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Modèles mathématiques des effets d’une mutation sur la valeur sélective d’une bactérie : estimation théorique et implémentation numérique
No full textThis thesis presents a mathematical study of genetic mutations and their effects on the fitness of individuals, a central concept in evolutionary biology. A mutation is a spontaneous or induced alteration of DNA, which can be deleterious, neutral, or beneficial. These mutations affect the fitness of individuals, i.e., their ability to survive and reproduce. The main question addressed is the estimation of the Density of Fitness Effects (DFE), which describes the distribution of the effects of mutations on fitness. Understanding the shape of the DFE is essential for predicting population evolution, genetic diversity, and the consequences of conservation programs. The starting point of this work is an experimental protocol developed by Lydia Robert et al. (2018), which allows real-time observation of mutation events in E. coli. These new data open up opportunities for improved estimation of the DFE, but also raise statistical challenges due to experimental noise .In the first part of this manuscript, we consider a stochastic model from [77] that describes the evolution of a bacterial population under mutation pressure. We develop a nonparametric estimation method based on Fourier estimators to recover the DFE, and establish convergence results for our estimator. The second part introduces two deterministic models for the evolution of fitness in a population structured by growth rate. These models allow us to study the asymptotic behavior of the population and provide mathematical insight into the dynamics of mutation accumulation observed in experiments. Finally, the third part applies various statistical methods to the experimental data, aiming to reconstruct the DFE from its empirical moments and to assess whether it is unimodal or multimodal. This work builds a bridge between experimental biology and the mathematical modeling of mutation effects.Cette thèse porte sur l'étude mathématique des mutations génétiques et de leurs effets sur la valeur sélective des individus, concept central en biologie évolutive. Une mutation est une modification spontanée ou provoquée de l'ADN, pouvant être délétère, neutre ou bénéfique. Ces mutations affectent la fitness des individus, c'est-à-dire leur capacité à survivre et à se reproduire. La question principale est la détermination de la Density of Fitness Effects (DFE), c'est-à-dire la répartition des effets des mutations sur la valeur sélective. Comprendre la DFE est crucial pour prédire l'évolution des populations, la diversité génétique ou encore les conséquences des programmes de conservation. Le point de départ de ce travail est un protocole expérimental développé par Lydia Robert et al. (2018), permettant d'observer en temps réel l'apparition de mutations chez E. coli. Ces nouvelles données rendent possible une estimation plus fine de la DFE, mais posent un défi statistique en raison du bruit expérimental. Dans une première partie, nous utilisons un modèle stochastique issu de [77] pour décrire l'évolution d'une population bactérienne sous l'effet des mutations, et nous développons une méthode non paramétrique d'estimation de la DFE basée sur des estimateurs de Fourier, pour laquelle nous prouvons des résultats de convergence. La deuxième partie introduit deux modèles déterministes représentant l'évolution de la fitness d'une population structurée par taux de croissance. Ces modèles permettent d'étudier les propriétés asymptotiques des solutions, et d'interpréter mathématiquement la dynamique des mutations observée expérimentalement. Enfin, la troisième partie applique plusieurs outils statistiques aux données expérimentales, dans le but de reconstruire la DFE à partir de ses moments, et de déterminer sa modalité. Ce travail offre un pont entre expérience biologique et modélisation mathématique des mutations
Post-Hoc Interpretation of POMDP Policies
No full textInternational audiencePolicies for partially observable Markov decision processes are rich objects, prescribing actions to take depending on the whole history of observations and actions. Typical representations of such policies are by hyperplanes in the space of belief states, or by finite-state controllers, which are arguably not easy to interpret.We propose to redescribe policies into mappings defined on features of the current belief state, built in a systematic manner from state features. Such a mapping can in turn be represented by an intelligible object, like a decision tree, thereby providing an interpretable representation of the policy as a whole. We moreover show how our approach allows to explain the decision taken by an agent at each step of an interaction with the environment. This provides an endto-end process, starting from a policy computed by any solver, and ending with an explanation of each decision made at execution time. We formally define our approach, investigate related computational problems, and report on experiments on several families of problems
Transport optimal martingale et jeux à champ moyen avec graphon
No full textThis thesis investigates two distinct directions: Martingale Optimal Transport (MOT) and Mean Field Games (MFG). Within the MOT framework, three independent projects are developed: maximal martingale Wasserstein inequality in Chapter 3, non-decreasing martingale couplings in Chapter 4, and numerical methods of weak optimal transport in Chapter 5. Under the MFG framework, the thesis includes a single project in Chapter 6: extended graphon mean field games in discrete time. In Chapter 3, we complete the analysis of the martingale Wasserstein inequality and prove that a maximal martingale Wasserstein inequality and prove that a maximal martingale Wasserstein inequality holds whatever the dimension d when integrability parameter rhoge 2. In Chapter 4, we observe numerically that the Hobson and Neuberger martingale coupling, which is known to satisfy some non-decreasing property, is still very close to maximize this integral of |y-x|^rho for rhoin(0,2) and minimize it for rho>2. We investigate the optimality of this coupling from a theoretical perspective. We find some decomposition of the second marginal which is in one-to-one correspondence with martingale couplings that are non-decreasing in a generalized sense. In Chapter 5, we apply the Frank-Wolfe algorithm to approximate the Weak Optimal Transport and Weak Martingale Optimal Transport problems from an optimization perspective. In Chapter 6, we studied extended graphon mean field games in discrete time setting, incorporating joint state-action interactions within the graphon aggregate. We establish the existence and uniqueness of graphon Nash Equilibrium under various assumptions and provide a numerical example of optimal investment with a relative performance criterion.Cette thèse s'intéresse à deux sujets distincts: le Transport Optimal Martingale (MOT) et la théorie des Jeux à champ moyen (MFG). Dans le cadre du MOT, trois projets indépendants sont développés : l'inégalité de Wasserstein martingale maximale dans le Chapitre 3, les couplages martingales croissants dans Chapitre 4, et les méthodes numériques pour le transport optimal faible au Chapitre 5. Dans le cadre des MFG, la thèse comprend un projet unique : les jeux à champ moyen étendus sur un graphon en temps discret, au Chapitre 6. Au Chapitre 3, nous achevons l'analyse de l'inégalité de Wasserstein martingale et démontrons qu'une inégalité maximale de Wasserstein martingale est valable quelle que soit la dimension d avec un paramètre d'intégrabilité rho geq 2. Au Chapitre 4, nous observons numériquement que le couplage martingale de Hobson et Neuberger, connu pour satisfaire une certaine propriété de croissance, permet d'approcher le maximum de l'intégrale de |y-x|^rho pour rho in (0,2), et le minimum pour rho > 2. Nous étudions l'optimalité de ce couplage d'un point de vue théorique. Nous identifions une décomposition de la seconde loi marginale qui est en bijection avec des couplages martingales croissants au sens généralisé. Au Chapitre 5, nous appliquons l'algorithme de Frank-Wolfe pour approcher les problèmes de Weak Optimal Transport et de Weak Martingale Optimal Transport dans une perspective d'optimisation. Au Chapitre 6, nous étudions des jeux de champ moyen étendus sur un graphon en temps discret, en intégrant des interactions conjointes entre les états et les actions dans l'agrégat du graphon. Nous établissons l'existence et l'unicité de l'équilibre de Nash sur le graphon sous diverses hypothèses et proposons un exemple numérique illustrant un investissement optimal avec un critère de performance relative
Clinical physiology of circadian rhythms: A systematic and hierarchized content analysis of circadian questionnaires
No full textInternational audienceCircadian rhythms are near-24 h patterns of physiology and behavior associated with several physical and mental health outcomes. Self-report questionnaires are routinely used and practical tools to assess circadian rhythms. However, the extent to which these questionnaires capture the relevant parameters and can be used interchangeably is unknown. We investigated different types of circadian manifestations using 14 circadian self-report questionnaires for adults. A systematic and hierarchical content analysis was combined with a visualization method. Jaccard indices were calculated to quantify the degree to which the questionnaires overlapped. Content analysis revealed 40 distinct manifestations, which we classified into five dimensions ("circadian phase," "circadian amplitude and stability," "nycthemeral timing," "nycthemeral regularity," and "circadian complaints"). The average Jaccard index was 0.150, indicating very weak content overlap. None of the 14 questionnaires explored all five dimensions. The Composite Scale of Morningness and Morningness-Eveningness Questionnaire exhibited greater, but still limited, overlap with the other questionnaires (Jaccard indices of 0.255 and 0.251, respectively), and are the best instruments for assessing the circadian phase. Nycthemeral timing, which must be analyzed to measure the circadian misalignment in clinical and research settings, is only explored by the Munich Chronotype Questionnaire, but that instrument does not evaluate circadian amplitude and stability and only partially assesses nycthemeral regularity. Based on our preliminary analysis, we make recommendations regarding the circumstances in which some circadian questionnaires could prove more useful than the others. The results might also aid the definition and investigation of circadian health at the crossroads of physiology and behavior
Preuves non-interactives : la nouvelle ère des chaînes compressées
No full textInternational audienceUn des défis majeurs de la blockchain réside dans la gestion de la complexité liée à la communication et au stockage.Pour garantir la sécurité de la blockchain, il est nécessaire de conserver intégralement les données de consensus, dontla taille augmente linéairement avec la taille de la chaîne, ce qui compromet la pérennité de la blockchain. Les PreuvesNon-Interactives de Preuve de Travail (NIPoPoWs) apportent une solution à ce problème, à condition que le systèmeconserve un nombre constant de participants. Nous proposons pour la première fois une construction qui répond rigou-reusement aux exigences d’une NIPoPoW, capable de résister à un adversaire contrôlant jusqu’à un tiers des ressourcesdans un environnement dynamique. Nous montrons la concision, la sécurité et l’actualisabilité de ce système, tandisque nos résultats expérimentaux confirment une réduction exponentielle de la taille de la blockchain Bitcoin
Extending the Quasidifferential Framework: From Fixed-Key to Expected Differential Probability
No full textInternational audienceBeyne and Rijmen proposed in 2022 a systematic and generic framework to study the fixed-key probability of differential characteristics. One of the main challenges for implementing this framework is the ability to efficiently handle very large quasidifferential transition matrices (QDTMs) for big (e.g. 8-bit) S-boxes. Our first contribution is a new MILP model capable of efficiently representing such matrices, by exploiting the inherent block structure of these objects. We then propose two extensions to the original framework. First, we demonstrate how to adapt the framework to the related-key setting. Next, we present a novel approach to compute the average expected probability of a differential characteristic that takes the key schedule into account. This method, applicable to both linear and non-linear key schedules, works in both the single-key and related-key settings. Furthermore, it provides a faster way to verify the validity of characteristics compared to computing the fixed-key probability. Using these extensions and our MILP model, we analyze various (related-key) differential characteristics from the literature. First, we prove the validity of several optimal related-key differential characteristics of AES. Next, we show that this approach permits to obtain more precise results than methods relying on key constraints for SKINNY. Finally, we examine the validity of a differential distinguisher used in two differential meet-in-the-middle attacks on SKINNY-128, demonstrating that its probability is significantly higher than initially estimated
Clustering with bandit feedback: breaking down the computation/information gap
No full textInternational audienceWe investigate the Clustering with Bandit feedback Problem (CBP). A learner interacts with an N-armed stochastic bandit with d-dimensional subGaussian feedback. There exists a hidden partition of the arms into K groups, such that arms within the same group, share the same mean vector. The learner's task is to uncover this hidden partition with the smallest budget - i.e. the least number of observation - and with a probability of error smaller than a prescribed constant δ. In this paper, (i) we derive a non asymptotic lower bound for the budget, and (ii) we introduce the computationally efficient ACB algorithm, whose budget matches the lower bound in most regimes. We improve on the performance of a uniform sampling strategy. Importantly, contrary to the batch setting, we establish that there is no computation-information gap in the bandit setting
Taking a Big Step: Large Learning Rates in Denoising Score Matching Prevent Memorization
No full textInternational audienceDenoising score matching plays a pivotal role in the performance of diffusion-based generative models. However, the empirical optimal score–the exact solution to the denoising score matching–leads to memorization, where generated samples replicate the training data. Yet, in practice, only a moderate degree of memorization is observed, even without explicit regularization. In this paper, we investigate this phenomenon by uncovering an implicit regularization mechanism driven by large learning rates. Specifically, we show that in the small-noise regime, the empirical optimal score exhibits high irregularity. We then prove that, when trained by stochastic gradient descent with a large enough learning rate, neural networks cannot stably converge to a local minimum with arbitrarily small excess risk. Consequently, the learned score cannot be arbitrarily close to the empirical optimal score, thereby mitigating memorization. To make the analysis tractable, we consider one-dimensional data and two-layer neural networks. Experiments validate the crucial role of the learning rate in preventing memorization, even beyond the one-dimensional setting
regularity of the minimizers for the inviscid total variation and Bingham fluid problems for data
No full textInternational audienceThe Bingham fluid model for viscoplastic materials involves the minimization of a nondifferentiable functional. The regularity of the associated solution is investigated here. The simplified scalar case is considered first: the total variation minimization problem. Our main result proves for a convex domain that a right-hand side gives a solution . Homogeneous Dirichlet conditions involve an additional trace term, then implies . In the case of the inviscid vector Bingham fluid model, boundary conditions are difficult to handle, but we prove the local regularity of the solution for . The proofs rely on several generalizations of a lemma due to Br\'ezis and on the viscous approximation. We obtain Euler--Lagrange characterizations of the solution. Homogeneous Dirichlet conditions on the viscous problem lead in the vanishing viscosity limit to relaxed boundary conditions of frictional type