Affiliation:
1. Department of Statistics, University of Chicago , 5747 South Ellis Avenue, Chicago 60637, IL , USA
Abstract
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.
Funder
National Science Foundation
Alfred Sloan Foundation
Publisher
Oxford University Press (OUP)
Subject
Applied Mathematics,Computational Theory and Mathematics,Numerical Analysis,Statistics and Probability,Analysis
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