Stability of the Shannon–Stam inequality via the Föllmer process

Abstract

AbstractWe 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 $$X,Y \in {\mathbb {R}}^d$$ X , Y ∈ R d , the deficit in the Shannon–Stam inequality is bounded from below by the expression $$\begin{aligned} C \left( \mathrm {D}\left( X||G\right) + \mathrm {D}\left( Y||G\right) \right) , \end{aligned}$$ C D X | | G + D Y | | G , where $$\mathrm {D}\left( \cdot ~ ||G\right) $$ D · | | G 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.