Minimax estimation of functionals in sparse vector model with correlated observations

TL;DR

This paper develops minimax optimal methods for estimating the norm and testing hypotheses for sparse vectors observed with correlated Gaussian noise, where only partial covariance information is available, emphasizing the role of the covariance's Frobenius norm.

Contribution

It introduces new minimax optimal estimation and testing procedures under partial covariance information in sparse vector models, highlighting the impact of the Frobenius norm.

Findings

Minimax rates depend on the Frobenius norm of the covariance matrix.

Proposed methods are optimal under partial covariance information.

Estimation and testing are less influenced by ambient dimension.

Abstract

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 ${\boldsymbol \Sigma}$.