Extend mutual information convergence beyond strong log-concavity via isoperimetry

Determine whether the convergence rates for mutual information between the initial state and the current state established in this work for the Langevin diffusion and the Unadjusted Langevin Algorithm under strong log-concavity continue to hold under weaker assumptions such as isoperimetry (e.g., log-Sobolev-type conditions) that are sufficient for mixing time guarantees.

Background

The paper establishes exponential decay of mutual information along the Langevin diffusion for strongly log-concave targets and polynomial decay for log-concave targets, and exponential decay for ULA under strong log-concavity and smoothness. In mixing time analysis, isoperimetric inequalities such as log-Sobolev are often sufficient for fast convergence, even without strong log-concavity.

The authors highlight the gap between assumptions used for mixing times and those currently used for their mutual information results, posing whether analogous mutual information convergence can be proved under isoperimetric conditions.