Testing Ising Models

TL;DR

This paper develops efficient algorithms for testing properties like independence and goodness-of-fit in Ising models, overcoming the high-dimensional sample complexity barriers faced in general distribution testing.

Contribution

It introduces the first sample and time-efficient testing methods tailored for Ising models, a fundamental class of Markov Random Fields, avoiding the curse of dimensionality.

Findings

Efficient testers for independence in Ising models.

Sample complexity is polynomial, not exponential.

Variance bounds for functions of Ising models are established.

Abstract

Given samples from an unknown multivariate distribution $p$, is it possible to distinguish whether $p$ is the product of its marginals versus $p$ being far from every product distribution? Similarly, is it possible to distinguish whether $p$ equals a given distribution $q$ versus $p$ and $q$ being far from each other? These problems of testing independence and goodness-of-fit have received enormous attention in statistics, information theory, and theoretical computer science, with sample-optimal algorithms known in several interesting regimes of parameters. Unfortunately, it has also been understood that these problems become intractable in large dimensions, necessitating exponential sample complexity. Motivated by the exponential lower bounds for general distributions as well as the ubiquity of Markov Random Fields (MRFs) in the modeling of high-dimensional distributions, we initiate the study of distribution testing on structured multivariate distributions, and in particular the prototypical example of MRFs: the Ising Model. We demonstrate that, in this structured setting, we can avoid the curse of dimensionality, obtaining sample and time efficient testers for independence and goodness-of-fit. One of the key technical challenges we face along the way is bounding the variance of functions of the Ising model.