Approximating the total variation distance between spin systems

TL;DR

This paper introduces a new reduction method to approximate the total variation distance between Gibbs distributions of spin systems, connecting it to sampling and counting, with applications to various Ising models.

Contribution

It presents a novel reduction linking TV-distance approximation to sampling and counting, and analyzes the complexity of marginal TV-distance approximation in spin systems.

Findings

Reduction connects TV-distance approximation to sampling and counting.

Approximation is feasible for certain models in the uniqueness regime.

Marginal TV-distance approximation remains hard even with polynomial-time algorithms.

Abstract

Spin systems form an important class of undirected graphical models. For two Gibbs distributions $\mu$ and $\nu$ induced by two spin systems on the same graph $G = (V, E)$, we study the problem of approximating the total variation distance $d_{TV}(\mu,\nu)$ 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 $d_{TV}(\mu_S,\nu_S)$ between two marginal distributions on an arbitrary subset $S \subseteq V$. We prove that this problem remains hard even when both $\mu$ and $\nu$ admit polynomial-time sampling and approximate counting algorithms.