We consider the observations of an unknown $s$-sparse vector ${\boldsymbol \theta}$ corrupted by Gaussian noise with zero mean and unknown covariance matrix ${\boldsymbol \Sigma}$. We propose minimax optimal methods of estimating the $\ell_2$ norm of ${\boldsymbol \theta}$ and testing the hypothesis $H_0: {\boldsymbol \theta}=0$ against sparse alternatives when only partial information about ${\boldsymbol \Sigma}$ is available, such as an upper bound on its Frobenius norm and the values of its diagonal entries to within an unknown scaling factor. We show that the minimax rates of the estimation and testing are leveraged not by the dimension of the problem but by the value of the Frobenius norm of ${\boldsymbol \Sigma}$.

翻译：我们考虑一个未知的$s$-稀疏向量${\boldsymbol \theta}$的观测值，该观测值被均值为零、协方差矩阵为${\boldsymbol \Sigma}$未知的高斯噪声所污染。当仅能获得关于${\boldsymbol \Sigma}$的部分信息（例如其Frobenius范数的上界及其对角线条目在一个未知缩放因子范围内的值）时，我们提出了估计${\boldsymbol \theta}$的$\ell_2$范数以及检验零假设$H_0: {\boldsymbol \theta}=0$对抗稀疏备择假设的极小极大最优方法。我们证明了估计和检验的极小极大速率并非由问题的维度决定，而是由${\boldsymbol \Sigma}$的Frobenius范数值所主导。