## The Annals of Statistics

### Property testing in high-dimensional Ising models

#### 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 in certain regimes.

#### Article information

Source
Ann. Statist., Volume 47, Number 5 (2019), 2472-2503.

Dates
First available in Project Euclid: 3 August 2019

https://projecteuclid.org/euclid.aos/1564797854

Digital Object Identifier
doi:10.1214/18-AOS1754

Mathematical Reviews number (MathSciNet)
MR3988763

Subjects
Primary: 62F03: Hypothesis testing 62H15: Hypothesis testing

#### Citation

Neykov, Matey; Liu, Han. Property testing in high-dimensional Ising models. Ann. Statist. 47 (2019), no. 5, 2472--2503. doi:10.1214/18-AOS1754. https://projecteuclid.org/euclid.aos/1564797854

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#### Supplemental materials

• Supplementary Material to “Property testing in high-dimensional Ising models”. The supplement contains several auxiliary results, minimax risk lower bound proofs for ferromagnets (including that of Theorem 2.4), proofs for the correlation screening algorithm, hardness results for general Ising models and the proofs for the correlation testing algorithms for general models.