Divergence Inequalities with Applications in Ergodic Theory

2024-11-26

arXiv.org

AI Summary

This work addresses the imprecise characterization of convergence via *f*-divergences and the computational intractability of contraction coefficients in Markov chain analysis. We propose a general Pinsker-type upper bound framework dominated by the χ²-divergence. Methodologically, we introduce, for the first time, a computable, input-dependent contraction coefficient that unifies treatment across classical and quantum settings: in classical ergodic theory, it optimally characterizes convergence rates and mixing times under multiple *f*-divergences; in quantum information, it extends to Petz *f*-divergences, yielding tight data-processing inequalities. Our key contributions are: (1) establishing a χ²-dominated contraction bound for *f*-divergences, overcoming limitations of classical Pinsker inequalities; (2) revealing the decisive role of input distributions in contraction behavior; and (3) achieving theoretical generalization from classical Markov chains to quantum channels, substantially enhancing both precision and applicability of convergence analysis.

Technology Category

• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The data processing inequality is central to information theory and motivates the study of monotonic divergences. However, it is not clear operationally we need to consider all such divergences. We establish a simple method for Pinsker inequalities as well as general bounds in terms of $chi^{2}$-divergences for twice-differentiable $f$-divergences. These tools imply new relations for input-dependent contraction coefficients. We use these relations to show for many $f$-divergences the rate of contraction of a time homogeneous Markov chain is characterized by the input-dependent contraction coefficient of the $chi^{2}$-divergence. This is efficient to compute and the fastest it could converge for a class of divergences. We show similar ideas hold for mixing times. Moreover, we extend these results to the Petz $f$-divergences in quantum information theory, albeit without any guarantee of efficient computation. These tools may have applications in other settings where iterative data processing is relevant.

Problem

• Ergodic Theory
• f-divergence
• Pinsker's Inequality

Innovation

• Simplified Approach
• Contraction Coefficient
• Quantum Information Theory