Adaptive robust estimation in sparse vector model

TL;DR

This paper develops adaptive estimators for sparse vector models that achieve optimal rates across various noise distributions and levels of sparsity, addressing the challenge of unknown noise variance and distribution.

Contribution

It introduces new adaptive estimation methods that attain optimal rates considering unknown noise level, distribution, and sparsity, extending robust estimation techniques.

Findings

Optimal adaptive rates depend on noise distribution and sparsity.

Rates differ from non-adaptive minimax rates when distribution is unknown.

Estimation of noise variance is feasible without precise distribution knowledge.

Abstract

For the sparse vector model, we consider estimation of the target vector, of its L2-norm and of the noise variance. We construct adaptive estimators and establish the optimal rates of adaptive estimation when adaptation is considered with respect to the triplet "noise level - noise distribution - sparsity". We consider classes of noise distributions with polynomially and exponentially decreasing tails as well as the case of Gaussian noise. The obtained rates turn out to be different from the minimax non-adaptive rates when the triplet is known. A crucial issue is the ignorance of the noise variance. Moreover, knowing or not knowing the noise distribution can also influence the rate. For example, the rates of estimation of the noise variance can differ depending on whether the noise is Gaussian or sub-Gaussian without a precise knowledge of the distribution. Estimation of noise variance in our setting can be viewed as an adaptive variant of robust estimation of scale in the contamination model, where instead of fixing the "nominal" distribution in advance, we assume that it belongs to some class of distributions.