Abstract
We prove stability estimates for the Shannon–Stam inequality (also known as the entropy-power inequality) for log-concave random vectors in terms of entropy and transportation distance. In particular, we give the first stability estimate for general log-concave random vectors in the following form: for log-concave random vectors X,Y in {mathbb {R}}^d, the deficit in the Shannon–Stam inequality is bounded from below by the expression CDX||G+DY||G,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} C \\left( \\mathrm {D}\\left( X||G\\right) + \\mathrm {D}\\left( Y||G\\right) \\right) , \\end{aligned}$$\\end{document}where mathrm {D}left( cdot ~ ||Gright) denotes the relative entropy with respect to the standard Gaussian and the constant C depends only on the covariance structures and the spectral gaps of X and Y. In the case of uniformly log-concave vectors our analysis gives dimension-free bounds. Our proofs are based on a new approach which uses an entropy-minimizing process from stochastic control theory.
Highlights
Let μ be a probability measure on Rd and X ∼ μ
A crucial observation made in their work is that without further assumptions on the measures μ and ν, one should not expect meaningful stability results to hold
And is the main subject of the slicing conjecture, which hypothesizes that L X is uniformly bounded by a constant, independent of the dimension, for every isotropic log-concave vector X
Summary
Let μ be a probability measure on Rd and X ∼ μ. First observed by Shannon in [23], and known today as the entropy power inequality. To state yet another equivalent form of the inequality, for any positive-definite matrix, Σ, we set γΣ as the centered Gaussian measure on Rd with density dγΣ (x). In light of the equality cases, a small deficit in (3) should imply that X and Y are both close, in some sense, to a Gaussian. A crucial observation made in their work is that without further assumptions on the measures μ and ν, one should not expect meaningful stability results to hold. We focus on the class of log-concave measures which, as our method demonstrates, turns out to be natural in this context
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