Extension of the Poincaré-inequality-based convergence argument to general regularizers

Ascertain whether the convergence proof based on the Poincaré inequality for Gaussian measures, which establishes weak convergence of the marginal law in the entropic-regularized gradient flow setting, can be extended to handle general Gibbs regularizers U beyond the Gaussian case.

Background

The authors compare their results to prior work that used the Poincaré inequality for the Gaussian distribution to derive convergence of the gradient flow. While their paper provides convergence for general regularizers U via a new technique based on LaSalle’s invariance principle and the HWI inequality, they note that it is unclear whether the previous Poincaré-based argument can be generalized.

This leaves a methodological gap: understanding the scope of the Poincaré-based approach under non-Gaussian Gibbs measures would clarify the robustness and limitations of that technique relative to the authors’ approach.