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On measures strongly log-concave on a subspace

Pierre Bizeul

TL;DR

The paper extends stochastic localization to log-concave measures that are curved only along a subspace, introducing a projector-based inequality $\nabla^2 V \ge \eta P_E$ and analyzing the projected covariance $Q= P_{E^{\perp}}KP_{E^{\perp}}$. By restricting localization to the subspace and controlling the induced $k$-dimensional covariance $Q_t$, the authors derive a Poincaré bound $c_P(\mu) \lesssim \max\left(\frac{1}{\sqrt{\eta}}, \|Q\|_{op}^{1/2}\Psi_k\sqrt{\max(\log k,1)}\right)$ and a Gaussian-type concentration bound, with an isotropic specialization yielding $\alpha_\mu(r) \lesssim \exp(-c\min(r, r^2/(\psi_n^2\log n)))$. The approach hinges on a restricted stochastic localization process, martingale methods, and dimension-reduction through the projected covariance, offering a pathway toward KLS-type conclusions under partial curvature. The results emphasize that concentration can be amplified by focusing localization on the subspace where curvature acts, and they provide explicit dependencies on the subspace dimension and the projection of the covariance.

Abstract

In this work we study the concentration properties of log-concave measures that are curved only on a subspace of directions. Proofs uses an adapted version of the stochastic localization process.

On measures strongly log-concave on a subspace

TL;DR

The paper extends stochastic localization to log-concave measures that are curved only along a subspace, introducing a projector-based inequality and analyzing the projected covariance . By restricting localization to the subspace and controlling the induced -dimensional covariance , the authors derive a Poincaré bound and a Gaussian-type concentration bound, with an isotropic specialization yielding . The approach hinges on a restricted stochastic localization process, martingale methods, and dimension-reduction through the projected covariance, offering a pathway toward KLS-type conclusions under partial curvature. The results emphasize that concentration can be amplified by focusing localization on the subspace where curvature acts, and they provide explicit dependencies on the subspace dimension and the projection of the covariance.

Abstract

In this work we study the concentration properties of log-concave measures that are curved only on a subspace of directions. Proofs uses an adapted version of the stochastic localization process.
Paper Structure (4 sections, 14 theorems, 63 equations)

This paper contains 4 sections, 14 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Restricted stochastic localization
  3. Control of the covariance matrix
  4. Proof of the main theorem

Key Result

Theorem 1.1

Let $V:\mathbb R^n\mapsto\mathbb R$ be a $C^2$ convex potential such that $d\mu(x) = e^{-V(x)}dx$ is a probability measure. Suppose that there is $1\leq k \leq n$, a subspace $E$ of codimension $k$ and $\eta>0$ such that where $P_E$ is the orthogonal projector onto E. Let $K$ be the covariance matrix of $\mu$. Define $Q = P_{E^\perp} KP_{E^\perp}$

Theorems & Definitions (29)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1
  • Remark 2
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Proposition 2.1
  • proof
  • ...and 19 more