Sharp detection of low-dimensional structure in probability measures via dimensional logarithmic Sobolev inequalities

TL;DR

This paper introduces a novel approach for detecting low-dimensional structures in high-dimensional probability measures using dimensional logarithmic Sobolev inequalities, enhancing approximation and dimension reduction techniques.

Contribution

It establishes a connection between dimensional LSIs and measure approximation, extending prior work and providing improved bounds for non-Gaussian measures.

Findings

Minimizing dimensional LSI is equivalent to KL divergence minimization for Gaussian measures.

Dimensional LSI yields better majorants for gradient-based dimension reduction in non-Gaussian measures.

The approach improves bounds for the squared Hellinger distance using the dimensional Poincaré inequality.

Abstract

Identifying low-dimensional structure in high-dimensional probability measures is an essential pre-processing step for efficient sampling. We introduce a method for identifying and approximating a target measure $\pi$ as a perturbation of a given reference measure $\mu$ along a few significant directions of $\mathbb{R}^{d}$. The reference measure can be a Gaussian or a nonlinear transformation of a Gaussian, as commonly arising in generative modeling. Our method extends prior work on minimizing majorizations of the Kullback--Leibler divergence to identify optimal approximations within this class of measures. Our main contribution unveils a connection between the \emph{dimensional} logarithmic Sobolev inequality (LSI) and approximations with this ansatz. Specifically, when the target and reference are both Gaussian, we show that minimizing the dimensional LSI is equivalent to minimizing the KL divergence restricted to this ansatz. For general non-Gaussian measures, the dimensional LSI produces majorants that uniformly improve on previous majorants for gradient-based dimension reduction. We further demonstrate the applicability of this analysis to the squared Hellinger distance, where analogous reasoning shows that the dimensional Poincar\'e inequality offers improved bounds.