Stability of the Shannon–Stam inequality via the Föllmer process

Abstract

We prove stability estimates for the Shannon-Stam inequality (also known as the entropypower 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 ∈ R d , the deficit in the Shannon-Stam inequality is bounded from below by the expressionwhere 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.Our approach is based on ideas somewhat related to the ones which appear in [8]: the very highlevel plan of the proof is to embed the variables X, Y as the terminal points of some martingales and express the entropies of X, Y and X+Y as functions of the associates quadratic co-variation processes. One of the main benefits in using such an embedding is that the co-variation process of X + Y can be easily expressed in terms on the ones of X, Y , as demonstrated below. In [8]

Entropy and Information jump for log-concave vectors

“…This should be compared with a recent result of Eldan and Mikulincer ( [4], Theorem 3). They get a more general result, allowing µ and ν to have different covariance matrices, but in the case where µ and ν have the same covariance matrix, they get a worst dependence on the Poincaré constant.…”

A trajectorial approach to relative entropy dissipation of McKean$\boldsymbol{-}$Vlasov diffusions: gradient flows and HWBI inequalities

“…In the literature, similar trajectorial approaches have also been applied in the context of martingale inequalities [ABP + 13, BS15], functional inequalities [Cat04, Leh13, BCGL20, GLRT20], and their stability estimates [ELS20,EM20]. In particular, we refer to [BCGL20, Corollary 1.4] for a related HWI inequality derived from the entropic interpolation of the mean-field Schrödinger problem.…”

Optimal Couplings on Wiener Space and An Extension of Talagrand’s Transport Inequality

“…In fact, inequality (2) becomes an identity as soon as we introduce the additional constraint that the transport should be adapted to the natural filtration on Wiener space; this was first shown by R. Lassalle in [9]. On Wiener space, inequality (2) was first studied by Feyel and Üstünel [3].…”

“…In Talagrand's original version [13], the inequality is formulated on Euclidean space R n , including the case n = ∞; the Wasserstein distance is defined in terms of the Euclidean norm, and the reference measure P is the product of standard normal distributions. But the Lévy-Ciesielski construction of Brownian motion in terms of the Schauder functions shows that inequality (2) on Wiener space can be viewed as a direct translation of the Euclidean case for n = ∞, as explained in Section 3.…”

“…This top-down approach involves the computation of relative entropy in terms of the intrinsic drift of Q that was used in [4] and [5] for the analysis of time reversal and large deviations on Wiener space. The intrinsic drift b Q is such that Q can be viewed as a weak solution of the stochastic differential equation dW = dW Q + b Q (W )dt, that is, W Q is a Wiener process under Q. Coupling W Q with the coordinate process W under Q immediately yields inequality (2), and it solves the optimal transport problem for the Cameron-Martin norm if the coupling is required to be adapted.…”