Geometry and Duality of Alternating Markov Chains

2024-10-16

arXiv.org

AI Summary

This paper addresses the lack of geometric interpretation for half-step evolution in Markov chains by modeling it as alternating projections onto the convex set of joint probability distributions with respect to the reverse Kullback–Leibler (KL) divergence. Methodologically, it establishes, for the first time, a rigorous correspondence between half-step dynamics and projection operations under reverse KL divergence, characterizing even- and odd-step evolutions as two complementary projection sequences. The theoretical contributions are threefold: (1) revealing an information-theoretic duality between two classes of half-step chains; (2) providing a convex-geometric proof of this duality; and (3) unifying the description of two complementary Markovian dynamics, thereby extending the applicability of information geometry to stochastic process modeling. The framework furnishes a novel analytical tool for stochastic systems grounded in probability manifolds.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Characterize half-steps of Markov chains as alternating projections
• Use reverse Kullback-Leibler divergence for convex probability sets
• Prove duality between even and odd half-steps geometrically

Innovation

• Alternating projections for Markov chains
• Reverse Kullback-Leibler divergence usage
• Geometric proof of information-theoretic duality