▼ If the vertex set, X, of a simplicial complex, K, is a metric space, then K can be interpreted as a subset of the Wasserstein space of probability measures on X. Such spaces are called simplicial metric thickenings, and a prominent example is the Vietoris–Rips metric thickening. In this work we study these spaces from three perspectives: metric geometry, optimal transport, and category theory. Using the geodesic structure of Wasserstein space we give a novel proof of Hausmann's theorem for Vietoris–Rips metric thickenings. We also prove the first Morse lemma in Wasserstein space and relate it to the geodesic perspective. Finally we study the category of simplicial metric thickenings and determine effects of certain limits and colimits on homotopy type. Advisors/Committee Members: Adams, Henry (advisor), Peterson, Christopher (committee member), Patel, Amit (committee member), Eykholt, Richard (committee member).

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Cette thèse se concentre sur l'analyse de données présentées sous forme de mesures de probabilité sur R^d. L'objectif est alors de fournir une meilleure compréhension des outils statistiques usuels sur cet espace muni de la distance de Wasserstein. Une première notion naturelle est l'analyse statistique d'ordre un, consistant en l'étude de la moyenne de Fréchet (ou barycentre). En particulier, nous nous concentrons sur le cas de données (ou observations) discrètes échantillonnées à partir de mesures de probabilité absolument continues (a.c.) par rapport à la mesure de Lebesgue. Nous introduisons ainsi un estimateur du barycentre de mesures aléatoires, pénalisé par une fonction convexe, permettant ainsi d'imposer son a.c. Un autre estimateur est régularisé par l'ajout d'entropie lors du calcul de la distance de Wasserstein. Nous nous intéressons notamment au contrôle de la variance de ces estimateurs. Grâce à ces résultats, le principe de Goldenshluger et Lepski nous permet d'obtenir une calibration automatique des paramètres de régularisation. Nous appliquons ensuite ce travail au recalage de densités multivariées, notamment pour des données de cytométrie de flux. Nous proposons également un test d'adéquation de lois capable de comparer deux distributions multivariées, efficacement en terme de temps de calcul. Enfin, nous exécutons une analyse statistique d'ordre deux dans le but d'extraire les tendances géométriques globales d'un jeu de donnée, c'est-à-dire les principaux modes de variations. Pour cela nous proposons un algorithme permettant d'effectuer une analyse en composantes principales géodésiques dans l'espace de Wasserstein.

This thesis focuses on the analysis of data in the form of probability measures on R^d. The aim is to provide a better understanding of the usual statistical tools on this space endowed with the Wasserstein distance. The first order statistical analysis is a natural notion to consider, consisting of the study of the Fréchet mean (or barycentre). In particular, we focus on the case of discrete data (or observations) sampled from absolutely continuous probability measures (a.c.) with respect to the Lebesgue measure. We thus introduce an estimator of the barycenter of random measures, penalized by a convex function, making it possible to enforce its a.c. Another estimator is regularized by adding entropy when computing the Wasserstein distance. We are particularly interested in controlling the variance of these estimators. Thanks to these results, the principle of Goldenshluger and Lepski allows us to obtain an automatic calibration of the regularization parameters. We then apply this work to the registration of multivariate densities, especially for flow cytometry data. We also propose a test statistic that can compare two multivariate distributions, efficiently in terms of computational time. Finally, we perform a second-order statistical analysis to extract the global geometric tendency of a dataset, also called the main modes of variation. For that purpose, we propose algorithms…

Advisors/Committee Members: Bigot, Jérémie (thesis director), Papadakis, Nicolas (thesis director).

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Dans cette thèse, nous étudions un modèle de régression avec des entrées de type distribution et le problème de test d'hypothèse pour la détection de signaux dans un modèle de régression. Nos modèles ont été appliqués aux données de sensibilité auditive mesurées par otoémissions acoustiques, cette mesure biologique contenant potentiellement des informations annexes sur l'individu (age, sexe, population/espèce).Dans la première partie, un nouveau modèle de régression de distribution pour les distributions de probabilité est introduit. Ce modèle est basé sur un cadre de régression RKHS, dans lequel les noyaux universels sont construits à l'aide de distances de Wasserstein pour les distributions appartenant à l'espace Wasserstein de Ω, où Ω est un sous-espace compact de l'espace réel. Nous prouvons la propriété de noyau universel de ces noyaux et utilisons ce cadre pour effectuer des régressions sur des fonctions. Différents modèles de régression sont d'abord comparés à celui proposé sur des données fonctionnelles simulées. Nous appliquons ensuite notre modèle de régression aux réponses de distribution des émissions otoascoutiques évoquées transitoires (TEOAE) et aux prédicteurs réels de l'âge. Dans la deuxième partie, en considérant un modèle de régression, nous abordons la question du test de la nullité de la fonction de régression. Nous proposons tout d'abord une nouvelle procédure de test unique basée sur un noyau symétrique général et une estimation de la variance des observations. Les valeurs critiques correspondantes sont construites pour obtenir des tests non-asymptotiques de niveau α. Nous introduisons ensuite une procédure d'agrégation afin d'éviter le choix complexe du noyau et des paramètres de celui-ci. Les tests multiples vérifient les propriétés non asymptotiques et adaptatives au sens minimax sur plusieurs classes d'alternatives régulières.

In this thesis, we study a regression model with distribution entries and the testing hypothesis problem for signal detection in a regression model. We aim to apply these models in hearing sensitivity measured by the transient evoked otoacoustic emissions (TEOAEs) data to improve our knowledge in the auditory investigation. In the first part, a new distribution regression model for probability distributions is introduced. This model is based on a Reproducing Kernel Hilbert Space (RKHS) regression framework, where universal kernels are built using Wasserstein distances for distributions belonging to Ω) and Ω is a compact subspace of the real space. We prove the universal kernel property of such kernels and use this setting to perform regressions on functions. Different regression models are first compared with the proposed one on simulated functional data. We then apply our regression model to transient evoked otoascoutic emission (TEOAE) distribution responses and real predictors of the age. This part is a joint work with Loubes, J-M., Risser, L. and Balaresque, P..In the second part, considering a regression model, we address the question…

Advisors/Committee Members: Loubès, Jean-Michel (thesis director), Balaresque, Patricia (thesis director).

▼ In the modern era of high and infinite dimensional data, classical statistical methodology is often rendered inefficient and ineffective when confronted with such big data problems as arise in genomics, medical imaging, speech analysis, and many other areas of research. Many problems manifest when the practitioner is required to take into account the covariance structure of the data during his or her analysis, which takes on the form of either a high dimensional low rank matrix or a finite dimensional representation of an infinite dimensional operator acting on some underlying function space. Thus, novel methodology is required to estimate, analyze, and make inferences concerning such covariances. In this manuscript, we propose using tools from the concentration of measure literature–a theory that arose in the latter half of the 20th century from connections between geometry, probability, and functional analysis–to construct rigorous descriptive and inferential statistical methodology for covariance matrices and operators. A variety of concentration inequalities are considered, which allow for the construction of nonasymptotic dimension-free confidence sets for the unknown matrices and operators. Given such confidence sets a wide range of estimation and inferential procedures can be and are subsequently developed. For high dimensional data, we propose a method to search a concentration in- equality based confidence set using a binary search algorithm for the estimation of large sparse covariance matrices. Both sub-Gaussian and sub-exponential concentration inequalities are considered and applied to both simulated data and to a set of gene expression data from a study of small round blue-cell tumours. For infinite dimensional data, which is also referred to as functional data, we use a celebrated result, Talagrand’s concentration inequality, in the Banach space setting to construct confidence sets for covariance operators. From these confidence sets, three different inferential techniques emerge: the first is a k-sample test for equality of covariance operator; the second is a functional data classifier, which makes its decisions based on the covariance structure of the data; the third is a functional data clustering algorithm, which incorporates the concentration inequality based confidence sets into the framework of an expectation-maximization algorithm. These techniques are applied to simulated data and to speech samples from a set of spoken phoneme data. Lastly, we take a closer look at a key tool used in the construction of concentration based confidence sets: Rademacher symmetrization. The symmetrization inequality, which arises in the probability in Banach spaces literature, is shown to be connected with optimal transport theory and specifically the Wasserstein distance. This insight is used to improve the symmetrization inequality resulting in tighter concentration bounds to be used in the construction of nonasymptotic confidence sets. A variety of other applications are considered including tests for data…

▼ This thesis consists of two parts. In the first part, we study stability properties of Hamiltonian systems on the Wasserstein space. Let H be a Hamiltonian satisfying conditions imposed in the work of Ambrosio and Gangbo. We regularize H via Moreau-Yosida approximation to get H[subscript Tau] and denote by μ[subscript Tau] a solution of system with the new Hamiltonian H[subscript Tau] . Suppose H[subscript Tau] converges to H as τ tends to zero. We show μ[subscript Tau] converges to μ and μ is a solution of a Hamiltonian system which is corresponding to the Hamiltonian H. At the end of first part, we give a sufficient condition for the uniqueness of Hamiltonian systems. In the second part, we develop a general theory of differential forms on the Wasserstein space. Our main result is to prove an analogue of Green's theorem for 1-forms and show that every closed 1-form on the Wasserstein space is exact. If the Wasserstein space were a manifold in the classical sense, this result wouldn't be worthy of mention. Hence, the first cohomology group, in the sense of de Rham, vanishes. Advisors/Committee Members: Gangbo, Wilfrid (Committee Chair), Loss, Michael (Committee Member), Pan, Ronghua (Committee Member), Swiech, Andrzej (Committee Member), Tannenbaum, Allen (Committee Member).

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Cette thèse étudie le contrôle optimal de la dynamique de type McKean-Vlasov et ses applications en mathématiques financières. La thèse contient deux parties. Dans la première partie, nous développons la méthode de la programmation dynamique pour résoudre les problèmes de contrôle stochastique de type McKean-Vlasov. En utilisant les contrôles admissibles appropriés, nous pouvons reformuler la fonction valeur en fonction de la loi (resp. la loi conditionnelle) du processus comme seule variable d’état et obtenir la propriété du flot de la loi (resp. la loi conditionnelle) du processus, qui permettent d’obtenir en toute généralité le principe de la programmation dynamique. Ensuite nous obtenons l’équation de Bellman correspondante, en s’appuyant sur la notion de différentiabilité par rapport aux mesures de probabilité introduite par P.L. Lions [Lio12] et la formule d’Itô pour le flot de probabilité. Enfin nous montrons la propriété de viscosité et l’unicité de la fonction valeur de l’équation de Bellman. Dans le premier chapitre, nous résumons quelques résultats utiles du calcul différentiel et de l’analyse stochastique sur l’espace de Wasserstein. Dans le deuxième chapitre, nous considérons le contrôle optimal stochastique de système à champ moyen non linéaire en temps discret. Le troisième chapitre étudie le problème de contrôle optimal stochastique d’EDS de type McKean-Vlasov sans bruit commun en temps continu où les coefficients peuvent dépendre de la loi joint de l’état et du contrôle, et enfin dans le dernier chapitre de cette partie nous nous intéressons au contrôle optimal de la dynamique stochastique de type McKean-Vlasov en présence de bruit commun en temps continu. Dans la deuxième partie, nous proposons un modèle d’allocation de portefeuille robuste permettant l’incertitude sur la rentabilité espérée et la matrice de corrélation des actifs multiples, dans un cadre de moyenne-variance en temps continu. Ce problème est formulé comme un jeu différentiel à champ moyen. Nous montrons ensuite un principe de séparation pour le problème associé. Nos résultats explicites permettent de justifier quantitativement la sous-diversification, comme le montrent les études empiriques.

This thesis deals with the study of optimal control of McKean-Vlasov dynamics and its applications in mathematical finance. This thesis contains two parts. In the first part, we develop the dynamic programming (DP) method for solving McKean-Vlasov control problem. Using suitable admissible controls, we propose to reformulate the value function of the problem with the law (resp. conditional law) of the controlled state process as sole state variable and get the flow property of the law (resp. conditional law) of the process, which allow us to derive in its general form the Bellman programming principle. Then by relying on the notion of differentiability with respect to probability measures introduced by P.L. Lions [Lio12], and Itô’s formula along measure-valued processes, we obtain the corresponding Bellman equation. At last we show the…

Advisors/Committee Members: Pham, Huyên (thesis director).

▼ This work is concerned with the Almost Axisymmetric Flows with Forcing Terms which are derived from the inviscid Boussinesq equations. It is our hope that these flows will be useful in Meteorology to describe tropical cyclones. We show that these flows give rise to a collection of Monge-Ampere equations for which we prove an existence and uniqueness result. What makes these equations unusual is the boundary conditions they are expected to satisfy and the fact that the boundary is part of the unknown. Our study allows us to make inferences in a toy Almost Axisymmetric Flows with a forcing term model. Advisors/Committee Members: Gangbo, Wilfrid (Committee Chair), McCuan, John (Committee Member), Loss, Michael (Committee Member), Monteiro, Renato (Committee Member), Swiech, Andrzej (Committee Member).

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Cette thèse comporte deux parties distinctes, toutes les deux liées à la théorie du transport optimal. Dans la première partie, nous considérons une variété riemannienne, deux mesures à densité régulière et un coût de transport, typiquement la distance géodésique quadratique et nous nous intéressons à la régularité de l’application de transport optimal. Le critère décisif à cette régularité s’avère être le signe du tenseur de Ma-Trudinger-Wang (MTW). Nous présentons tout d’abord une synthèse des travaux réalisés sur ce tenseur. Nous nous intéressons ensuite au lien entre la géométrie des lieux d’injectivité et le tenseur MTW. Nous montrons que dans de nombreux cas, la positivité du tenseur MTW implique la convexité des lieux d’injectivité. La deuxième partie de cette thèse est liée aux équations aux dérivées partielles. Certaines peuvent être considérées comme des flots gradients dans l’espace de Wasserstein W2. C’est le cas de l’équation de Keller-Segel en dimension 2. Pour cette équation nous nous intéressons au problème de quantification de la masse lors de l’explosion des solutions ; cette explosion apparaît lorsque la masse initiale est supérieure à un seuil critique Mc. Nous cherchons alors à montrer qu’elle consiste en la formation d’un Dirac de masse Mc. Nous considérons ici un modèle particulaire en dimension 1 ayant le même comportement que l’équation de Keller-Segel. Pour ce modèle nous exhibons des bassins d’attractions à l’intérieur desquels l’explosion se produit avec seulement le nombre critique de particules. Finalement nous nous intéressons au profil d’explosion : à l’aide d’un changement d’échelle parabolique nous montrons que la structure de l’explosion correspond aux points critiques d’une certaine fonctionnelle.

This thesis consists in two distinct parts both related to the optimal transport theory.The first part deals with the regularity of the optimal transport map. The key tool is the Ma-Trundinger-Wang tensor and especially its positivity. We first give a review of the known results about the MTW tensor. We then explore the geometrical consequences of the MTW tensor on the injectivity domain. We prove that in many cases the positivity of MTW implies the convexity of the injectivity domain. The second part is devoted to the behaviour of a Keller-Segel solution in the super critical case. In particular we are interested in the mass quantization problem: we wish to quantify the mass aggregated when the blow-up occurs. In order to study the behaviour of the solution we consider a particle approximation of a Keller-Segel type equation in dimension 1. We define this approximation using the gradient flow interpretation of the Keller-Segel equation and the particular structure of the Wasserstein space in dimension 1. We show two kinds of results; we first prove a stability theorem for the blow-up mechanism: we exhibit basins of attraction in which the solution blows up with only the critical number of particles. We then prove a rigidity theorem for the blow-up mechanism: thanks to a parabolic…

Advisors/Committee Members: Villani, Cédric (thesis director).

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The main object of interest in this thesis is P(M) – the space of probability measures on a manifold endowed with the Wasserstein distance: In chapter 1 we give the most basic topological facts and introduce a locally convex topology on P∞ (the space of smooth positive densities) to identify this space as infinite dimensional manifold. In chapter 2 we develop further the Riemannian calculus on P resp. P∞ where the different approaches (calculus of variation, Riemannian geometry on spaces of smooth mappings) are shown to be equivalent on P∞ . In chapter 3 we restrict ourself tomeasures on the unit circle and give calculations of renormalized Laplacians on the respective Wasserstein spaces, seen as the Hilbert-Schmidt trace of the Hessian: This trace depends on a real parameter s and has an analytic continuation as a function of s ∈ C {1} which enables us to calculate evaluate at s = 0: The square-field operator of the Wassersein Laplacian equals the squared Wasserstein gradient times the volume of the unit circle. In chapter 4 we give an approximation of the Wasserstein space P ([0, 1]) by spaces of box-type measures which are geodesically convex and can be mapped isometrically via a mapping simplex , where a sticky diffusion process is constructed. We show that image of this processes constitute a tight family in C(R₊

P ([0, 1])) with respect to the Skorohod topology. In the last chapter we restrict ourselves to the space of histograms on the unit interval. We calculate the Wasserstein distances numerically and obtain a Riemannian metric on the simplex. We investigate explosion behaviour of the respective diffusion processes in dimension 1 and 2.

Advisors/Committee Members: Thalmaier, Anton [superviser].

▼ In the modern era of high and infinite dimensional data, classical statistical methodology is often rendered inefficient and ineffective when confronted with such big data problems as arise in genomics, medical imaging, speech analysis, and many other areas of research. Many problems manifest when the practitioner is required to take into account the covariance structure of the data during his or her analysis, which takes on the form of either a high dimensional low rank matrix or a finite dimensional representation of an infinite dimensional operator acting on some underlying function space. Thus, novel methodology is required to estimate, analyze, and make inferences concerning such covariances. In this manuscript, we propose using tools from the concentration of measure literature–a theory that arose in the latter half of the 20th century from connections between geometry, probability, and functional analysis–to construct rigorous descriptive and inferential statistical methodology for covariance matrices and operators. A variety of concentration inequalities are considered, which allow for the construction of nonasymptotic dimension-free confidence sets for the unknown matrices and operators. Given such confidence sets a wide range of estimation and inferential procedures can be and are subsequently developed. For high dimensional data, we propose a method to search a concentration in- equality based confidence set using a binary search algorithm for the estimation of large sparse covariance matrices. Both sub-Gaussian and sub-exponential concentration inequalities are considered and applied to both simulated data and to a set of gene expression data from a study of small round blue-cell tumours. For infinite dimensional data, which is also referred to as functional data, we use a celebrated result, Talagrand’s concentration inequality, in the Banach space setting to construct confidence sets for covariance operators. From these confidence sets, three different inferential techniques emerge: the first is a k-sample test for equality of covariance operator; the second is a functional data classifier, which makes its decisions based on the covariance structure of the data; the third is a functional data clustering algorithm, which incorporates the concentration inequality based confidence sets into the framework of an expectation-maximization algorithm. These techniques are applied to simulated data and to speech samples from a set of spoken phoneme data. Lastly, we take a closer look at a key tool used in the construction of concentration based confidence sets: Rademacher symmetrization. The symmetrization inequality, which arises in the probability in Banach spaces literature, is shown to be connected with optimal transport theory and specifically the Wasserstein distance. This insight is used to improve the symmetrization inequality resulting in tighter concentration bounds to be used in the construction of nonasymptotic confidence sets. A variety of other applications are considered including tests for data…