Distribution-free Detection of a Submatrix

TL;DR

This paper introduces a distribution-free method for detecting a submatrix with elevated values in large data matrices, using permutation calibration and rank-based variants, achieving comparable asymptotic performance to parametric tests.

Contribution

It demonstrates that permutation calibration achieves asymptotic performance similar to parametric methods in submatrix detection, and analyzes rank-based variants for nonparametric detection.

Findings

Permutation calibration matches parametric test performance asymptotically

Rank-based methods have quantifiable power loss

Method applicable in distribution-free settings

Abstract

We consider the problem of detecting the presence of a submatrix with larger-than-usual values in a large data matrix. This problem was considered in (Butucea and Ingster, 2013) under a one-parameter exponential family, and one of the test they analyzed is the scan test. Taking a nonparametric stance, we show that a calibration by permutation leads to the same (first-order) asymptotic performance. This is true for the two types of permutations we consider. We also study the corresponding rank-based variants and precisely quantify the loss in asymptotic power.