La géométrie de l’information est la théorie mathématique qui applique les méthodes de la géométrie différentielle dans le domaine des statistiques et de la théorie de l’information. C’est une technique très prometteuse pour l’analyse et l’illustration des algorithmes itératifs utilisés en communications numériques. Cette thèse porte sur l’application de cette technique ainsi que d’autre technique d’optimisation bien connue, l’algorithme itératif du point proximal, sur les algorithmes itératifs en général. Nous avons ainsi trouvé des interprétations géométriques (basée sur la géométrie de l’information) et proximales (basée sur l’algorithme du point proximal)intéressantes dans le cas d’un algorithme itératif de calcul de la capacité des canaux discrets sans mémoire, l’algorithme de Blahut-Arimoto. L’idée étant d’étendre cette application sur une classe d’algorithmes itératifs plus complexes. Nous avons ainsi choisi d’analyser l’algorithme de décodage itératif des modulations codées à bits entrelacés afin de trouver quelques interprétations et essayer de proposer des liens existant avec le critère optimal de maximum de vraisemblance et d’autres algorithmes bien connus dans le but d’apporter certaines améliorations par rapport au cas classique de cet algorithme, en particulier l’étude de la convergence.Mots-clefs : Géométrie de l’information, algorithme du point proximal, algorithme de Blahut-Arimoto, décodage itératif, Modulations codées à bits entrelacés, maximum de vraisemblance. / Information geometry is a mathematical theory that applies methods of differential geometryin the fields of statistics and information theory. It is a very promising technique foranalyzing iterative algorithms used in digital communications. In this thesis, we apply this technique, in addition to the proximal point algorithm, to iterative algorithms. First, we have found some geometrical and proximal point interpretations in the case of an iterative algorithmfor computing the capacity of discrete and memoryless channel, the Blahut-Arimoto algorithm.Interesting results obtained motivated us to extend this application to a larger class of iterative algorithms. Then, we have studied in details iterative decoding algorithm of Bit Interleaved Coded Modulation (BICM) in order to analyse and propose some ameliorations of the classical decoding case. We propose a proximal point interpretation of this iterative process and find the link with some well known decoding algorithms, the Maximum likelihood decoding.

Orientador: João Eloir Strapasson / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-23T13:44:50Z (GMT). No. of bitstreams: 1 Porto_JuliannaPineleSantos_M.pdf: 2346170 bytes, checksum: 9f8b7284329ef1eb2f319c2e377b7a3c (MD5) Previous issue date: 2013 / Resumo: A Geometria da Informação é uma área da matemática que utiliza ferramentas geométricas no estudo de modelos estatísticos. Em 1945, Rao introduziu uma métrica Riemanniana no espaço das distribuições de probabilidade usando a matriz de informação, dada por Ronald Fisher em 1921. Com a métrica associada a essa matriz, define-se uma distância entre duas distribuições de probabilidade (distância de Rao), geodésicas, curvaturas e outras propriedades do espaço. Desde então muitos autores veem estudando esse assunto, que está naturalmente ligado a diversas aplicações como, por exemplo, inferência estatística, processos estocásticos, teoria da informação e distorção de imagens. Neste trabalho damos uma breve introdução à geometria diferencial e Riemanniana e fazemos uma coletânea de alguns resultados obtidos na área de Geometria da Informação. Mostramos a distância de Rao entre algumas distribuições de probabilidade e damos uma atenção especial ao estudo da distância no espaço formado por distribuições Normais Multivariadas. Neste espaço, como ainda não é conhecida uma fórmula fechada para a distância e nem para a curva geodésica, damos ênfase ao cálculo de limitantes para a distância de Rao. Conseguimos melhorar, em alguns casos, o limitante superior dado por Calvo e Oller em 1990 / Abstract: Information Geometry is an area of mathematics that uses geometric tools in the study of statistical models. In 1945, Rao introduced a Riemannian metric on the space of the probability distributions using the information matrix provided by Ronald Fisher in 1921. With the metric associated with this matrix, we define a distance between two probability distributions (Rao's distance), geodesics, curvatures and other properties. Since then, many authors have been studying this subject, which is associated with various applications, such as: statistical inference, stochastic processes, information theory, and image distortion. In this work we provide a brief introduction to Differential and Riemannian Geometry and a survey of some results obtained in Information Geometry. We show Rao's distance between some probability distributions, with special atention to the study of such distance in the space of multivariate normal distributions. In this space, since closed forms for the distance and for the geodesic curve are not known yet, we focus on the calculus of bounds for Rao's distance. In some cases, we improve the upper bound provided by Calvo and Oller in 1990 / Mestrado / Matematica Aplicada / Mestra em Matemática Aplicada

This thesis is divided into two parts dealing with the optimization problems in Markov decision processes (MDPs) and different neural network-based numerical solvers for partial differential equations (PDEs). In Part I we analyze the optimization problem arising in (partially observable) Markov decision processes using tools from algebraic statistics and information geometry, which can be viewed as neighboring fields of applied algebra and differential geometry, respectively. Here, we focus on infinite horizon problems and memoryless stochastic policies. Markov decision processes provide a mathematical framework for sequential decision-making on which most current reinforcement learning algorithms are built. They formalize the task of optimally controlling the state of a system through appropriate actions. For fully observable problems, the action can be selected knowing the current state of the system. This case has been studied extensively and optimizing the action selection is known to be equivalent to solving a linear program over the (generalized) stationary distributions of the Markov decision process, which are also referred to as state-action frequencies. In Chapter 3, we study partially observable problems where an action must be chosen based solely on an observation of the current state, which might not fully reveal the underlying state. We characterize the feasible state-action frequencies of partially observable Markov decision processes by polynomial inequalities. In particular, the optimization problem in partially observable MDPs is described as a polynomially constrained linear objective program that generalizes the (dual) linear programming formulation of fully observable problems. We use this to study the combinatorial and algebraic complexity of this optimization problem and to upper bound the number of critical points over the individual boundary components of the feasible set. Furthermore, we show that our polynomial programming formulation can be used to effectively solve partially observable MDPs using interior point methods, numerical algebraic techniques, and convex relaxations. Gradient-based methods, including variants of natural gradient methods, have gained tremendous attention in the theoretical reinforcement learning community, where they are commonly referred to as (natural) policy gradient methods. In Chapter 4, we provide a unified treatment of a variety of natural policy gradient methods for fully observable problems by studying their state-action frequencies from the standpoint of information geometry. For a variety of NPGs and reward functions, we show that the trajectories in state-action space are solutions of gradient flows with respect to Hessian geometries, based on which we obtain global convergence guarantees and convergence rates. In particular, we show linear convergence for unregularized and regularized NPG flows with the metrics proposed by Morimura and co-authors and Kakade by observing that these arise from the Hessian geometries of the entropy and conditional entropy, respectively. Further, we obtain sublinear convergence rates for Hessian geometries arising from other convex functions like log-barriers. We provide experimental evidence indicating that our predicted rates are essentially tight. Finally, we interpret the discrete-time NPG methods with regularized rewards as inexact Newton methods if the NPG is defined with respect to the Hessian geometry of the regularizer. This yields local quadratic convergence rates of these methods for step size equal to the inverse penalization strength, which recovers existing results as special cases. Part II addresses neural network-based PDE solvers that have recently experienced tremendous growth in popularity and attention in the scientific machine learning community. We focus on two approaches that represent the approximation of a solution of a PDE as the minimization over the parameters of a neural network: the deep Ritz method and physically informed neural networks. In Chapter 5, we study the theoretical properties of the boundary penalty for these methods and obtain a uniform convergence result for the deep Ritz method for a large class of potentially nonlinear problems. For linear PDEs, we estimate the error of the deep Ritz method in terms of the optimization error, the approximation capabilities of the neural network, and the strength of the penalty. This reveals a trade-off in the choice of the penalization strength, where too little penalization allows large boundary values, and too strong penalization leads to a poor solution of the PDE inside the domain. For physics-informed networks, we show that when working with neural networks that have zero boundary values also the second derivatives of the solution are approximated whereas otherwise only lower-order derivatives are approximated. In Chapter 6, we propose energy natural gradient descent, a natural gradient method with respect to second-order information in the function space, as an optimization algorithm for physics-informed neural networks and the deep Ritz method. We show that this method, which can be interpreted as a generalized Gauss-Newton method, mimics Newton’s method in function space except for an orthogonal projection onto the tangent space of the model. We show that for a variety of PDEs, natural energy gradients converge rapidly and approximations to the solution of the PDE are several orders of magnitude more accurate than gradient descent, Adam and Newton’s methods, even when these methods are given more computational time.:Chapter 1. Introduction 1 1.1 Notation and conventions 7 Part I. Geometry of Markov decision processes 11 Chapter 2. Background on Markov decision processes 12 2.1 State-action frequencies 19 2.2 The advantage function and Bellman optimality 23 2.3 Rational structure of the reward and an explicit line theorem 26 2.4 Solution methods for Markov decision processes 35 Chapter 3. State-action geometry of partially observable MDPs 44 3.1 The state-action polytope of fully observables systems 45 3.2 State-action geometry of partially observable systems 54 3.3 Number and location of critical points 69 3.4 Reward optimization in state-action space (ROSA) 83 Chapter 4. Geometry and convergence of natural policy gradient methods 94 4.1 Natural gradients 96 4.2 Natural policy gradient methods 101 4.3 Convergence of natural policy gradient flows 107 4.4 Locally quadratic convergence for regularized problems 128 4.5 Discussion and outlook 131 Part II. Neural network-based PDE solvers 133 Chapter 5. Theoretical analysis of the boundary penalty method for neural network-based PDE solvers 134 5.1 Presentation and discussion of the main results 137 5.2 Preliminaries regarding Sobolev spaces and neural networks 146 5.3 Proofs regarding uniform convergence for the deep Ritz method 150 5.4 Proofs of error estimates for the deep Ritz method 156 5.5 Proofs of implications of exact boundary values in residual minimization 167 Chapter 6. Energy natural gradients for neural network-based PDE solvers 174 6.1 Energy natural gradients 176 6.2 Experiments 183 6.3 Conclusion and outlook 192 Bibliography 193

Dans cette thèse, nous développons des modèles et des méthodes statistiques pour le traitement d’images de microscopie confocale de la peau dans le but de détecter une maladie de la peau appelée lentigo. Une première contribution consiste à proposer un modèle statistique paramétrique pour représenter la texture dans le domaine des ondelettes. Plus précisément, il s’agit d’une distribution gaussienne généralisée dont on montre que le paramètre d’échelle est caractéristique des tissus sousjacents. La modélisation des données dans le domaine de l’image est un autre sujet traité dans cette thèse. A cette fin, une distribution gamma généralisée est proposée. Notre deuxième contribution consiste alors à développer un estimateur efficace des paramètres de cette loi à l’aide d’une descente de gradient naturel. Finalement, un modèle d’observation de bruit multiplicatif est établi pour expliquer la distribution gamma généralisée des données. Des méthodes d’inférence bayésienne paramétrique sont ensuite développées avec ce modèle pour permettre la classification d’images saines et présentant un lentigo. Les algorithmes développés sont appliqués à des images réelles obtenues d’une étude clinique dermatologique. / In this work, we develop statistical models and processing methods for confocal microscopy images. The first contribution consists of a parametric statistical model to represent textures in the wavelet domain. Precisely, a generalized Gaussian distribution is proposed, whose scale parameter is shown to be discriminant of the underlying tissues. The thesis deals also with modeling data in the image domain using the generalized gamma distribution. The second contribution develops an efficient parameter estimator for this distribution based on a natural gradient approach. The third contribution establishes a multiplicative noise observation model to explain the distribution of the data. Parametric Bayesian inference methods are subsequently developed based on this model to classify healthy and lentigo images. All algorithms developed in this thesis have been applied to real images from a dermatologic clinical study.

We study minimization problems with respect to a one-parameter family of generalized relative entropies. These relative entropies, which we call relative -entropies (denoted I (P; Q)), arise as redundancies under mismatched compression when cumulants of compression lengths are considered instead of expected compression lengths. These parametric relative entropies are a generalization of the usual relative entropy (Kullback-Leibler divergence). Just like relative entropy, these relative -entropies behave like squared Euclidean distance and satisfy the Pythagorean property. We explore the geometry underlying various statistical models and its relevance to information theory and to robust statistics. The thesis consists of three parts. In the first part, we study minimization of I (P; Q) as the first argument varies over a convex set E of probability distributions. We show the existence of a unique minimizer when the set E is closed in an appropriate topology. We then study minimization of I on a particular convex set, a linear family, which is one that arises from linear statistical constraints. This minimization problem generalizes the maximum Renyi or Tsallis entropy principle of statistical physics. The structure of the minimizing probability distribution naturally suggests a statistical model of power-law probability distributions, which we call an -power-law family. Such a family is analogous to the exponential family that arises when relative entropy is minimized subject to the same linear statistical constraints. In the second part, we study minimization of I (P; Q) over the second argument. This minimization is generally on parametric families such as the exponential family or the - power-law family, and is of interest in robust statistics ( > 1) and in constrained compression settings ( < 1). In the third part, we show an orthogonality relationship between the -power-law family and an associated linear family. As a consequence of this, the minimization of I (P; ), when the second argument comes from an -power-law family, can be shown to be equivalent to a minimization of I ( ; R), for a suitable R, where the first argument comes from a linear family. The latter turns out to be a simpler problem of minimization of a quasi convex objective function subject to linear constraints. Standard techniques are available to solve such problems, for example, via a sequence of convex feasibility problems, or via a sequence of such problems but on simpler single-constraint linear families.

Numerous studies on Queueing systems, such as Internet traffic flows, have shown to be bursty, self-similar and/or long-range dependent, because of the heavy (long) tails for the various distributions of interest, including intermittent intervals and queue lengths. Other studies have addressed vacation in no-customers’ queueing system or when the server fails. These patterns are important for capacity planning, performance prediction, and optimization of networks and have a negative impact on their effective functioning. Heavy-tailed distributions have been commonly used by telecommunication engineers to create workloads for simulation studies, which, regrettably, may show peculiar queueing characteristics. To cost-effectively examine the impacts of different network patterns on heavy- tailed queues, new and reliable analytical approaches need to be developed. It is decided to establish a brand-new analytical framework based on optimizing entropy functionals, such as those of Shannon, Rényi, Tsallis, and others that have been suggested within statistical physics and information theory, subject to suitable linear and non-linear system constraints. In both discrete and continuous time domains, new heavy tail analytic performance distributions will be developed, with a focus on those exhibiting the power law behaviour seen in many Internet scenarios. The exposition of two major revolutionary approaches, namely the unification of information geometry and classical queueing systems and unifying information length theory with transient queueing systems. After conclusions, open problems arising from this thesis and limitations are introduced as future work.

Nous proposons dans cette thèse de caractériser quantitativement la variabilité à différentes échelles au cours de l'embryogenèse. Pour ce faire, nous utilisons une combinaison de modèles mathématiques et de résultats expérimentaux. Dans la première partie, nous utilisons une petite cohorte d'oursins digitaux pour construire une représentation prototypique du lignage cellulaire, reliant les caractéristiques des cellules individuelles avec les dynamiques à l'échelle de l'embryon tout entier. Ce modèle probabiliste multi-niveau et empirique repose sur les symétries des embryons et sur les identités cellulaires; cela permet d'identifier un niveau de granularité générique pour observer les distributions de caractéristiques cellulaires individuelles. Le prototype est défini comme le barycentre de la cohorte dans la variété statistique correspondante. Parmi plusieurs résultats, nous montrons que la variabilité intra-individuelle est impliquée dans la reproductibilité du développement embryonnaire. Dans la seconde partie, nous considérons les mécanismes sources de variabilité au cours du développement et leurs relations à l'évolution. En nous appuyant sur des résultats expérimentaux montrant une pénétrance incomplète et une expressivité variable de phénotype dans une lignée mutante du poisson zèbre, nous proposons une clarification des différents niveaux de variabilité biologique reposant sur une analogie formelle avec le cadre mathématique de la mécanique quantique. Nous trouvons notamment une analogie formelle entre l'intrication quantique et le schéma Mendélien de transmission héréditaire. Dans la troisième partie, nous étudions l'organisation biologique et ses relations aux trajectoires développementales. En adaptant les outils de la topologie algébrique, nous caractérisons des invariants du réseaux de contacts cellulaires extrait d'images de microscopie confocale d'épithéliums de différentes espèces et de différents fonds génétiques. En particulier, nous montrons l'influence des histoires individuelles sur la distribution spatiales des cellules dans un tissu épithélial. / We propose in this thesis to characterize variability quantitatively at various scales during embryogenesis. We use a combination of mathematical models and experimental results. In the first part, we use a small cohort of digital sea urchin embryos to construct a prototypical representation of the cell lineage, which relates individual cell features with embryo-level dynamics. This multi-level data-driven probabilistic model relies on symmetries of the embryo and known cell types, which provide a generic coarse-grained level of observation for distributions of individual cell features. The prototype is defined as the centroid of the cohort in the corresponding statistical manifold. Among several results, we show that intra-individual variability is involved in the reproducibility of the developmental process. In the second part, we consider the mechanisms sources of variability during development and their relations to evolution. Building on experimental results showing variable phenotypic expression and incomplete penetrance in a zebrafish mutant line, we propose a clarification of the various levels of biological variability using a formal analogy with quantum mechanics mathematical framework. Surprisingly, we find a formal analogy between quantum entanglement and Mendel’s idealized scheme of inheritance. In the third part, we study biological organization and its relations to developmental paths. By adapting the tools of algebraic topology, we compute invariants of the network of cellular contacts extracted from confocal microscopy images of epithelia from different species and genetic backgrounds. In particular, we show the influence of individual histories on the spatial distribution of cells in epithelial tissues.

Nous proposons dans cette thèse de caractériser quantitativement la variabilité à différentes échelles au cours de l'embryogenèse. Pour ce faire, nous utilisons une combinaison de modèles mathématiques et de résultats expérimentaux. Dans la première partie, nous utilisons une petite cohorte d'oursins digitaux pour construire une représentation prototypique du lignage cellulaire, reliant les caractéristiques des cellules individuelles avec les dynamiques à l'échelle de l'embryon tout entier. Ce modèle probabiliste multi-niveau et empirique repose sur les symétries des embryons et sur les identités cellulaires; cela permet d'identifier un niveau de granularité générique pour observer les distributions de caractéristiques cellulaires individuelles. Le prototype est défini comme le barycentre de la cohorte dans la variété statistique correspondante. Parmi plusieurs résultats, nous montrons que la variabilité intra-individuelle est impliquée dans la reproductibilité du développement embryonnaire. Dans la seconde partie, nous considérons les mécanismes sources de variabilité au cours du développement et leurs relations à l'évolution. En nous appuyant sur des résultats expérimentaux montrant une pénétrance incomplète et une expressivité variable de phénotype dans une lignée mutante du poisson zèbre, nous proposons une clarification des différents niveaux de variabilité biologique reposant sur une analogie formelle avec le cadre mathématique de la mécanique quantique. Nous trouvons notamment une analogie formelle entre l'intrication quantique et le schéma Mendélien de transmission héréditaire. Dans la troisième partie, nous étudions l'organisation biologique et ses relations aux trajectoires développementales. En adaptant les outils de la topologie algébrique, nous caractérisons des invariants du réseaux de contacts cellulaires extrait d'images de microscopie confocale d'épithéliums de différentes espèces et de différents fonds génétiques. En particulier, nous montrons l'influence des histoires individuelles sur la distribution spatiales des cellules dans un tissu épithélial. / We propose in this thesis to characterize variability quantitatively at various scales during embryogenesis. We use a combination of mathematical models and experimental results. In the first part, we use a small cohort of digital sea urchin embryos to construct a prototypical representation of the cell lineage, which relates individual cell features with embryo-level dynamics. This multi-level data-driven probabilistic model relies on symmetries of the embryo and known cell types, which provide a generic coarse-grained level of observation for distributions of individual cell features. The prototype is defined as the centroid of the cohort in the corresponding statistical manifold. Among several results, we show that intra-individual variability is involved in the reproducibility of the developmental process. In the second part, we consider the mechanisms sources of variability during development and their relations to evolution. Building on experimental results showing variable phenotypic expression and incomplete penetrance in a zebrafish mutant line, we propose a clarification of the various levels of biological variability using a formal analogy with quantum mechanics mathematical framework. Surprisingly, we find a formal analogy between quantum entanglement and Mendel’s idealized scheme of inheritance. In the third part, we study biological organization and its relations to developmental paths. By adapting the tools of algebraic topology, we compute invariants of the network of cellular contacts extracted from confocal microscopy images of epithelia from different species and genetic backgrounds. In particular, we show the influence of individual histories on the spatial distribution of cells in epithelial tissues.

My thesis addresses a systematic approach to stochastic models in population genetics; in particular, the Wright-Fisher models affected only by the random genetic drift. I used various mathematical methods such as Probability, PDE, and Geometry to answer an important question: \"How do genetic change factors (random genetic drift, selection, mutation, migration, random environment, etc.) affect the behavior of gene frequencies or genotype frequencies in generations?”. In a Hardy-Weinberg model, the Mendelian population model of a very large number of individuals without genetic change factors, the answer is simple by the Hardy-Weinberg principle: gene frequencies remain unchanged from generation to generation, and genotype frequencies from the second generation onward remain also unchanged from generation to generation. With directional genetic change factors (selection, mutation, migration), we will have a deterministic dynamics of gene frequencies, which has been studied rather in detail. With non-directional genetic change factors (random genetic drift, random environment), we will have a stochastic dynamics of gene frequencies, which has been studied with much more interests. A combination of these factors has also been considered. We consider a monoecious diploid population of fixed size N with n + 1 possible alleles at a given locus A, and assume that the evolution of population was only affected by the random genetic drift. The question is that what the behavior of the distribution of relative frequencies of alleles in time and its stochastic quantities are. When N is large enough, we can approximate this discrete Markov chain to a continuous Markov with the same characteristics. In 1931, Kolmogorov first introduced a nice relation between a continuous Markov process and diffusion equations. These equations called the (backward/forward) Kolmogorov equations which have been first applied in population genetics in 1945 by Wright. Note that these equations are singular parabolic equations (diffusion coefficients vanish on boundary). To solve them, we use generalized hypergeometric functions. To know more about what will happen after the first exit time, or more general, the behavior of whole process, in joint work with J. Hofrichter, we define the global solution by moment conditions; calculate the component solutions by boundary flux method and combinatorics method. One interesting property is that some statistical quantities of interest are solutions of a singular elliptic second order linear equation with discontinuous (or incomplete) boundary values. A lot of papers, textbooks have used this property to find those quantities. However, the uniqueness of these problems has not been proved. Littler, in his PhD thesis in 1975, took up the uniqueness problem but his proof, in my view, is not rigorous. In joint work with J. Hofrichter, we showed two different ways to prove the uniqueness rigorously. The first way is the approximation method. The second way is the blow-up method which is conducted by J. Hofrichter. By applying the Information Geometry, which was first introduced by Amari in 1985, we see that the local state space is an Einstein space, and also a dually flat manifold with the Fisher metric; the differential operator of the Kolmogorov equation is the affine Laplacian which can be represented in various coordinates and on various spaces. Dynamics on the whole state space explains some biological phenomena.

Nous proposons dans cette thèse de caractériser quantitativement la variabilité à différentes échelles au cours de l'embryogenèse. Pour ce faire, nous utilisons une combinaison de modèles mathématiques et de résultats expérimentaux. Dans la première partie, nous utilisons une petite cohorte d'oursins digitaux pour construire une représentation prototypique du lignage cellulaire, reliant les caractéristiques des cellules individuelles avec les dynamiques à l'échelle de l'embryon tout entier. Ce modèle probabiliste multi-niveau et empirique repose sur les symétries des embryons et sur les identités cellulaires; cela permet d'identifier un niveau de granularité générique pour observer les distributions de caractéristiques cellulaires individuelles. Le prototype est défini comme le barycentre de la cohorte dans la variété statistique correspondante. Parmi plusieurs résultats, nous montrons que la variabilité intra-individuelle est impliquée dans la reproductibilité du développement embryonnaire. Dans la seconde partie, nous considérons les mécanismes sources de variabilité au cours du développement et leurs relations à l'évolution. En nous appuyant sur des résultats expérimentaux montrant une pénétrance incomplète et une expressivité variable de phénotype dans une lignée mutante du poisson zèbre, nous proposons une clarification des différents niveaux de variabilité biologique reposant sur une analogie formelle avec le cadre mathématique de la mécanique quantique. Nous trouvons notamment une analogie formelle entre l'intrication quantique et le schéma Mendélien de transmission héréditaire. Dans la troisième partie, nous étudions l'organisation biologique et ses relations aux trajectoires développementales. En adaptant les outils de la topologie algébrique, nous caractérisons des invariants du réseaux de contacts cellulaires extrait d'images de microscopie confocale d'épithéliums de différentes espèces et de différents fonds génétiques. En particulier, nous montrons l'influence des histoires individuelles sur la distribution spatiales des cellules dans un tissu épithélial. / We propose in this thesis to characterize variability quantitatively at various scales during embryogenesis. We use a combination of mathematical models and experimental results. In the first part, we use a small cohort of digital sea urchin embryos to construct a prototypical representation of the cell lineage, which relates individual cell features with embryo-level dynamics. This multi-level data-driven probabilistic model relies on symmetries of the embryo and known cell types, which provide a generic coarse-grained level of observation for distributions of individual cell features. The prototype is defined as the centroid of the cohort in the corresponding statistical manifold. Among several results, we show that intra-individual variability is involved in the reproducibility of the developmental process. In the second part, we consider the mechanisms sources of variability during development and their relations to evolution. Building on experimental results showing variable phenotypic expression and incomplete penetrance in a zebrafish mutant line, we propose a clarification of the various levels of biological variability using a formal analogy with quantum mechanics mathematical framework. Surprisingly, we find a formal analogy between quantum entanglement and Mendel’s idealized scheme of inheritance. In the third part, we study biological organization and its relations to developmental paths. By adapting the tools of algebraic topology, we compute invariants of the network of cellular contacts extracted from confocal microscopy images of epithelia from different species and genetic backgrounds. In particular, we show the influence of individual histories on the spatial distribution of cells in epithelial tissues.