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

TL;DR

This paper establishes sharp minimax rates for detecting sparse signals in heteroscedastic Gaussian models, revealing phase transitions and bridging gaps in the Euclidean case, with practical tests matching theoretical bounds.

Contribution

It provides the first complete characterization of minimax separation rates for sparse detection in heteroscedastic Gaussian models, including phase transitions and optimal testing procedures.

Findings

Minimax bounds for signal detection are tight and match.

Phase transitions depend on sparsity, metric, and heteroscedasticity.

Optimal tests achieve the derived bounds.

Abstract

Given a heterogeneous Gaussian sequence model with unknown mean $\theta \in \mathbb R^d$ and known covariance matrix $\Sigma = \operatorname{diag}(\sigma_1^2,\dots, \sigma_d^2)$, we study the signal detection problem against sparse alternatives, for known sparsity $s$. Namely, we characterize how large $\epsilon^*>0$ should be, in order to distinguish with high probability the null hypothesis $\theta=0$ from the alternative composed of $s$-sparse vectors in $\mathbb R^d$, separated from $0$ in $L^t$ norm ($t \in [1,\infty]$) by at least $\epsilon^*$. We find minimax upper and lower bounds over the minimax separation radius $\epsilon^*$ 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 $\epsilon^*$ with respect to the level of sparsity, to the $L^t$ metric, and to the heteroscedasticity profile of $\Sigma$. In the case of the Euclidean (i.e. $L^2$) separation, we bridge the remaining gaps in the literature.