Max Fathi
YOU?
Author Swipe
View article: On cutoff via rigidity for high dimensional curved diffusions
On cutoff via rigidity for high dimensional curved diffusions Open
We consider overdamped Langevin diffusions in Euclidean space, with curvature equal to the spectral gap. This includes the Ornstein–Uhlenbeck process as well as non Gaussian and non product extensions with convex interaction, such as the D…
View article: Exponential convergence for ultrafast diffusion equations with log-concave weights
Exponential convergence for ultrafast diffusion equations with log-concave weights Open
We study the asymptotic behavior of a weighted ultrafast diffusion PDE on the real line, with a log-concave and log-lipschitz weight, and prove exponential convergence to equilibrium. This result goes beyond the compact setting studied in …
View article: On cutoff via rigidity for high dimensional curved diffusions
On cutoff via rigidity for high dimensional curved diffusions Open
We consider overdamped Langevin diffusions in Euclidean space, with curvature equal to the spectral gap. This includes the Ornstein-Uhlenbeck process as well as non Gaussian and non product extensions with convex interaction, such as the D…
View article: Stochastic proof of the sharp symmetrized Talagrand inequality
Stochastic proof of the sharp symmetrized Talagrand inequality Open
A new proof of the sharp symmetrized form of Talagrand’s transport-entropy inequality is given. Compared to stochastic proofs of other Gaussian functional inequalities, the new idea here is a certain coupling induced by time-reversed marti…
View article: Growth estimates on optimal transport maps via concentration inequalities
Growth estimates on optimal transport maps via concentration inequalities Open
We give an alternative proof and some extensions of results of Carlier, Figalli and Santambrogio on polynomial upper bounds on the Brenier map between probability measures under various conditions on the densities. The proofs are based on …
View article: Stochastic proof of the sharp symmetrized Talagrand inequality
Stochastic proof of the sharp symmetrized Talagrand inequality Open
We give a new proof of the sharp symmetrized form of Talagrand's transport-entropy inequality. Compared to stochastic proofs of other Gaussian functional inequalities, the new idea here is a certain coupling induced by time-reversed martin…
View article: Stability estimates for the sharp spectral gap bound under a curvature-dimension condition
Stability estimates for the sharp spectral gap bound under a curvature-dimension condition Open
We study stability of the sharp spectral gap bounds for metric-measure spaces satisfying a curvature bound. Our main result, new even in the smooth setting, is a sharp quantitative estimate showing that if the spectral gap of an space is a…
View article: Stability of the Poincaré-Korn inequality
Stability of the Poincaré-Korn inequality Open
We resolve a question of Carrapatoso et al. on Gaussian optimality for the sharp constant in Poincaré-Korn inequalities, under a moment constraint. We also prove stability, showing that measures with near-optimal constant are quantitativel…
View article: Stability of Klartag's improved Lichnerowicz inequality
Stability of Klartag's improved Lichnerowicz inequality Open
In a recent work, Klartag gave an improved version of Lichnerowicz' spectral gap bound for uniformly log-concave measures, which improves on the classical estimate by taking into account the covariance matrix. We analyze the equality cases…
View article: Some obstructions to contraction theorems on the half-sphere
Some obstructions to contraction theorems on the half-sphere Open
Caffarelli's contraction theorem states that probability measures with uniformly logconcave densities on R d can be realized as the image of a standard Gaussian measure by a globally Lipschitz transport map. We discuss some counterexamples…
View article: HWI inequalities in discrete spaces via couplings
HWI inequalities in discrete spaces via couplings Open
HWI inequalities are interpolation inequalities relating entropy, Fisher information and optimal transport distances. We adapt an argument of Y. Wu for proving the Gaussian HWI inequality via a coupling argument to the discrete setting, es…
View article: HWI inequalities in discrete spaces via couplings
HWI inequalities in discrete spaces via couplings Open
HWI inequalities are interpolation inequalities relating entropy, Fisher information and optimal transport distances. We adapt an argument of Y. Wu for proving the Gaussian HWI inequality via a coupling argument to the discrete setting, es…
View article: Transportation onto log-Lipschitz perturbations
Transportation onto log-Lipschitz perturbations Open
We establish sufficient conditions for the existence of globally Lipschitz transport maps between probability measures and their log-Lipschitz perturbations, with dimension-free bounds. Our results include Gaussian measures on Euclidean sp…
View article: Bounds in L1 Wasserstein distance on the normal approximation of general M-estimators
Bounds in L1 Wasserstein distance on the normal approximation of general M-estimators Open
We derive quantitative bounds on the rate of convergence in $L^1$ Wasserstein\ndistance of general M-estimators, with an almost sharp (up to a logarithmic\nterm) behavior in the number of observations. We focus on situations where the\nest…
View article: Recent progress on limit theorems for large stochastic particle systems
Recent progress on limit theorems for large stochastic particle systems Open
This article presents a selection of recent results in the mathematical study of physical systems described by a large number of particles, with various types of interactions (mean-field, moderate, nearest-neighbor). Limit theorems are obt…
View article: Relaxing the Gaussian assumption in shrinkage and SURE in high dimension
Relaxing the Gaussian assumption in shrinkage and SURE in high dimension Open
Shrinkage estimation is a fundamental tool of modern statistics, pioneered by\nCharles Stein upon his discovery of the famous paradox involving the\nmultivariate Gaussian. A large portion of the subsequent literature only\nconsiders the ef…
View article: Hypocoercivity with Schur complements
Hypocoercivity with Schur complements Open
We propose an approach to obtaining explicit estimates on the resolvent of hypocoercive operators by using Schur complements, rather than from an exponential decay of the evolution semigroup combined with a time integral. We present applic…
View article: Stability estimates for the sharp spectral gap bound under a curvature-dimension condition
Stability estimates for the sharp spectral gap bound under a curvature-dimension condition Open
We study stability of the sharp spectral gap bounds for metric-measure spaces satisfying a curvature bound. Our main result, new even in the smooth setting, is a sharp quantitative estimate showing that if the spectral gap of an RCD$(N-1, …
View article: Bounds in $L^1$ Wasserstein distance on the normal approximation of general M-estimators
Bounds in $L^1$ Wasserstein distance on the normal approximation of general M-estimators Open
We derive quantitative bounds on the rate of convergence in $L^1$ Wasserstein distance of general M-estimators, with an almost sharp (up to a logarithmic term) behavior in the number of observations. We focus on situations where the estima…
View article: Stability of eigenvalues and observable diameter in RCD$(1,\infty)$ spaces
Stability of eigenvalues and observable diameter in RCD$(1,\infty)$ spaces Open
We study stability of the spectral gap and observable diameter for metricmeasure spaces satisfying the RCD(1, $\infty$) condition. We show that if such a space has an almost maximal spectral gap, then it almost contains a Gaussian componen…
View article: Stability estimates for invariant measures of diffusion processes, with applications to stability of moment measures and Stein kernels
Stability estimates for invariant measures of diffusion processes, with applications to stability of moment measures and Stein kernels Open
We investigate stability of invariant measures of diffusion processes with respect to $L^p$ distances on the coefficients, under an assumption of log-concavity. The method is a variant of a technique introduced by Crippa and De Lellis to s…
View article: Higher-order Stein kernels for Gaussian approximation
Higher-order Stein kernels for Gaussian approximation Open
We introduce higher-order Stein kernels relative to the standard Gaussian measure, which generalize the usual Stein kernels by involving higher-order derivatives of test functions. We relate the associated discrepancies to various metrics …
View article: Relaxing the Gaussian assumption in Shrinkage and SURE in high dimension
Relaxing the Gaussian assumption in Shrinkage and SURE in high dimension Open
Shrinkage estimation is a fundamental tool of modern statistics, pioneered by Charles Stein upon his discovery of the famous paradox involving the multivariate Gaussian. A large portion of the subsequent literature only considers the effic…
View article: Self-improvement of the Bakry-Emery criterion for Poincar{é} inequalities and Wasserstein contraction using variable curvature bounds
Self-improvement of the Bakry-Emery criterion for Poincar{é} inequalities and Wasserstein contraction using variable curvature bounds Open
We study Poincar{é} inequalities and long-time behavior for diffusion processes on R^n under a variable curvature lower bound, in the sense of Bakry-Emery. We derive various estimates on the rate of convergence to equilibrium in L^1 optima…
View article: Stein kernels and moment maps
Stein kernels and moment maps Open
We describe a construction of Stein kernels using moment maps, which are solutions to a variant of the Monge-Ampère equation. As a consequence, we show how regularity bounds on these maps control the rate of convergence in the classical ce…
View article: A proof of the Caffarelli contraction theorem via entropic regularization
A proof of the Caffarelli contraction theorem via entropic regularization Open
We give a new proof of the Caffarelli contraction theorem, which states that the Brenier optimal transport map sending the standard Gaussian measure onto a uniformly log-concave probability measure is Lipschitz. The proof combines a recent…