Detection problems in the spiked matrix models

TL;DR

This paper investigates detection in spiked random matrix models, improving PCA with data transformations, establishing phase transition thresholds, and proposing distribution-agnostic hypothesis tests and rank estimation algorithms.

Contribution

It generalizes PCA improvements for non-Gaussian noise, derives sharp eigenvalue phase transitions, and introduces distribution-independent hypothesis testing and rank estimation methods.

Findings

Entrywise transformation improves PCA in non-Gaussian noise.

Sharp phase transition thresholds for eigenvalues are established.

A distribution-agnostic hypothesis test is proposed.

Abstract

We study the statistical decision process of detecting the low-rank signal from various signal-plus-noise type data matrices, known as the spiked random matrix models. We first show that the principal component analysis can be improved by entrywise pre-transforming the data matrix if the noise is non-Gaussian, generalizing the known results for the spiked random matrix models with rank-1 signals. As an intermediate step, we find out sharp phase transition thresholds for the extreme eigenvalues of spiked random matrices, which generalize the Baik-Ben Arous-P\'{e}ch\'{e} (BBP) transition. We also prove the central limit theorem for the linear spectral statistics for the spiked random matrices and propose a hypothesis test based on it, which does not depend on the distribution of the signal or the noise. When the noise is non-Gaussian noise, the test can be improved with an entrywise transformation to the data matrix with additive noise. We also introduce an algorithm that estimates the rank of the signal when it is not known a priori.