A Quantitative Log-Sobolev Inequality for a Two Parameter Family of Functions

Abstract

We prove a sharp, dimension-free stability result for the classical logarithmic Sobolev inequality for a two parameter family of functions. Roughly speaking, our family consists of a certain class of log C 1,1 functions. Moreover, we show how to enlarge this space at the expense of the dimensionless constant and the sharp exponent. As an application we obtain new bounds on the entropy.

Quantitative logarithmic Sobolev inequalities and stability estimates

“…Theorem 1 improves upon the recent [22] where stronger conditions on the Hessian of the density f are considered (in particular parts of the class P(λ)), with weaker dependence of the constant. The work [22] actually investigates how far an admissible density is from saturating the logarithmic Sobolev inequality as measured with Wasserstein distance, providing a control of the deficit δ LSI (ν) in the logarithmic Sobolev inequality by the (quadratic) Kantorovich-Wasserstein distance W 2 (ν, γ). Within the class P(λ), this is easily achieved via Theorem 1 together with the Talagrand quadratic transportation cost inequality [33] (cf.…”

Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp–Lieb inequalities

“…Refined Gaussian logarithmic Sobolev inequalities have been considered for certain classes of test measures ν : measures ν satisfying lower and upper curvature bounds as in [8] and [32], measures ν satisfying a (weaker) Poincaré inequality as in [22]. Under these additional assumptions on ν, the goal is then to obtain better constants in the logarithmic Sobolev inequality, mimicking in a sense the phenomenon observed in the Poincaré inequality when considering test functions orthogonal to the first eigenfunctions.…”

“…Under these additional assumptions on ν, the goal is then to obtain better constants in the logarithmic Sobolev inequality, mimicking in a sense the phenomenon observed in the Poincaré inequality when considering test functions orthogonal to the first eigenfunctions. In Indrei-Marcon [32], the deficit is controlled by the Wasserstein distance for the class of centered functions with upper and lower bounded curvature. The authors in [8] also give new bounds in terms of conditionally centered vectors.…”

Wasserstein stability of the entropy power inequality for log-concave random vectors

“…When µ satisfies the Bakry-Émery criterion (4), we have a dimension-free quantitative stability result for the logarithmic Sobolev inequality 1 2 I(µ γ) ≥ D(µ γ). This is an improvement upon the main result of Indrei and Marcon [16], who consider the subset of densities satisfying (4) for some parameter η > 0, whose Hessians are also uniformly upper bounded. Unfortunately, this improvement is already obsolete, as Fathi, Indrei and Ledoux [7] have recently shown that a similar result holds for all probability measures with positive spectral gap.…”