Statistics Papers

Document Type

Journal Article

Date of this Version

2017

Publication Source

The Annals of Statistics

Volume

45

Issue

4

Start Page

1403

Last Page

1430

DOI

10.1214/16-AOS1488

Abstract

We study in this paper computational and statistical boundaries for submatrix localization. Given one observation of (one or multiple nonoverlapping) signal submatrix (of magnitude λ and size km×kn) embedded in a large noise matrix (of size m × n), the goal is to optimal identify the support of the signal submatrix computationally and statistically.

Two transition thresholds for the signal-to-noise ratio λ/σ are established in terms of m, n, km and kn. The first threshold, SNRc, corresponds to the computational boundary. We introduce a new linear time spectral algorithm that identifies the submatrix with high probability when the signal strength is above the threshold SNRc. Below this threshold, it is shown that no polynomial time algorithm can succeed in identifying the submatrix, under the hidden clique hypothesis. The second threshold, SNRs, captures the statistical boundary, below which no method can succeed in localization with probability going to one in the minimax sense. The exhaustive search method successfully finds the submatrix above this threshold. In marked contrast to submatrix detection and sparse PCA, the results show an interesting phenomenon that SNRc is always significantly larger than SNRs, which implies an essential gap between statistical optimality and computational efficiency for submatrix localization.

Copyright/Permission Statement

The original and published work is available at: https://projecteuclid.org/euclid.aos/1498636861#abstract

Keywords

computational boundary, computational complexity, detection, planted clique, lower bounds, minimax, signal to noise ratio, statistical boundary, submatrix localization

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Date Posted: 27 November 2017

This document has been peer reviewed.