Publications

2024

• Statistical limits of correlation detection in trees
  
  • Ganassali Luca
  • Massoulié Laurent
  • Semerjian Guilhem
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2024, 34 (4), pp.3701-3734. In this paper we address the problem of testing whether two observed trees (t, t ′ ) are sampled either independently or from a joint distribution under which they are correlated. This problem, which we refer to as correlation detection in trees, plays a key role in the study of graph alignment for two correlated random graphs. Motivated by graph alignment, we investigate the conditions of existence of one-sided tests, i.e. tests which have vanishing type I error and non-vanishing power in the limit of large tree depth.<p>For the correlated Galton-Watson model with Poisson offspring of mean λ &gt; 0 and correlation parameter s ∈ (0, 1), we identify a phase transition in the limit of large degrees at s = √ α, where α ∼ 0.3383 is Otter's constant. Namely, we prove that no such test exists for s ≤ √ α, and that such a test exists whenever s &gt; √ α, for λ large enough. This result sheds new light on the graph alignment problem in the sparse regime (with O(1) average node degrees) and on the performance of the MPAlign method studied in [13, 20], proving in particular the conjecture of [20] that MPAlign succeeds in the partial recovery task for correlation parameter s &gt; √ α provided the average node degree λ is large enough. As a byproduct, we identify a new family of orthogonal polynomials for the Poisson-Galton-Watson measure which enjoy remarkable properties. These polynomials may be of independent interest for a variety of problems involving graphs, trees or branching processes, beyond the scope of graph alignment.</p> (10.1214/23-AAP2048)
  DOI : 10.1214/23-AAP2048
• An Efficient SSP-based Methodology for Assessing Climate Risks of a Large Credit Portfolio
  
  • Bourgey Florian
  • Gobet Emmanuel
  • Jiao Ying
  , 2024. We examine climate-related exposure within a large credit portfolio, addressing transition and physical risks. We design a modeling methodology that begins with the Shared Socioeconomic Pathways (SSP) scenarios and ends with describing the losses of a portfolio of obligors. The SSP scenarios impact the physical risk of each obligor via a DICE-inspired damage function and their transition risk through production, requiring optimal adjustment. To achieve optimal production, the obligor optimizes various energy sources to align its greenhouse gas (GHG) emission trajectories with SSP objectives, while accounting for uncertainties in consumption trajectories. Ultimately, we obtain a Gaussian factor model whose dimension is of the order of the number of obligors. Two efficient dimension reduction methods (Polynomial Chaos Expansion and Principal Component Analysis) provide a fast and accurate method for analyzing credit portfolio losses.
• Random features models: a way to study the success of naive imputation
  
  • Ayme Alexis
  • Boyer Claire
  • Dieuleveut Aymeric
  • Scornet Erwan
  , 2024. Constant (naive) imputation is still widely used in practice as this is a first easy-to-use technique to deal with missing data. Yet, this simple method could be expected to induce a large bias for prediction purposes, as the imputed input may strongly differ from the true underlying data. However, recent works suggest that this bias is low in the context of high-dimensional linear predictors when data is supposed to be missing completely at random (MCAR). This paper completes the picture for linear predictors by confirming the intuition that the bias is negligible and that surprisingly naive imputation also remains relevant in very low dimension. To this aim, we consider a unique underlying random features model, which offers a rigorous framework for studying predictive performances, whilst the dimension of the observed features varies. Building on these theoretical results, we establish finite-sample bounds on stochastic gradient (SGD) predictors applied to zero-imputed data, a strategy particularly well suited for large-scale learning. If the MCAR assumption appears to be strong, we show that similar favorable behaviors occur for more complex missing data scenarios.
• Mean-field limit of particle systems with absorption
  
  • Guo Gaoyue
  • Tomasevic Milica
  , 2024. We consider a particle system of singular interaction where particles are removed from the system once they hit some barrier. We show the wellposedness of the particle system and its mean-field limit and prove the propagation of chaos.
• Convex compact surfaces with no bound on their synthetic Ricci curvature
  
  • Vernicos Constantin
  Bulletin de la société mathématique de France, Société Mathématique de France, 2024, 152 (2), pp.185-198. L'utilisation de la notion de réfraction dans le cadre des espaces vectoriels normés permet de construite un exemple de surface convexe et compacte qui n'est pas de courbure de Ricci minorée telle que défini par Lott-Villani et Sturm. (10.24033/bsmf.2888)
  DOI : 10.24033/bsmf.2888
• Nonlinear system identification with control-based continuation of bifurcation curves
  
  • Mélot Adrien
  • Denimal Goy Enora
  • Renson Ludovic
  , 2024, pp.1-2. We propose a methodology to carry out nonlinear system identification based on bifurcations. The proposed approach relies on fold bifurcation curves identified experimentally through control-based continuation and an optimization framework to minimize the distance between the experimental and numerical curves computed with bifurcation tracking analyses. The approach is demonstrated on a nonlinear model of base-excited energy harvester with magnetic force nonlinearity.
• Eigenvalue Methods for Sparse Tropical Polynomial Systems
  
  • Akian Marianne
  • Béreau Antoine
  • Gaubert Stéphane
  , 2024. We develop an analogue of eigenvalue methods to construct solutions of sys- tems of tropical polynomial equalities and inequalities. We show that solutions can be ob- tained by solving parametric mean payoff games, arising to approriate linearizations of the systems using tropical Macaulay matrices. We implemented specific algorithms adapted to the large scale parametric games that arise in this way, and present numerical experiments.
• Stochastic Localization via Iterative Posterior Sampling
  
  • Grenioux Louis
  • Noble Maxence
  • Gabrié Marylou
  • Durmus Alain Oliviero
  Proceedings of the 41st International Conference on Machine Learning, PMLR, 2024, 235, pp.16337--16376. Building upon score-based learning, new interest in stochastic localization techniques has recently emerged. In these models, one seeks to noise a sample from the data distribution through a stochastic process, called observation process, and progressively learns a denoiser associated to this dynamics. Apart from specific applications, the use of stochastic localization for the problem of sampling from an unnormalized target density has not been explored extensively. This work contributes to fill this gap. We consider a general stochastic localization framework and introduce an explicit class of observation processes, associated with flexible denoising schedules. We provide a complete methodology, $\textit{Stochastic Localization via Iterative Posterior Sampling}$ (SLIPS), to obtain approximate samples of this dynamics, and as a by-product, samples from the target distribution. Our scheme is based on a Markov chain Monte Carlo estimation of the denoiser and comes with detailed practical guidelines. We illustrate the benefits and applicability of SLIPS on several benchmarks, including Gaussian mixtures in increasing dimensions, Bayesian logistic regression and a high-dimensional field system from statistical-mechanics.
• Structural optimization for controlling isolated response curves
  
  • Mélot Adrien
  • Denimal Goy Enora
  • Renson Ludovic
  , 2024, pp.1-3. We introduce a numerical analysis framework for controlling isolated response curves (isolas), i.e., curves that form closed loops and are not connected to the main branch of solutions. The methodology relies on bifurcation tracking analyses to monitor the evolution of node-collocation bifurcations in a codimension-2 space. Singularity theory is employed to differentiate points of isolas formation and merger from codimension-2 bifurcations. An optimization problem is defined to advance or delay the formation or merger of isolas.
• Incentivized Learning in Principal-Agent Bandit Games
  
  • Scheid Antoine
  • Tiapkin Daniil
  • Boursier Etienne
  • Capitaine Aymeric
  • Mahdi El
  • Moulines Éric
  • Jordan Michael I
  • Durmus Alain
  , 2024. This work considers a repeated principal-agent bandit game, where the principal can only interact with her environment through the agent. The principal and the agent have misaligned objectives and the choice of action is only left to the agent. However, the principal can influence the agent's decisions by offering incentives which add up to his rewards. The principal aims to iteratively learn an incentive policy to maximize her own total utility. This framework extends usual bandit problems and is motivated by several practical applications, such as healthcare or ecological taxation, where traditionally used mechanism design theories often overlook the learning aspect of the problem. We present nearly optimal (with respect to a horizon T) learning algorithms for the principal's regret in both multi-armed and linear contextual settings. Finally, we support our theoretical guarantees through numerical experiments.
• Consistent Long-Term Forecasting of Ergodic Dynamical Systems
  
  • Kostic Vladimir
  • Inzerili Prune
  • Lounici Karim
  • Novelli Pietro
  • Pontil Massimiliano
  , 2024.
• AutoFreeFem: Automatic code generation with FreeFEM++ and LaTex output for shape and topology optimization of non-linear multi-physics problems
  
  • Allaire Grégoire
  • Gfrerer Michael H
  , 2024. For an educational purpose we develop the Python package AutoFreeFem which generates all ingredients for shape optimization with non-linear multi-physics in FreeFEM++ and also outputs the expressions for use in Latex. As an input, the objective function and the weak form of the problem have to be specified only once. This ensures consistency between the simulation code and its documentation. In particular, AutoFreeFem provides the linearization of the state equation, the adjoint problem, the shape derivative, as well as a basic implementation of the level-set based mesh evolution method for shape optimization. For the computation of shape derivatives we utilize the mathematical Lagrangian approach for differentiating PDE-constrained shape functions. Differentiation is done symbolically using Sympy. In numerical experiments we verify the accuracy of the computed derivatives. Finally, we showcase the capabilities of AutoFreeFem by considering shape optimization of a non-linear diffusion problem, linear and non-linear elasticity problems, a thermo-elasticity problem and a fluid-structure interaction problem.
• On a branched transport model for type-I superconductors
  
  • Goldman Michael
  , 2024. This is an extended abstract of the talk I gave in Cortona for the conference Geometric Measure Theory and applications, 2024.
• Approximation of stochastic models for epidemics on large multi-level graphs
  
  • Kubasch Madeleine
  , 2024. We study an SIR model with two levels of mixing, namely a uniformly mixing global level, and a local level with two layers of household and workplace contacts, respectively. More precisely, we aim at proposing reduced models which approximate well the epidemic dynamics at hand, while being more prone to mathematical analysis and/or numerical exploration.We investigate the epidemic impact of the workplace size distribution. Our simulation study shows that if the average workplace size is kept fixed, the variance of the workplace size distribution is a good indicator of its influence on key epidemic outcomes. In addition, this allows to design an efficient teleworking strategy. Next, we demonstrate that a deterministic, uniformly mixing SIR model calibrated using the epidemic growth rate yields a parsimonious approximation of the household-workplace model.However, the accuracy of this reduced model deteriorates over time and lacks theoretical guarantees. Hence, we study the large population limit of the stochastic household-workplace model, which we formalize as a measure-valued process with continuous state space. In a general setting, we establish convergence to the unique deterministic solution of a measure-valued equation. In the case of exponentially distributed infectious periods, a stronger reduction to a finite dimensional dynamical system is obtained.Further, in order to gain a finer insight on the impact of the model parameters on the performance of both reduced models, we perform a sensitivity study. We show that the large population limit of the household-workplace model can approximate well the epidemic even if some assumptions on the contact network are relaxed. Similarly, we quantify the impact of epidemic parameters on the capacity of the uniformly mixing reduced model to predict key epidemic outcomes.Finally, we consider density-dependent population processes in general. We establish a many-to-one formula which reduces the typical lineage of a sampled individual to a time-inhomogeneous spinal process. In addition, we use a coupling argument to quantify the large population convergence of a spinal process.
• Continuous Torque Formulation for Dual Magnetostatics using Shape Differentiation
  
  • Gauthey Thomas
  • Allaire Grégoire
  • Hage-Hassan Maya
  • Mininger Xavier
  , 2024. This paper proposes a continuous formulation of the torque for the 2D non-linear dual magnetostatics based on the virtual work principle. Shape differentiation for a class of rigid body rotation of the rotor is performed, thus giving local forces and, ultimately, the torque. This process is detailed, and a comparison is proposed with the torque expression obtained from the primal form of 2D magnetostatics.
• Path-dependent processes from signatures
  
  • Abi Jaber Eduardo
  • Gérard Louis-Amand
  • Huang Yuxing
  , 2024. We provide explicit series expansions to certain stochastic path-dependent integral equations in terms of the path signature of the time augmented driving Brownian motion. Our framework encompasses a large class of stochastic linear Volterra and delay equations and in particular the fractional Brownian motion with a Hurst index H in (0, 1). Our expressions allow to disentangle an infinite dimensional Markovian structure and open the door to straightforward and simple approximation schemes, that we illustrate numerically.
• Risk measures in finance, Backtesting, Sensitivity and Robustness
  
  • Rivoire Manon
  , 2024. In Chapter 1, we focus on two time transformations: the time-translation and the time-scaling and on the related properties called the stationarity and the self-similarity. We prove the stationarity and self-similarity properties of the processes first in a the very general framework of the Hilbert spaces; then in a the more specific framework of the the Gaussian Hilbert space where the properties are proved in distribution (weak sense) and in a trajectory sense (strict sense). We also provide examples of such processes called standard Brownian motion and fractional Brownian motion (fBm), in the univariate and multivariate frameworks (mfBm). In Chapter 2, we propose to describe price trajectories using fractional geometric Brownian motions. This allows adding correlations between logarithmic returns to express long-range dependency. Logarithmic returns are then described using self-similar Gaussian processes with stationary and correlated increments, the fBm's and mfBm's. In this framework, risk measures that are based on the loss distribution, can then be accurately predicted taking into account the long-range dependency. We focus on predicting the most commonly used risk measure by regulators, called Value-at-Risk (VaR). We introduce a model that provides a Gaussian approximation of Value-at-Risk (VaR) for the assets portfolio under fractional dynamics (mfBm). We provide a quantification of the error of approximation and we carry out backtesting experiments on simulated and market data. In Chapter 3, we propose to model the loss distribution with a heavy-tailed distribution that better takes into account the extreme events, called the Pareto distribution that presents interesting properties of scaling and stability by conditioning and to replace VaR by Expected-Shortfall which is more sensitive to the tail risk. The objective is to explore non-asymptotic robust methods for estimating ES in heavy-tailed distributions such that the Median-of-Means, the Trimmed-Means, and the Lee-Valiant estimators that we compare to the empirical mean estimator (asymptotic). We study their bias and their convergence rate.
• Interpretable and Causal Analysis for Multivariate Time Series
  
  • Dhaou Amin
  , 2024. Advances in artificial intelligence have led to the development of increasingly complex models for solving a wide range of tasks. In critical applications such as industry and medicine, it has become necessary to propose "interpretable" models that clearly establish the decision-making process, thus promoting understanding of these models and their decisions and, consequently, their user acceptance. These objectives fall within the field of eXplainable Artificial Intelligence (XAI), which has been attracting growing interest in recent years.Time-series data, which measure the evolution of variables over time, such as sensor readings or data monitoring, provide valuable information on the system's behavior. By identifying patterns in these data, we can understand the interactions between variables, improve forecasting accuracy, and design better intervention strategies. This thesis studies the analysis of high-dimensional time-series data, focusing on explaining local system deviations from normal operation and, on the global scale, modeling the underlying dynamics of the system to predict its evolution.This work has two main objectives. The first objective is to develop an interpretable algorithm that identifies the root causes of both normal and abnormal behavior in time series data. Various techniques are used to identify root causes, but they suffer from limitations in their ability to handle high dimensions and to distinguish causality from correlations.To overcome these limitations, an approach based on the concept of Granger causality [Granger 1988], which extracts interpretable and causal relationships in the form of rules, has been developed. The resulting algorithm is designed to handle different data types (numerical, categorical), provide users with interpretable explanations of the problem, and develop predictive rules to defuse the event in advance.The second objective aims to develop a forecasting model that not only predicts future values but also reveals the underlying dynamic of the time series influencing those predictions. This field, called symbolic regression, fosters transparency for users by explaining the model's reasoning. Regression models with parsimonious penalization are widely used in this field for their ability to learn complex dynamics in high-dimensional settings. Nevertheless, their forecasting performances can be limited, especially for complex, non-linear data. To address this, we propose a novel approach that combines penalized regression with forecasting error correction within a time series forecasting framework for improved learning of underlying dynamics.By achieving these goals, this research has the potential to significantly improve our ability to analyze and understand time series data. This will result in better forecasts, a better understanding of the system, and the development of more effective intervention strategies.
• Perfect simulation of Markovian load balancing queueing networks in equilibrium
  
  • Graham Carl
  , 2024. We define a wide class of Markovian load balancing networks of identical single-server infinite-buffer queues. These networks may implement classic parallel server or work stealing load balancing policies, and may be asymmetric, for instance due to topological constraints. The invariant laws are usually not known even up to normalizing constant. We provide three perfect simulation algorithms enabling Monte Carlo estimation of quantities of interest in equilibrium. The state space is infinite, and the algorithms use a dominating process provided by the network with uniform routing, in a coupling preserving a preorder which is related to the increasing convex order. It constitutes an order up to permutation of the coordinates, strictly weaker than the product order. The use of a preorder is novel in this context. The first algorithm is in direct time and uses Palm theory and acceptance rejection. Its duration is finite, a.s., but has infinite expectation. The two other algorithms use dominated coupling from the past; one achieves coalescence by simulating the dominating process into the past until it reaches the empty state, the other, valid for exchangeable policies, is a back-off sandwiching method. Their durations have some exponential moments.
• QuadWire: An extended one dimensional model for efficient mechanical simulations of bead-based additive manufacturing processes
  
  • Preumont Laurane
  • Viano Rafaël
  • Weisz-Patrault Daniel
  • Margerit Pierre
  • Allaire Grégoire
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2024, 427, pp.117010. This paper presents the basis of a new mechanical model named QuadWire dedicated to efficient simulations of bead-based additive manufacturing processes in which elongated beads undergoing significant cooling and eigenstrain are assembled to form 3D parts. The key contribution is to use a multi-particular approach containing 4 particles per material point to develop an extended 1D model capable of capturing complex 3D mechanical states, while significantly reducing computation time with respect to conventional approaches. Indeed, 3D models usually require at least 3 to 4 elements across the bead section, which results in fine discretization along the tangential direction to avoid conditioning issues, and therefore very fine mesh of the entire 3D part. In the QuadWire model, the bead height and thickness are internal dimensions, enabling a significantly coarser mesh along the tangential direction. Thus, although the QuadWire has 12 degrees of freedom per material point instead of 3 for classical models, the total number of degrees of freedom is reduced by several orders of magnitude for large parts. The proposed model is classically developed within the framework of the principle of virtual power and standard generalized hyperelastic media (i.e finite strain theory), which necessitates a thermodynamic analysis. Furthermore, the proposed approach includes native and manageable kinematic constraints between successive beads so that the stress state properly evolves during fabrication. Finite element analysis is used for numerical implementation under infinitesimal strain assumption for the sake of simplicity, and the QuadWire stiffness parameters are optimized so that the mechanical response fit conventional 3D approaches. To validate and demonstrate the capabilities of the proposed strategy, the evolution of displacements and stresses in fused deposition modeling of polylactide is simulated. (10.1016/j.cma.2024.117010)
  DOI : 10.1016/j.cma.2024.117010
• Improved High-Probability Bounds for the Temporal Difference Learning Algorithm via Exponential Stability
  
  • Samsonov Sergey
  • Tiapkin Daniil
  • Naumov Alexey
  • Moulines Eric
  , 2023, vol. 247 of PMLR, pp.4511-4547. In this paper we consider the problem of obtaining sharp bounds for the performance of temporal difference (TD) methods with linear function approximation for policy evaluation in discounted Markov decision processes. We show that a simple algorithm with a universal and instance-independent step size together with Polyak-Ruppert tail averaging is sufficient to obtain near-optimal variance and bias terms. We also provide the respective sample complexity bounds. Our proof technique is based on refined error bounds for linear stochastic approximation together with the novel stability result for the product of random matrices that arise from the TD-type recurrence. (10.48550/arXiv.2310.14286)
  DOI : 10.48550/arXiv.2310.14286
• Automated approach for source location in shallow water
  
  • Niclas Angèle
  • Garnier Josselin
  , 2024. This work proposes a fully automated method for recovering the location of a source and medium parameters in shallow water. The scenario involves an unknown source emitting lowfrequency sound waves in a shallow water environment and a single hydrophone recording the signal. Using the spectrogram of each modal component obtained by a warping method, we investigate how to recover the modal travel times and we provide stability estimates. A penalized minimization algorithm is then presented to estimate the source location and medium parameters. The proposed method is tested on different experimental data, demonstrating its effectiveness in real-world scenarios.
• On One-Dimensional Bose Gases with Two-Body and (Critical) Attractive Three-Body Interactions
  
  • Nguyen Dinh-Thi
  • Ricaud Julien
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2024, 56 (3), pp.3203-3251. We consider a one-dimensional, trapped, focusing Bose gas where $N$ bosons interact with each other via both a two-body interaction potential of the form $aN^{\alpha-1} U(N^\alpha(x-y))$ and an attractive three-body interaction potential of the form $-bN^{2\beta-2} W(N^\beta(x-y,x-z))$, where $a\in\mathbb{R}$, $b,\alpha>0$, $0<\beta<1$, $U, W \geq 0$, and $\int_{\mathbb{R}}U(x) \mathop{}\!\mathrm{d}x = 1 = \iint_{\mathbb{R}^2} W(x,y) \mathop{}\!\mathrm{d}x \mathop{}\!\mathrm{d}y$. The system is stable either for any $a\in\mathbb{R}$ as long as $b<\mathfrak{b} := 3\pi^2/2$ (the critical strength of the 1D focusing quintic nonlinear Schrödinger equation) or for $a \geq 0$ when $b=\mathfrak{b}$. In the former case, fixing $b \in (0,\mathfrak{b})$, we prove that in the mean-field limit the many-body system exhibits the Bose$\unicode{x2013}$Einstein condensation on the cubic-quintic NLS ground states. When assuming $b=b_N \nearrow \mathfrak{b}$ and $a=a_N \to 0$ as $N \to\infty$, with the former convergence being slow enough and "not faster" than the latter, we prove that the ground state of the system is fully condensed on the (unique) solution to the quintic NLS equation. In the latter case $b=\mathfrak{b}$ fixed, we obtain the convergence of many-body energy for small $\beta$ when $a > 0$ is fixed. Finally, we analyze the behavior of the many-body ground states when the convergence $b_N \nearrow \mathfrak{b}$ is "faster" than the slow enough convergence $0<a_N \searrow 0$. (10.1137/22M1535139)
  DOI : 10.1137/22M1535139
• Motion by curvature and large deviations for an interface dynamics on Z 2
  
  • Dagallier B
  Probability and Mathematical Physics, MSP, 2024, 5 (3), pp.609-734. We study large deviations for a Markov process on curves in Z 2 mimicking the motion of an interface. Our dynamics can be tuned with a parameter β, which plays the role of an inverse temperature, and coincides at β = ∞ with the zero-temperature Ising model with Glauber dynamics, where curves correspond to the boundaries of droplets of one phase immersed in a sea of the other one. We prove that contours typically follow a motion by curvature with an influence of the parameter β, and establish large deviations bounds at all large enough β &lt; ∞. The diffusion coefficient and mobility of the model are identified and correspond to those predicted in the literature. (10.2140/pmp.2024.5.609)
  DOI : 10.2140/pmp.2024.5.609
• On non-uniqueness of phase retrieval for functions with disconnected support
  
  • Novikov Roman
  • Xu Tianli
  , 2024. We show that the phase retrieval problem is not uniquely solvable even for functions with strongly disconnected compact support in multidimensions.