An Information-Theoretic View of Stochastic Localization

TL;DR

This paper offers an elementary, information-theoretic proof of Eldan's stochastic localization theorem, which decomposes probability measures into a small number of near-product measures based on entropy and covariance.

Contribution

It provides a simplified, information-theoretic proof of Eldan's decomposition theorem, enhancing understanding and accessibility.

Findings

Elementary proof of Eldan's theorem using information theory

Decomposition characterized by entropy and covariance

Simplifies the understanding of stochastic localization methods

Abstract

Given a probability measure $\mu$ over ${\mathbb R}^n$, it is often useful to approximate it by the convex combination of a small number of probability measures, such that each component is close to a product measure. Recently, Ronen Eldan used a stochastic localization argument to prove a general decomposition result of this type. In Eldan's theorem, the `number of components' is characterized by the entropy of the mixture, and `closeness to product' is characterized by the covariance matrix of each component. We present an elementary proof of Eldan's theorem which makes use of an information theory (or estimation theory) interpretation. The proof is analogous to the one of an earlier decomposition result known as the `pinning lemma.'