Statistical and Computational Limits for Sparse Matrix Detection

TL;DR

This paper characterizes the fundamental statistical and computational limits for detecting high-dimensional sparse matrices contaminated by noise, revealing a surprising computational barrier when sparsity exceeds a certain threshold.

Contribution

It provides a tight characterization of detection limits and uncovers a computational barrier linked to the planted clique problem for certain sparsity levels.

Findings

Detection is statistically feasible below a certain sparsity threshold.

Computational hardness emerges when sparsity exceeds the cubic root of matrix size.

The results extend to sparse covariance matrix models.

Abstract

This paper investigates the fundamental limits for detecting a high-dimensional sparse matrix contaminated by white Gaussian noise from both the statistical and computational perspectives. We consider $p\times p$ matrices whose rows and columns are individually $k$-sparse. We provide a tight characterization of the statistical and computational limits for sparse matrix detection, which precisely describe when achieving optimal detection is easy, hard, or impossible, respectively. Although the sparse matrices considered in this paper have no apparent submatrix structure and the corresponding estimation problem has no computational issue at all, the detection problem has a surprising computational barrier when the sparsity level $k$ exceeds the cubic root of the matrix size $p$: attaining the optimal detection boundary is computationally at least as hard as solving the planted clique problem. The same statistical and computational limits also hold in the sparse covariance matrix model, where each variable is correlated with at most $k$ others. A key step in the construction of the statistically optimal test is a structural property for sparse matrices, which can be of independent interest.