Relationship between discrete-state Gibbs sampler techniques and continuous log-concave entropy methods

Investigate and characterize the extent to which mixing-time results and analytical techniques for Gibbs samplers on discrete state spaces (e.g., conductance-based or coupling methods) relate to and can be transferred to entropy-contraction-based analyses for Gibbs samplers targeting continuous log-concave distributions.

Background

The literature on Gibbs sampler convergence in discrete state spaces is extensive and often uses different tools (e.g., conductance or combinatorial methods) than those employed for continuous log-concave targets. The present work builds on relative entropy contraction and transport-entropy techniques suited to continuous settings.

The authors note uncertainty about the relationship between these bodies of work, marking a gap in understanding how discrete-space results and techniques connect to or inform continuous log-concave analyses.

References

However, both the aforementioned contexts are quite different from ours, and it is unclear how much the associated results and techniques are related to the ones employed herein.

— Entropy contraction of the Gibbs sampler under log-concavity  (2410.00858 - Ascolani et al., 2024) in Section 1.2 (Related works)

Neither results are directly comparable with uniform sampling from a convex body and it is not clear if there methods can be used to recover our results.

— Improved $\ell_0$-Isoperimetry for Convex Bodies via Mass Transport  (2608.27854 - Fernandez, 28 Aug 2026) in Section 1, subsection “$\ell_0$ isoperimetry and polynomial mixing time of CHAR”