Open Access
June 2020 Statistical and computational limits for sparse matrix detection
T. Tony Cai, Yihong Wu
Ann. Statist. 48(3): 1593-1614 (June 2020). DOI: 10.1214/19-AOS1860

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 of sparse matrices, which can be of independent interest.

Citation

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T. Tony Cai. Yihong Wu. "Statistical and computational limits for sparse matrix detection." Ann. Statist. 48 (3) 1593 - 1614, June 2020. https://doi.org/10.1214/19-AOS1860

Information

Received: 1 January 2018; Revised: 1 January 2019; Published: June 2020
First available in Project Euclid: 17 July 2020

zbMATH: 07241604
MathSciNet: MR4124336
Digital Object Identifier: 10.1214/19-AOS1860

Subjects:
Primary: 62H15
Secondary: 62C20

Keywords: computational limits , Minimax rates , Sparse covariance matrix , sparse detection

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 3 • June 2020
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