Weak Detection in the Spiked Wigner Model with General Rank

TL;DR

This paper develops a spectral-based hypothesis test for detecting signals in noisy matrices, optimizing detection under Gaussian noise and improving performance with non-Gaussian noise through transformations, also estimating signal rank.

Contribution

It introduces a distribution-agnostic spectral test for signal detection and an algorithm for rank estimation in noisy matrix models.

Findings

Test minimizes combined error for small SNR under Gaussian noise

Entrywise transformations improve detection with non-Gaussian noise

Algorithm accurately estimates signal rank

Abstract

We study the statistical decision process of detecting the signal from a `signal+noise' type matrix model with an additive Wigner noise. We propose a hypothesis test based on the linear spectral statistics of the data matrix, which does not depend on the distribution of the signal or the noise. The test is optimal under the Gaussian noise if the signal-to-noise ratio is small, as it minimizes the sum of the Type-I and Type-II errors. Under the non-Gaussian noise, the test can be improved with an entrywise transformation to the data matrix. We also introduce an algorithm that estimates the rank of the signal when it is not known a priori.