Sparse signal detection in heteroscedastic Gaussian sequence models: Sharp minimax rates

Abstract

Given a heterogeneous Gaussian sequence model with unknown mean $\mathit{\theta }\in {\mathbb{R}}^d$ and known covariance matrix $\mathrm{\Sigma }=diag({\mathit{\sigma }}_1^2,\dots ,{\mathit{\sigma }}_d^2)$, we study the signal detection problem against sparse alternatives, for known sparsity s. Namely, we characterize how large ${\mathit{ϵ}}^{\ast }>0$ should be, in order to distinguish with high probability the null hypothesis $\mathit{\theta }=0$ from the alternative composed of s-sparse vectors in ${\mathbb{R}}^d$, separated from 0 in $L^t$ norm ($t\in [1,\mathrm{\infty }]$) by at least ${\mathit{ϵ}}^{\ast }$. We find non-asymptotic minimax upper and lower bounds over the minimax separation radius ${\mathit{ϵ}}^{\ast }$ 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 ${\mathit{ϵ}}^{\ast }$ 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.