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.

翻译：给定一个异方差高斯序列模型，其未知均值 $\theta \in \mathbb R^d$ 和已知协方差矩阵 $\Sigma = \operatorname{diag}(\sigma_1^2,\dots, \sigma_d^2)$，我们研究在已知稀疏度 $s$ 下针对稀疏备择假设的信号检测问题。即，我们刻画 $\epsilon^*>0$ 应多大，才能以高概率区分原假设 $\theta=0$ 与由 $\mathbb R^d$ 中 $s$-稀疏向量组成的备择假设，这些向量在 $L^t$ 范数（$t \in [1,\infty]$）下与 $0$ 至少相距 $\epsilon^*$。我们给出了极小极大分离半径 $\epsilon^*$ 的极小极大上界和下界，并证明它们始终匹配。我们还推导了实现这些界的相应极小极大检验。我们的结果揭示了关于 $\epsilon^*$ 相对于稀疏度水平、$L^t$ 度量以及 $\Sigma$ 的异方差轮廓的行为的新相变。在欧几里得（即 $L^2$）分离的情况下，我们填补了文献中的剩余空白。