Publications

2023

• A Geometric proof for the Polygonal Isoperimetric Inequality
  
  • Bogosel Beniamin
  The Mathematical Intelligencer, Springer Verlag, 2023. Gradients of the perimeter and area of a polygon have straightforward geometric interpretations. The use of optimality conditions for constrained problems and basic ideas in triangle geometry show that polygons with prescribed area minimizing the perimeter must be regular. (10.1007/s00283-024-10366-x)
  DOI : 10.1007/s00283-024-10366-x
• Compression bidirectionnelle pour l'apprentissage fédéré hétérogène
  
  • Philippenko Constantin
  , 2023. The last two decades have witnessed an unprecedented increase in computational power, leading to a vast surge in the volume of available data. As a consequence, machine learning algorithms have evolved to adapt to this new situation. Especially, many modern applications now use a network of clients to store the data and compute the models: efficient learning in this framework is harder, especially under communication constraints. This is why, a new approach, federated learning, has been developed in recent years: the data is kept on the original server and a central server orchestrates the training. This thesis aims to address two fundamental aspects of federated learning. The first goal is to analyze the trade-offs of distributed learning with communication constraints, with the objective of reducing its energy cost and environmental footprint. The second goal is to tackle problems resulting from heterogeneity among clients. This thesis focuses on bidirectional compression and summarizes my contributions to this field of research.In our first contribution, we focus on the intertwined effect of compression and client (statistical) heterogeneity. We introduce a framework of algorithms, named Artemis, that tackles the problem of learning in a federated setting with communication constraints. In our second contribution, we move the focus toward feedback loops to reduce the impact of compression. We introduce an algorithm, coined MCM; it builds upon Artemis and introduces a new paradigm that preserves the central model from down compression. This mechanism allows to carry out bidirectional compression while asymptotically achieving the rates of convergence of unidirectional compression. In our third contribution, we go beyond the classical worst-case assumption on the variance of compressors and provide a fine-grained analysis of the impact of compression within the fundamental learning framework of least-squares regression. Within this setting, we highlight differences in convergence between several unbiased compression schemes having the same variance increase.
• Entropic Wasserstein component analysis
  
  • Collas Antoine
  • Vayer Titouan
  • Flamary Rémi
  • Breloy Arnaud
  , 2023. Dimension reduction (DR) methods provide systematic approaches for analyzing high-dimensional data. A key requirement for DR is to incorporate global dependencies among original and embedded samples while preserving clusters in the embedding space. To achieve this, we combine the principles of optimal transport (OT) and principal component analysis (PCA). Our method seeks the best linear subspace that minimizes reconstruction error using entropic OT, which naturally encodes the neighborhood information of the samples. From an algorithmic standpoint, we propose an efficient block-majorization-minimization solver over the Stiefel manifold. Our experimental results demonstrate that our approach can effectively preserve high-dimensional clusters, leading to more interpretable and effective embeddings. Python code of the algorithms and experiments is available online.
• Procédé de contrôle d'un système et produit programme d'ordinateur associé
  
  • Dhaou Amin
  • Bertoncello Antoine
  • Gourvénec Sébastien
  • Garnier Josselin
  • Le Pennec Erwan
  , 2023.
• Transparent scatterers and transmission eigenvalues of infinite multiplicity
  
  • Grinevich Piotr
  • Novikov Roman
  , 2023. We give a short review of old and recent results on scatterers with transmission eigenvalues of infinite multiplicity, including transparent scatterers. Our examples include potentials from the Schwartz class and multipoint potentials of Bethe - Peierls type. This talk is based, in particular, on the works Grinevich, Novikov ( Comm. Math. Phys. 174, 409-446 (1995), EJMCA 9(4), 17-25 (2021), Russian Math. Surveys 77, 1021-1028 (2022)).
• Synchronization in a Kuramoto Mean Field Game
  
  • Carmona Rene
  • Cormier Quentin
  • Soner H. Mete
  Communications in Partial Differential Equations, Taylor & Francis, 2023, 48 (9), pp.1214-1244. The classical Kuramoto model is studied in the setting of an infinite horizon mean field game. The system is shown to exhibit both synchronization and phase transition. Incoherence below a critical value of the interaction parameter is demonstrated by the stability of the uniform distribution. Above this value, the game bifurcates and develops self-organizing time homogeneous Nash equilibria. As interactions become stronger, these stationary solutions become fully synchronized. Results are proved by an amalgam of techniques from nonlinear partial differential equations, viscosity solutions, stochastic optimal control and stochastic processes. (10.1080/03605302.2023.2264611)
  DOI : 10.1080/03605302.2023.2264611
• Commentary on Galanis et al .: When age could make the difference—let's not sweep violence under the keyboard
  
  • Luquiens Amandine
  • Lopez-Castroman Jorge
  Addiction, Wiley, 2023, 118 (9), pp.1699-1700. (10.1111/add.16272)
  DOI : 10.1111/add.16272
• A refined Weissman estimator for extreme quantiles
  
  • Allouche Michaël
  • El Methni Jonathan
  • Girard Stéphane
  Extremes, Springer Verlag (Germany), 2023, 26, pp.545-572. Weissman extrapolation methodology for estimating extreme quantiles from heavy-tailed distributions is based on two estimators: an order statistic to estimate an intermediate quantile and an estimator of the tail-index. The common practice is to select the same intermediate sequence for both estimators. In this work, we show how an adapted choice of two different intermediate sequences leads to a reduction of the asymptotic bias associated with the resulting refined Weissman estimator. The asymptotic normality of the latter estimator is established and a data-driven method is introduced for the practical selection of the intermediate sequences. Our approach is compared to Weissman estimator and to six bias reduced estimators of extreme quantiles on a large scale simulation study. It appears that the refined Weissman estimator outperforms its competitors in a wide variety of situations, especially in the challenging high bias cases. Finally, an illustration on an actuarial real data set is provided. (10.1007/s10687-022-00452-8)
  DOI : 10.1007/s10687-022-00452-8
• Mathematical modelling and analysis of Impermanent Loss and Fees in Uniswap v3
  
  • Echenim Mnacho
  • Gobet Emmanuel
  • Maurice Anne-Claire
  , 2023.
• Generative methods for sampling transition paths in molecular dynamics
  
  • Lelièvre Tony
  • Robin Geneviève
  • Sekkat Inass
  • Stoltz Gabriel
  • Victorino Cardoso Gabriel
  ESAIM: Proceedings, EDP Sciences, 2023, 73, pp.238-256. Molecular systems often remain trapped for long times around some local minimum of the potential energy function, before switching to another one -- a behavior known as metastability. Simulating transition paths linking one metastable state to another one is difficult by direct numerical methods. In view of the promises of machine learning techniques, we explore in this work two approaches to more efficiently generate transition paths: sampling methods based on generative models such as variational autoencoders, and importance sampling methods based on reinforcement learning. (10.1051/proc/202373238)
  DOI : 10.1051/proc/202373238
• Walking forward and backward in Euler schemes and random number generators
  
  • Cohort Pierre
  • Gobet Emmanuel
  • Mrad Mohamed
  , 2023.
• A Piecewise Deterministic Markov Process Approach Modeling a Dry Friction Problem with Noise
  
  • Garnier Josselin
  • Lu Ziyu
  • Mertz Laurent
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2023, 83 (4), pp.1392-1421. Understanding and predicting the dynamical properties of systems involving dry friction is a major concern in physics and engineering. It abounds in many mechanical processes, from the sound produced by a violin to the screeching of chalk on a blackboard to human infant crawling dynamics and friction-based locomotion of a multitude of living organisms (snakes, bacteria, scallops) to the displacement of mechanical structures (building, bridges, nuclear plants, massive industrial infrastructures) under earthquakes and beyond. Surprisingly, even for low-dimensional systems, the modeling of dry friction in the presence of random forcing has not been elucidated. In this paper, we propose a piecewise deterministic Markov process approach modeling a system with dry friction including different coefficients for the static and dynamic forces. In this mathematical framework, we derive the corresponding Kolmogorov equations to compute statistical quantities of interest related to the distributions of the static (sticked) and dynamic phases. We show ergodicity and provide a representation formula of the stationary measure using independent identically distributed portions of the trajectory (excursions). We also obtain deterministic characterizations of the Laplace transforms of the probability density functions of the durations of the static and dynamic phases. In particular, the analysis of the power spectral density of the velocity reveals a critical value of the noise correlation time below which the correlations of the dynamic behaviors coincide with those of the white noise limit. The existence of such a critical value was already mentioned in the physical literature [Geffert and Just, Phys. Rev. E, 95 (2017), 062111]. (10.1137/22M1480847)
  DOI : 10.1137/22M1480847
• Exponential ergodicity of a degenerate age-size piecewise deterministic process
  
  • Madrid Ignacio
  Acta Applicandae Mathematicae, Springer Verlag, 2023, 187 (5). We study the long-time behaviour of the first-moment semigroup of a non conservative piecewise deterministic measure-valued stochastic process with support on R 2 + driven by a deterministic flow between random jump times, with a transition kernel which has a degenerate form. Using a Doob h-transform where the function h is taken as an eigenfunction of the associated generator, we can bring ourselves back to the study of a conservative process whose exponential ergodicity is proven via Harris' Theorem. Particular attention is given to the proof of Doeblin minoration condition. The main difficulty is the degeneracy of one of the two variables, and the deterministic dependency between the two variables, which make it no trivial to uniformly bound the expected value of the trajectories with respect to a non-degenerate measure in a two-dimensional space, which is particularly hard in a non-compact setting. Here, we propose a general method to construct explicit trajectories which explore the space state with positive probability and witch permit to prove a petite-set condition for the compact sets of the state space. An application to an age-structured growthfragmentation process modelling bacterial growth is also shown. (10.1007/s10440-023-00597-z)
  DOI : 10.1007/s10440-023-00597-z
• Propagation of chaos for stochastic particle systems with singular mean-field interaction of $L^q − L^p$ type
  
  • Tomašević Milica
  Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2023, 28 (none), pp.13. In this work, we prove the well-posedness and propagation of chaos for a stochastic particle system in mean-field interaction under the assumption that the interacting kernel belongs to a suitable $L_t^q-L_x^p$ space. Contrary to the large deviation principle approach recently proposed in [2], the main ingredient of the proof here are the \textit{Partial Girsanov transformations} introduced in [3] and developed in a general setting in this work. (10.1214/23-ECP539)
  DOI : 10.1214/23-ECP539
• The Ulam-Hammersley problem for multiset permutations
  
  • Gerin Lucas
  , 2023. We obtain the asymptotic behaviour of the longest increasing/non-decreasing subsequences in a random uniform multiset permutation in which each element in {1,...,n} occurs k times, where k may depend on n. This generalizes the famous Ulam-Hammersley problem of the case k=1. The proof relies on poissonization and a connection with variants of the Hammersley-Aldous-Diaconis particle system.
• Un algorithme multiéchelle pour déformer les objets de façon réalisteapplication à la modélisation de la croissance du cerveau foetal
  
  • Gaudfernau Fleur
  • Allassonière Stéphanie
  • Le Pennec Erwan
  , 2023. Nous proposons d'établir une trajectoire continue du plissement cortical du cerveau au cours de la grossesse en utilisant la régression géodésique dans le cadre du Large Deformation Diffeomorphic Metric Mapping. Deux problèmes se posent lors de l'estimation de transformations de grandes dimensions, à savoir le risque élevé de bloquer l'optimisation dans un minimum local irréaliste et le fait que les déformations sont contraintes à une unique échelle spatiale. Pour résoudre ces problèmes, nous introduisons une stratégie d'optimisation coarse-to-fine basée sur des reparamétrisations multiéchelles des objets et des déformations. Nos expériences sur des maillages de la surface corticale du foetus montrent que la stratégie multiéchelle génère des représentations plus naturelles du plissement cortical, ce qui offre des perspectives intéressantes pour l'analyse quantitative de la croissance saine ou anormale du cerveau.
• Solving Irreducible Stochastic Mean-Payoff Games and Entropy Games by Relative Krasnoselskii-Mann Iteration
  
  • Akian Marianne
  • Gaubert Stéphane
  • Naepels Ulysse
  • Terver Basile
  , 2023, 272. We analyse an algorithm solving stochastic mean-payoff games, combining the ideas of relative value iteration and of Krasnoselskii-Mann damping. We derive parameterized complexity bounds for several classes of games satisfying irreducibility conditions. We show in particular that an ε-approximation of the value of an irreducible concurrent stochastic game can be computed in a number of iterations in O(|log(ε)|) where the constant in the O(⋅) is explicit, depending on the smallest non-zero transition probabilities. This should be compared with a bound in O(ε^{-1}|log(ε)|) obtained by Chatterjee and Ibsen-Jensen (ICALP 2014) for the same class of games, and to a O(ε^{-1}) bound by Allamigeon, Gaubert, Katz and Skomra (ICALP 2022) for turn-based games. We also establish parameterized complexity bounds for entropy games, a class of matrix multiplication games introduced by Asarin, Cervelle, Degorre, Dima, Horn and Kozyakin. We derive these results by methods of variational analysis, establishing contraction properties of the relative Krasnoselskii-Mann iteration with respect to Hilbert’s semi-norm. (10.4230/LIPIcs.MFCS.2023.10)
  DOI : 10.4230/LIPIcs.MFCS.2023.10
• An acoustic-transport splitting method for the barotropic Baer-Nunziato two-phase flow model
  
  • Ait-Ameur Katia
  • Kokh Samuel
  • Massot Marc
  • Pelanti Marica
  • Pichard Teddy
  ESAIM: Proceedings and Surveys, EDP Sciences, 2023, 72, pp.93-116. This work focuses on the numerical approximation of the barotropic Baer-Nunziato twophase flow model. The scheme relies on an operator splitting method corresponding to a separate treatment of fast propagation phenomena due to the acoustic waves on the one hand and slow propagation phenomena due to the fluid motion on the other. We propose to extend the implicit-explicit schemes developed in [7]. These methods enable the use of time steps that are no longer constrained by the sound velocity thanks to an implicit treatment of the acoustic waves, and maintain accuracy in the subsonic regime thanks to an explicit treatment of the material waves. In the present setting, a particular attention will be also given to the discretization of the non conservative terms in the Baer-Nunziato model. We prove that the proposed numerical strategy preserve positive values of the volume fractions and densities and we illustrate its behaviour with several relevant test cases. (10.1051/proc/202372093)
  DOI : 10.1051/proc/202372093
• Blow-up for a stochastic model of chemotaxis driven by conservative noise on $\mathbb{R}^2$
  
  • Mayorcas Avi
  • Tomašević Milica
  Journal of Evolution Equations, Springer Verlag, 2023, 23, pp.57. Abstract We establish criteria on the chemotactic sensitivity $\chi $ for the non-existence of global weak solutions (i.e., blow-up in finite time) to a stochastic Keller–Segel model with spatially inhomogeneous, conservative noise on $\mathbb{R}^2$. We show that if $\chi$ is sufficiently large then blow-up occurs with probability 1. In this regime, our criterion agrees with that of a deterministic Keller–Segel model with increased viscosity. However, for $\chi $ in an intermediate regime, determined by the variance of the initial data and the spatial correlation of the noise, we show that blow-up occurs with positive probability. (10.1007/s00028-023-00900-3)
  DOI : 10.1007/s00028-023-00900-3
• Neural Networks: From the Perceptron to Deep Nets
  
  • Gabrié Marylou
  • Ganguli Surya
  • Lucibello Carlo
  • Zecchina Riccardo
  , 2023, pp.477-497. (10.1142/9789811273926_0024)
  DOI : 10.1142/9789811273926_0024
• Spherical Harmonics and Discontinuous Galerkin Finite Element Methods for the Three-Dimensional Neutron Transport Equation: Application to Core and Lattice Calculation
  
  • Assogba Kenneth
  • Bourhrara Lahbib
  • Zmijarevic Igor
  • Allaire Grégoire
  • Galia Antonio
  Nuclear Science and Engineering, Academic Press, 2023, 197 (8), pp.1584-1599. The spherical harmonics or P N method is intended to approximate the neutron angular flux by a linear combination of spherical harmonics of degree at most N. In this work, the P N method is combined with discontinuous Galerkin finite elements method and yield to a full discretization of the multigroup neutron transport equation. The employed method is able to handle all geometries describing the fuel elements without any simplification nor homogenisation. Moreover the use of a matrix assembly-free method avoids building large sparse matrices, which enables to produce high-order solutions in small computational time and less storage usage. The resulting transport solver called NYMO has a wide range of applications: it can be used for a core calculation as well as for a precise 281-groups lattice calculation accounting anisotropic scattering. To assess the accuracy of this numerical scheme, it was applied to 3D reactor core and fuel assembly calculations. These calculations point out that the proposed P N-DG method is capable of producing precise solutions while the developed solver is able to handle complex 3D core and assemblies geometries. (10.1080/00295639.2022.2154546)
  DOI : 10.1080/00295639.2022.2154546
• Multi-Output Gaussian Processes for Inverse Uncertainty Quantification in Neutron Noise Analysis
  
  • Lartaud Paul
  • Humbert Philippe
  • Garnier Josselin
  Nuclear Science and Engineering, Academic Press, 2023, 197 (8), pp.1928-1951. In a fissile material, the inherent multiplicity of neutrons born through induced fissions leads to correlations in their detection statistics. The correlations between neutrons can be used to trace back some characteristics of the fissile material. This technique, known as neutron noise analysis, has applications in nuclear safeguards or waste identification. It provides a nondestructive examination method for an unknown fissile material. This is an example of an inverse problem where the cause is inferred from observations of the consequences. However, neutron correlation measurements are often noisy because of the stochastic nature of the underlying processes. This makes the resolution of the inverse problem more complex since the measurements are strongly dependent on the material characteristics. A minor change in the material properties can lead to very different outputs. Such an inverse problem is said to be ill posed. For an ill-posed inverse problem, the inverse uncertainty quantification is crucial. Indeed, seemingly low noise in the data can lead to strong uncertainties in the estimation of the material properties. Moreover, the analytical framework commonly used to describe neutron correlations relies on strong physical assumptions, and is thus inherently biased. This paper addresses dual goals. First, surrogate models are used to improve neutron correlation predictions and quantify the errors on those predictions. Then the inverse uncertainty quantification is performed to include the impact of measurement error alongside the residual model bias. (10.1080/00295639.2022.2143705)
  DOI : 10.1080/00295639.2022.2143705
• Tropical Complementarity Problems and Nash Equilibria
  
  • Allamigeon Xavier
  • Gaubert Stéphane
  • Meunier Frédéric
  SIAM Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2023, 37 (3), pp.1645-1665. Linear complementarity programming is a generalization of linear programming which encompasses the computation of Nash equilibria for bimatrix games. While the latter problem is PPAD-complete, we show that the tropical analogue of the complementarity problem associated with Nash equilibria can be solved in polynomial time. Moreover, we prove that the Lemke–Howson algorithm carries over the tropical setting and performs a linear number of pivots in the worst case. A consequence of this result is a new class of (classical) bimatrix games for which Nash equilibria computation can be done in polynomial time. (10.1137/21M1446861)
  DOI : 10.1137/21M1446861
• Compressed and distributed least-squares regression: convergence rates with applications to Federated Learning
  
  • Philippenko Constantin
  • Dieuleveut Aymeric
  , 2023. In this paper, we investigate the impact of compression on stochastic gradient algorithms for machine learning, a technique widely used in distributed and federated learning. We underline differences in terms of convergence rates between several unbiased compression operators, that all satisfy the same condition on their variance, thus going beyond the classical worst-case analysis. To do so, we focus on the case of least-squares regression (LSR) and analyze a general stochastic approximation algorithm for minimizing quadratic functions relying on a random field. We consider weak assumptions on the random field, tailored to the analysis (specifically, expected H\"older regularity), and on the noise covariance, enabling the analysis of various randomizing mechanisms, including compression. We then extend our results to the case of federated learning. More formally, we highlight the impact on the convergence of the covariance $\mathfrak{C}_{\mathrm{ania}}$ of the additive noise induced by the algorithm. We demonstrate despite the non-regularity of the stochastic field, that the limit variance term scales with $\mathrm{Tr}(\mathfrak{C}_{\mathrm{ania}} H^{-1})/K$ (where $H$ is the Hessian of the optimization problem and $K$ the number of iterations) generalizing the rate for the vanilla LSR case where it is $\sigma^2 \mathrm{Tr}(H H^{-1}) / K = \sigma^2 d / K$ (Bach and Moulines, 2013). Then, we analyze the dependency of $\mathfrak{C}_{\mathrm{ania}}$ on the compression strategy and ultimately its impact on convergence, first in the centralized case, then in two heterogeneous FL frameworks. (10.1214/15-AOS1391)
  DOI : 10.1214/15-AOS1391
• Mean field games with incomplete information
  
  • Bertucci Charles
  , 2023. This paper is concerned with mean field games in which the players do not know the repartition of the other players. First a case in which the players do not gain information is studied. Results of existence and uniqueness are proved and discussed. Then, a case in which the players observe the payments is investigated. A master equation is derived and partial results of uniqueness are given for this more involved case.