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Hyper-contractivity and entropy decay in discrete time

Justin Salez

TL;DR

This paper investigates the relationship between hyper-contractivity and entropy decay for measure-preserving kernels $T$ on a probability space. It proves a one-step, static implication: if $T$ satisfies $\|T\|_{p\to q}\le 1$ with $1\le p\le q$, then for all $\mu$ we have $\mathrm H(\mu T\,|\pi)\le (p/q)\,\mathrm H(\mu\,|\pi)$, with no reversibility or regularity required. The result can be iterated for discrete-time processes to obtain geometric rates in relative entropy and provides a stronger instantaneous bound at fixed time compared to the classical dynamic link between hypercontractivity and entropy decay along Markov semigroups; the authors also supply a concise duality-based proof that avoids semigroup techniques. Overall, the work offers a general and elementary static framework connecting functional-analytic regularization (hyper-contractivity) with information-theoretic decay (entropy).

Abstract

Consider a measure-preserving transition kernel $T$ on an arbitrary probability space $(\mathbb X,\mathcal cA,π)$. In this level of generality, we prove that a one-step hyper-contractivity estimate of the form $\|T\|_{p\to q}\le 1$ with $p< q$ implies a one-step entropy contraction estimate of the form ${\mathrm H}(μT\,|\,π)\le θ\, {\mathrm H}(μ\,|\,π)$, with $θ=p/q$. Neither reversibility, nor any sort of regularity is required. This static implication is simultaneously simpler and stronger than the celebrated dynamic relation between exponential hyper-contractivity and exponential entropy decay along continuous-time Markov semi-groups.

Hyper-contractivity and entropy decay in discrete time

TL;DR

This paper investigates the relationship between hyper-contractivity and entropy decay for measure-preserving kernels on a probability space. It proves a one-step, static implication: if satisfies with , then for all we have , with no reversibility or regularity required. The result can be iterated for discrete-time processes to obtain geometric rates in relative entropy and provides a stronger instantaneous bound at fixed time compared to the classical dynamic link between hypercontractivity and entropy decay along Markov semigroups; the authors also supply a concise duality-based proof that avoids semigroup techniques. Overall, the work offers a general and elementary static framework connecting functional-analytic regularization (hyper-contractivity) with information-theoretic decay (entropy).

Abstract

Consider a measure-preserving transition kernel on an arbitrary probability space . In this level of generality, we prove that a one-step hyper-contractivity estimate of the form with implies a one-step entropy contraction estimate of the form , with . Neither reversibility, nor any sort of regularity is required. This static implication is simultaneously simpler and stronger than the celebrated dynamic relation between exponential hyper-contractivity and exponential entropy decay along continuous-time Markov semi-groups.
Paper Structure (6 sections, 1 theorem, 21 equations)

This paper contains 6 sections, 1 theorem, 21 equations.

Table of Contents

  1. Introduction
  2. Transition kernels.
  3. Hyper-contractivity.
  4. Entropy contraction.
  5. Result and discussion
  6. Proof of the theorem

Key Result

Theorem 1

For any parameters $1\le p\le q$, the hyper-contractivity estimate (def:hyper-contractivity) implies the entropy contraction estimate (def:entropycontraction), with $\theta=p/q$.

Theorems & Definitions (2)

  • Theorem 1: Main result
  • Remark 1: Converse