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Jordan Serres

Publications and source records attributed to Jordan Serres.

13 recordsLinked to original sources
arXiv2026

Alexandrov spaces with non negative curvature and displacement convexity of the entropy tensor

On a smooth Riemannian manifold, Aishwarya, Rotem and Shenfeld characterised nonnegative sectional curvature as the matrix displacement convexity of an entropy tensor, the Lagrangian, matrix-valued refinement of Shenfeld's entropy matrix. In order to extend the entropy tensor to a finite-dimensional Alexandrov space of curvature bounded below, we construct a parallel trivialisation satisfying both the cocycle property and the second variation formula. The construction is strongly inspired by Petrunin's synthetic parallel transport. The entropy tensor defined is taken in block-diagonal form; on smooth manifolds the resulting convexity property still characterises nonnegative sectional curvature exactly. We show that the smooth equivalence persists synthetically: an Alexandrov space has nonnegative curvature if and only if its entropy tensor is matrix displacement convex.

math.DG↗
arXiv2026

$K-$means with learned metrics

We study the Fr\'echet $k-$means of a metric measure space when both the measure and the distance are unknown and have to be estimated. We prove a general result that states that the $k-$means are continuous with respect to the measured Gromov-Hausdorff topology. In this situation, we also prove a stability result for the Voronoi clusters they determine. We do not assume uniqueness of the set of $k-$means, but when it is unique, the results are stronger. This framework provides a unified approach to proving consistency for a wide range of metric learning procedures. As concrete applications, we obtain new consistency results for several important estimators that were previously unestablished, even when $k=1$. These include $k-$means based on: (i) Isomap and Fermat geodesic distances on manifolds, (ii) difussion distances, (iii) Wasserstein distances computed with respect to learned ground metrics. Finally, we consider applications beyond the statistical inference paradigm like (iv) first passage percolation and (v) discrete approximations of length spaces.

math.ST↗
arXiv2025

A Generalization of Caffarelli's Contraction Theorem to Nearly Spherical Manifolds

We show that every nearly spherical manifold can be realized as the volume-preserving image of a round sphere, via the Brenier-McCann optimal transport map. This theorem extends Caffarelli's contraction theorem to nearly spherical manifolds and yields, as a corollary, a proof of a perturbative form of Milman's conjecture. The proof is based on a novel stability result for optimal transport maps on the sphere.

math.AP↗
arXiv2025

Finite sample bounds for barycenter estimation in geodesic spaces

We study the problem of estimating the barycenter of a distribution given i.i.d. data in a geodesic space. Assuming an upper curvature bound in Alexandrov's sense and a support condition ensuring the strong geodesic convexity of the barycenter problem, we establish finite-sample error bounds in expectation and with high probability. Our results generalize Hoeffding- and Bernstein-type concentration inequalities from Euclidean to geodesic spaces. Building on these concentration inequalities, we derive statistical guarantees for two efficient algorithms for the computation of barycenters.

math.ST↗
arXiv2024

Generalized perimeters and gradient estimates

We use a variational formulation to define a generalized notion of perimeter, from which we derive abstract isoperimetric Cheeger's inequalities via gradient estimates on solutions of Poisson equations. Our abstract framework unifies many existing results and in particular allows us to recover the $W_1-W^{1,1}$ transport inequality, which strengthens the usual transport-information inequality. Conversely, we also prove that Cheeger's inequality implies certain first order Calder{\'o}n-Zygmund-type gradient estimates.

math.FA↗
arXiv2024

Isoperimetric and geometric inequalities in quantitative form: Stein's method approach

We adapt Stein's method to isoperimetric and geometric inequalities. The main challenge is the treatment of boundary terms. We address this by using an elliptic PDE with an oblique boundary condition. We apply our geometric formulation of Stein's method to obtain stability of the Brock-Weinstock inequality, stability of the isoperimetric inequality under a constraint on Steklov's first non-zero eigenvalue, and stability for the combination of weighted and unweighted perimeters. All stability results are formulated with respect to the $\alpha$-Zolotarev distance, $\alpha$ $\in$ (0, 1], that we introduce to interpolate between the Fraenkel asymmetry and the Kantorovich distance.

math.AP↗
arXiv2024

A generalization of Gr\"unbaum's inequality in RCD$(0,N)$-spaces

We generalize Gr\"unbaum's classical inequality in convex geometry to curved spaces with nonnegative Ricci curvature, precisely, to $\mathrm{RCD}(0,N)$-spaces with $N \in (1,\infty)$ as well as weighted Riemannian manifolds of $\mathrm{Ric}_N \ge 0$ for $N \in (-\infty,-1) \cup \{\infty\}$. Our formulation makes use of the isometric splitting theorem; given a convex set $\Omega$ and the Busemann function associated with any straight line, the volume of the intersection of $\Omega$ and any sublevel set of the Busemann function that contains a barycenter of $\Omega$ is bounded from below in terms of $N$. We also extend this inequality beyond uniform distributions on convex sets. Moreover, we establish some rigidity results by using the localization method, and the stability problem is also studied.

math.MG↗
arXiv2023

Concentration of empirical barycenters in metric spaces

Barycenters (aka Fr\'echet means) were introduced in statistics in the 1940's and popularized in the fields of shape statistics and, later, in optimal transport and matrix analysis. They provide the most natural extension of linear averaging to non-Euclidean geometries, which is perhaps the most basic and widely used tool in data science. In various setups, their asymptotic properties, such as laws of large numbers and central limit theorems, have been established, but their non-asymptotic behaviour is still not well understood. In this work, we prove finite sample concentration inequalities (namely, generalizations of Hoeffding's and Bernstein's inequalities) for barycenters of i.i.d. random variables in metric spaces with non-positive curvature in Alexandrov's sense. As a byproduct, we also obtain PAC guarantees for a stochastic online algorithm that computes the barycenter of a finite collection of points in a non-positively curved space. We also discuss extensions of our results to spaces with possibly positive curvature.

math.ST↗
arXiv2022

Behavior of the Poincar{\'e} constant along the Polchinski renormalization flow

We control the behavior of the Poincar{\'e} constant along the Polchinski renormalization flow using a dynamic version of $\Gamma$-calculus. We also treat the case of higher order eigenvalues. Our method generalizes a method introduced by B. Klartag and E. Putterman to analyze the evolution of log-concave distributions along the heat flow. Furthermore, we apply it to general $\Phi$ 4-measures and discuss the interpretation in terms of transport maps.

math.FA↗
arXiv2022

Stability of higher order eigenvalues in dimension one

We study stability of the eigenvalues of the generator of a one dimensional reversible diffusion process satisfying some natural conditions. The proof is based on Stein's method. In particular, these results are applied to the Normal distribution (via the Ornstein-Uhlenbeck process), to Gamma distributions (via the Laguerre process) and to Beta distributions (via Jacobi process).

math.CA↗
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↗
arXiv2022

Stability of Poincar{é} constant

We study stability of the sharp Poincar{é} constant of the invariant probability measure of a reversible diffusion process satisfying some natural conditions. The proof is based on the spectral interpretation of Poincar{é} inequalities and Stein's method. In particular, these results are applied to the gamma distributions and to strictly log-concave measures in dimension one, giving stability for Brascamp-Lieb inequalities.

math.CA↗