Sparse Signal Detection in Heteroscedastic Gaussian Sequence Models: Sharp Minimax Rates

Abstract

Given a heterogeneous Gaussian sequence model with unknown mean θ ∈ R d and known covariance matrix Σ = diag(σ 2 1 , . . . , σ 2 d ), we study the signal detection problem against sparse alternatives, for known sparsity s. Namely, we characterize how large ǫ * > 0 should be, in order to distinguish with high probability the null hypothesis θ = 0 from the alternative composed of s-sparse vectors in R d , separated from 0 in L t norm (t ≥ 1) by at least ǫ * . We find minimax upper and lower bounds over the minimax separation radius ǫ * and prove that they are always matching. We also derive the corresponding minimax tests achieving these bounds. Our results reveal new phase transitions regarding the behavior of ǫ * with respect to the level of sparsity, to the L t metric, and to the heteroscedasticity profile of Σ. In the case of the Euclidean (i.e. L 2 ) separation, we bridge the remaining gaps in the literature.