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

Given a heterogeneous Gaussian sequence model with unknown mean θ ∈ mathbb Rd and known covariance matrix Sigma = operatornamediag(σ12,dots, σd2), we study the signal detection problem against sparse alternatives, for known sparsity s. Namely, we characterize how large ε^*>0 should be, in order to distinguish with high probability the null hypothesis θ=0 from the alternative composed of s-sparse vectors in mathbb Rd, separated from 0 in Lt norm (t ∈ [1,∞]) by at least ε^*. We find minimax upper and lower bounds over the minimax separation radius ε^* 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 ε^* with respect to the level of sparsity, to the Lt metric, and to the heteroscedasticity profile of Sigma. In the case of the Euclidean (i.e. L2) separation, we bridge the remaining gaps in the literature.