Geometry and Duality of Alternating Markov Chains

TL;DR

This work studies the geometry of alternating Markov chains, where half-step resampling using fixed conditionals defines a product chain on $\mathcal{X} \times \mathcal{Y}$. It shows that these half-steps are realized as alternating projections onto convex sets with respect to the reverse KL divergence, made tractable by a log-denormalization that recasts the reverse KL as a Bregman divergence context. A central result expresses $D_{\rm{RKL}}(\pi_{\mathrm{ES}}, \pi_t)$ as the sum of the next-step increment and the current-step increment, yielding monotone entropy decay and enabling transfer of entropy-decay properties between the primal and dual chains. The approach connects to Sinkhorn-type iterative schemes and Edwards-Sokal couplings, offering a geometric, convex-analysis perspective on entropy decay for Markov chains and providing tools for analyzing half-step dynamics in related systems.

Abstract

In this note, we realize the half-steps of a general class of Markov chains as alternating projections with respect to the reverse Kullback-Leibler divergence between convex sets of joint probability distributions. Using this characterization, we provide a geometric proof of an information-theoretic duality between the Markov chains defined by the even and odd half-steps of the alternating projection scheme.

Theorems & Definitions (19)

• Definition 1.1: Generalised Edwards-Sokal Coupling
• Definition 1.2
• Definition 1.3
• Lemma 1.4
• Lemma 1.5
• Definition 1.6: Denormalization/log-denormalization (modification of ohara2017doublyautoparallelstructureprobability, Definition 2)
• Theorem 1.7
• Theorem 1.8: Duality
• Definition 2.1: Legendre function (cesa2006prediction, Section 11.2)
• Definition 2.2: Bregman divergence (cesa2006prediction, Section 11.2)