Property Testing in High Dimensional Ising models

TL;DR

This paper investigates the fundamental limits of testing basic properties in high-dimensional Ising models, proposing new correlation-based methods that are both statistically optimal and computationally efficient for different types of interactions.

Contribution

It introduces new bounds for property testing in Ising models and develops correlation-based tests tailored for ferromagnetic and general models, improving efficiency and understanding.

Findings

Correlation screening tests match theoretical bounds in ferromagnets.

Property testing is more challenging in general Ising models.

New methods outperform existing algorithms in specific settings.

Abstract

This paper explores the information-theoretic limitations of graph property testing in zero-field Ising models. Instead of learning the entire graph structure, sometimes testing a basic graph property such as connectivity, cycle presence or maximum clique size is a more relevant and attainable objective. Since property testing is more fundamental than graph recovery, any necessary conditions for property testing imply corresponding conditions for graph recovery, while custom property tests can be statistically and/or computationally more efficient than graph recovery based algorithms. Understanding the statistical complexity of property testing requires the distinction of ferromagnetic (i.e., positive interactions only) and general Ising models. Using combinatorial constructs such as graph packing and strong monotonicity, we characterize how target properties affect the corresponding minimax upper and lower bounds within the realm of ferromagnets. On the other hand, by studying the detection of an antiferromagnetic (i.e., negative interactions only) Curie-Weiss model buried in Rademacher noise, we show that property testing is strictly more challenging over general Ising models. In terms of methodological development, we propose two types of correlation based tests: computationally efficient screening for ferromagnets, and score type tests for general models, including a fast cycle presence test. Our correlation screening tests match the information-theoretic bounds for property testing in ferromagnets.