arXiv subjects

Max Fathi

Publications and source records attributed to Max Fathi.

At least 19 recordsLinked to original sources

arXiv2026

Quantitative stability in distribution for the Sobolev inequality under curvature dimension condition

The goal of this note is to investigate quantitative stability properties of the critical Sobolev inequality in ${\sf CD}(N-1,N)$ metric measure spaces. Assuming that the optimal constant for the inequality is almost the same as the one of the round sphere, we show that the cumulative distribution of any almost extremal function is close, in Wasserstein distance, to the one of an Aubin-Talenti bubble on the round sphere. We obtain similar results for the log Sobolev inequality and the spectral gap under various curvature and dimension assumptions. In all cases we obtain a quantitative stability with sharp exponent.

math.AP↗

arXiv2025

Quantitative rigidity of the Wasserstein contraction under convolution

The aim of this paper is to investigate the contraction properties of $p$-Wasserstein distances with respect to convolution in Euclidean spaces both qualitatively and quantitatively. We connect this question to the question of uniform convexity of the Kantorovich functional on which there was substantial recent progress (mostly for $p=2$ and partially for $p>1$). Motivated by this connection we extend these uniform convexity results to the case $p=1$, which is of independent interest.

math.AP↗

arXiv2025

Exponential convergence for ultrafast diffusion equations with log-concave weights

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 [22]. This equation is motivated by the gradient flow approach to the problem of quantization of measures introduced in [11].

math.AP↗

arXiv2025

On the cutoff phenomenon for fast diffusion and porous medium equations

The cutoff phenomenon, conceptualized at the origin for finite Markov chains, states that for a parametric family of evolution equations, started from a point, the distance towards a long time equilibrium may become more and more abrupt for certain choices of initial conditions, when the parameter tends to infinity. This threshold phenomenon can be seen as a critical competition between trend to equilibrium and worst initial condition. In this note, we investigate this phenomenon beyond stochastic processes, in the context of the analysis of nonlinear partial differential equations, by proving cutoff for the fast diffusion and porous medium Fokker-Planck equations on the Euclidean space, when the dimension tends to infinity. We formulate the phenomenon using quadratic Wasserstein distance, as well as using specific relative entropy and Fisher information. Our high dimensional asymptotic analysis uses the exact solvability of the model involving Barenblatt profiles. It includes the Ornstein-Uhlenbeck dynamics as a special linear case.

math.AP↗

arXiv2024

On cutoff via rigidity for high dimensional curved diffusions

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 Dyson process from random matrix theory. We show that a cutoff phenomenon or abrupt convergence to equilibrium occurs in high dimension, at a critical time equal to the logarithm of the dimension divided by twice the spectral gap. This cutoff holds for Wasserstein distance, total variation, relative entropy, and Fisher information. A key observation is a relation to a spectral rigidity, linked to the presence of a Gaussian factor. A novelty is the extensive usage of functional inequalities, even for short-time regularization, and the reduction to Wasserstein. The proofs are short and conceptual. Since the product condition is satisfied, an Lp cutoff holds for all p. We moreover discuss a natural extension to Riemannian manifolds, a link with logarithmic gradient estimates in short-time for the heat kernel, and ask about stability by perturbation. Finally, beyond rigidity but still for diffusions, a discussion around the recent progress on the product condition for non-negatively curved diffusions leads us to introduce a new curvature product condition.

math.PR↗

arXiv2024

Growth estimates on optimal transport maps via concentration inequalities

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 the monotonicity of the map and various concentration inequalities, as already used by Colombo and Fathi to prove quadratic growth for transport maps from the standard Gaussian onto log-concave measures.

math.AP↗

arXiv2024

Stochastic proof of the sharp symmetrized Talagrand inequality

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 martingale representations.

math.PR↗

arXiv2024

Stability of the Poincar\'e-Korn inequality

We resolve a question of Carrapatoso et al. on Gaussian optimality for the sharp constant in Poincar\'e-Korn inequalities, under a moment constraint. We also prove stability, showing that measures with near-optimal constant are quantitatively close to standard Gaussian.

math.AP↗

arXiv2024

Stability of Klartag's improved Lichnerowicz inequality

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 in Klartag's bound, showing that it can be further improved whenever the measure has no Gaussian factor. Additionally, we give a quantitative improvement for log-concave measures with finite Fisher information.

math.FA↗

arXiv2024

Some obstructions to contraction theorems on the half-sphere

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 and obstructions that prevent a similar result from holding on the half-sphere endowed with a uniform measure, answering a question of Beck and Jerison.

math.FA↗

arXiv2023

HWI inequalities in discrete spaces via couplings

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, establishing new interpolation inequalities for the discrete hypercube and the discrete torus. In particular, we obtain an improvement of the modified logarithmic Sobolev inequality for the discrete hypercube of Bobkov and Tetali.

math.PR↗

arXiv2023

Transportation onto log-Lipschitz perturbations

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 spaces and uniform measures on spheres as source measures. More generally, we prove results for source measures on manifolds satisfying strong curvature assumptions. These seem to be the first examples of dimension-free Lipschitz transport maps in non-Euclidean settings, which are moreover sharp on the sphere. We also present some applications to functional inequalities, including a new dimension-free Gaussian isoperimetric inequality for log-Lipschitz perturbations of the standard Gaussian measure. Our proofs are based on the Langevin flow construction of transport maps of Kim and Milman.

math.PR↗

arXiv2022

Stability estimates for the sharp spectral gap bound under a curvature-dimension condition

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$(N-1, N)$ space is almost minimal, then the pushforward of the measure by an eigenfunction associated with the spectral gap is close to a Beta distribution. The proof combines estimates on the eigenfunction obtained via a new $L^1$-functional inequality for RCD spaces with Stein's method for distribution approximation. We also derive analogous, almost sharp, estimates for infinite and negative values of the dimension parameter.

math.MG↗

arXiv2021

Bounds in $L^1$ Wasserstein distance on the normal approximation of general M-estimators

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 estimator does not have an explicit expression as a function of the data. The general method may be applied even in situations where the observations are not independent. Our main application is a rate of convergence for cross validation estimation of covariance parameters of Gaussian processes.

math.ST↗

arXiv2021

Stability of eigenvalues and observable diameter in RCD$(1,\infty)$ spaces

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 component, and the Laplacian has eigenvalues that are close to any integers, with dimension-free quantitative bounds. Under the additional assumption that the space admits a needle disintegration, we show that the spectral gap is almost maximal iff the observable diameter is almost maximal, again with quantitative dimension-free bounds.

math.MG↗

arXiv2020

Stability estimates for invariant measures of diffusion processes, with applications to stability of moment measures and Stein kernels

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 study transport equations. As an application, we prove a partial extension of an inequality of Ledoux, Nourdin and Peccati relating transport distances and Stein discrepancies to a non-Gaussian setting via the moment map construction of Stein kernels.

math.PR↗

arXiv2020

Relaxing the Gaussian assumption in Shrinkage and SURE in high dimension

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 efficiency of shrinkage, and that of an associated procedure known as Stein's Unbiased Risk Estimate, or SURE, in the Gaussian setting of that original work. We investigate what extensions to the domain of validity of shrinkage and SURE can be made away from the Gaussian through the use of tools developed in the probabilistic area now known as Stein's method. We show that shrinkage is efficient away from the Gaussian under very mild conditions on the distribution of the noise. SURE is also proved to be adaptive under similar assumptions, and in particular in a way that retains the classical asymptotics of Pinsker's theorem. Notably, shrinkage and SURE are shown to be efficient under mild distributional assumptions, and particularly for general isotropic log-concave measures.

math.ST↗

arXiv2020

Self-improvement of the Bakry-Emery criterion for Poincar{\'e} inequalities and Wasserstein contraction using variable curvature bounds

We study Poincar{\'e} 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 optimal transport distance, as well as bounds on the constant in the Poincar{\'e} inequality in several situations of interest, including some where curvature may be negative. In particular, we prove a self-improvement of the Bakry-Emery estimate for Poincar{\'e} inequalities when curvature is positive but not constant.

math.FA↗