , where is f defined as in Equation ( 6.4):

As stated by Theorem 6.4:

In our two examples, the convergence of E(X n )/(2f (S n )) towards 1 seems fast. From n = 5, it is difficult to distinguish E(X n ) from 2f (S n ). In both cases, the sequence (E(X n )) n∈N provides a very precise estimation of f in the high values.

We remark that, although (E(X n )) n∈N precisely characterizes the distribution tail of X 0 , it conveys a much less precise information about the distribution at small values.

In particular, the sequence (E(X n )) n∈N does not uniquely determine the law of X 0 . An example of two different processes associated to the same sequence (E(X n )) n∈N is shown on Figure 6.3. When the initial process X 0 admits a probability density function, so do the X n ; we denote by p n their density function. In Figure 6.4 is displayed p 10 , the probability density function of X 10 , for the Pareto-like distribution.

It is discontinuous, although the initial probability density function is continuous over R + . Indeed, the iterative definition X n+1 = |X n -E(X n )| does not preserve the continuity of the density function.

In general, we observe that the density function p n presents two peaks, one around zero and the other one around E(X n ). The mass of each peak converges to 1/2 when n goes to infinity. The tail of p n , on the other hand, is equal to: The integer S is small, typically of the order 5. This observation is not new; it has in particular been used to reduce the computational time needed to solve classification tasks with deep networks [START_REF] Jaderberg | Speeding up convolutional neural networks with low rank expansions[END_REF]. Although S is in general not equal to 1, the first term of the decomposition, ψ 1,f req ν ψ 1,spat k,l , already gives an idea of the operations performed by second-layer filters.

The spatial components ψ 1,spat are displayed on Figure 7.5. They fall into two categories: zero-mean filters which again tend to look like wavelets (subfigure (a)) and filters with non-zero mean, the majority of which seem to perform local averages (subfigure(b)).

The frequential components ψ 1,f req ν are shown on Figure 7.6. Each component is seen as a function from R 2 to R: to a point (x, y) ∈ R 2 we associate the value ψ 1,f req ν , for ν the index of the first-layer filter whose characteristic frequency is the closest to (x, y).

Some filters are almost constant; they seem to perform averages in the frequential domain (last filter of the fifth row or second filter of the third row). Others, in polar coordinates (ρ, θ), do not vary much in r but present oscillations in θ (sixth filter of the first row or third filter of the seventh row). These filters are similar to the ones of the second-layer of scattering networks used for image analysis: in these networks, the second layer is a joint wavelet transform in both spatial and frequential variables (which allows to achieve invariance or stability to the action