Hypercontractivity Via the Entropy Method.

Eric Blais,Li-Yang Tan

doi:10.4086/toc.2013.v009a029

Abstract

The Hypercontractive Inequality of Bonami (1968, 1970) and Gross (1975) is equivalent to the following statement: for every q > 2 and every function f : {−1,1}n→ R of Fourier degree at most m, ‖ f‖q ≤ (q−1)m/2‖ f‖2 . The original proof of this inequality is analytical. Friedgut and Rodl (2001) gave an alternative proof of the slightly weaker Hypercontractive Inequality ‖ f‖4 ≤ 28m/4‖ f‖2 by combining tools from information theory and combinatorics. Specifically, they recast the problem as a statement about multi-hypergraphs, generalized Shearer’s lemma, and used probabilistic arguments to obtain the inequality. We show that Shearer’s Lemma and elementary arguments about the entropy of random variables are sufficient to recover the optimal Hypercontractive Inequality for all even integers q. ACM Classification: G.2, G.3, F.1.3 AMS Classification: 68Q87

Highlights

We show that Shearer’s Lemma and elementary arguments about the entropy of random variables are sufficient to recover the optimal Hypercontractive Inequality for all even integers q
The Hypercontractive Inequality plays a fundamental role in the analysis of Boolean functions
Discovered independently1 by Aline Bonami [3, 4] and several years later by Leonard Gross [14], the inequality is concerned with the noise operator Tρ that acts on functions f : {−1, 1}n → R via Tρ f (x) = E[ f (y)], where y is a ρ-correlated copy of x (i. e., y is drawn from the product distribution where E[yixi] = ρ for all i ∈ [n])

Summary

Introduction

The Hypercontractive Inequality plays a fundamental role in the analysis of Boolean functions. Ehud Friedgut and Vojtech Rödl [12] obtained one such alternative proof by exploiting a novel connection between the Hypercontractive Inequality and Shannon entropy Their main result is an information-theoretic/combinatorial proof of the inequality. We give positive answers to both questions: we show that Friedgut and Rödl’s argument can be simplified and sharpened by reasoning directly about the Fourier spectrum of f , without requiring the translation to hypergraphs or generalization of Shearer’s lemma With this direct approach, we obtain a simple proof of the optimal Hypercontractive. There is a short and elegant inductive proof of the q = 4 special case of Theorem 1.1 that requires only the Cauchy-Schwarz inequality This proof first appeared in the literature in [19], Bonami’s original paper [4] contains a proof along similar lines. This proof still uses the inductive step as a black-box (i. e., the fact that the Hypercontractive Inequality tensorizes, requiring Minkowski’s inequality), and remains very much an analytic proof

Basics of Shannon entropy

Hypercontractivity via the entropy method