Minimax rates for sparse signal detection under correlation

TL;DR

This paper characterizes the optimal rates for detecting sparse signals in Gaussian models with correlated observations, revealing how correlation levels influence detection difficulty and phase transitions.

Contribution

It generalizes previous results to equicorrelated models, identifies phase transitions related to correlation and sparsity, and explores structured correlations and multiple effects.

Findings

Strong correlation can aid detection, while moderate correlation hampers it.

Phase transitions occur at specific sparsity levels related to correlation.

Group structures significantly alter the minimax separation rates.

Abstract

We fully characterize the nonasymptotic minimax separation rate for sparse signal detection in the Gaussian sequence model with $p$ equicorrelated observations, generalizing a result of Collier, Comminges, and Tsybakov. As a consequence of the rate characterization, we find that strong correlation is a blessing, moderate correlation is a curse, and weak correlation is irrelevant. Moreover, the threshold correlation level yielding a blessing exhibits phase transitions at the $\sqrt{p}$ and $p-\sqrt{p}$ sparsity levels. We also establish the emergence of new phase transitions in the minimax separation rate with a subtle dependence on the correlation level. Additionally, we study group structured correlations and derive the minimax separation rate in a model including multiple random effects. The group structure turns out to fundamentally change the detection problem from the equicorrelated case and different phenomena appear in the separation rate.