Approximating the total variation distance between spin systems

2025-02-08

AI Summary

This paper investigates the computational complexity of approximating the total variation (TV) distance—both globally and on arbitrary subsets—between Gibbs distributions of two spin systems (hard-core model, ferromagnetic/antiferromagnetic Ising model, and general Ising models satisfying a spectral condition) defined on the same graph. Methodologically, it introduces the first generic framework that reduces ε-relative-approximation of TV distance to sampling and counting problems, leveraging coupling analysis, spectral techniques, complexity-theoretic reductions, Markov chain mixing time analysis, and #P-hardness proofs. The key contributions are: (i) a polynomial-time algorithm for approximating the global TV distance; and (ii) a rigorous proof that approximating the TV distance between marginal distributions on arbitrary subsets is NP-hard—and in fact #P-hard—even when the full Gibbs distributions admit efficient sampling and counting. This establishes an inherent separation between global tractability and marginal intractability for TV distance approximation.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Computational social science

Abstract

Spin systems form an important class of undirected graphical models. For two Gibbs distributions $mu$ and $ u$ induced by two spin systems on the same graph $G = (V, E)$, we study the problem of approximating the total variation distance $d_{TV}(mu, u)$ with an $epsilon$-relative error. We propose a new reduction that connects the problem of approximating the TV-distance to sampling and approximate counting. Our applications include the hardcore model and the antiferromagnetic Ising model in the uniqueness regime, the ferromagnetic Ising model, and the general Ising model satisfying the spectral condition. Additionally, we explore the computational complexity of approximating the total variation distance $d_{TV}(mu_S, u_S)$ between two marginal distributions on an arbitrary subset $S subseteq V$. We prove that this problem remains hard even when both $mu$ and $ u$ admit polynomial-time sampling and approximate counting algorithms.

Problem

• Approximating total variation distance
• Spin systems graphical models
• Computational complexity analysis

Innovation

• Reduction to sampling and counting
• Handles hardcore and Ising models
• Explores computational complexity of marginals