Rényi--Sobolev Inequalities and Connections to Spectral Graph Theory

Abstract: In this paper, we generalize the log-Sobolev inequalities to R\'enyi--Sobolev inequalities by replacing the entropy with the two-parameter entropy, which is a generalized version of entropy and closely related to R\'enyi divergences. We derive the sharp nonlinear dimension-free version of this kind of inequalities. Interestingly, the resultant inequalities show a transition phenomenon depending on the parameters. We then connect R\'enyi--Sobolev inequalities to contractive and data-processing inequalities, concentration inequalities, and spectral graph theory. Our proofs in this paper are based on the information-theoretic characterization of the R\'enyi--Sobolev inequalities, as well as the method of types.