Optimal adaptive estimation of linear functionals under sparsity

TL;DR

This paper develops an adaptive estimator for linear functionals in Gaussian models that effectively handles unknown sparsity levels and noise variance, achieving near-optimal convergence rates.

Contribution

It introduces a new adaptive estimation method that attains near-minimax rates without prior knowledge of sparsity or noise variance, and proves these rates are optimal.

Findings

Estimator achieves near-minimax rate with unknown s and sigma^2

Optimal adaptive rate cannot be improved when s is unknown

Method performs well in sparse Gaussian sequence models

Abstract

We consider the problem of estimation of a linear functional in the Gaussian sequence model where the unknown vector theta in R^d belongs to a class of s-sparse vectors with unknown s. We suggest an adaptive estimator achieving a non-asymptotic rate of convergence that differs from the minimax rate at most by a logarithmic factor. We also show that this optimal adaptive rate cannot be improved when s is unknown. Furthermore, we address the issue of simultaneous adaptation to s and to the variance sigma^2 of the noise. We suggest an estimator that achieves the optimal adaptive rate when both s and sigma^2 are unknown.