Interplay of minimax estimation and minimax support recovery under sparsity

TL;DR

This paper introduces a new scaled minimaxity concept for high-dimensional sparse linear regression, providing improved lower bounds, sharp results, and a novel optimal estimator that outperforms classical minimax methods.

Contribution

It proposes a new scaled minimaxity framework, deriving better bounds and constructing an optimal estimator for sparse high-dimensional regression.

Findings

Improved lower bounds for scaled minimax estimation.

Construction of a new optimal estimator for sparse regression.

Demonstration of smaller estimation errors than classical minimax bounds.

Abstract

In this paper, we study a new notion of scaled minimaxity for sparse estimation in high-dimensional linear regression model. We present more optimistic lower bounds than the one given by the classical minimax theory and hence improve on existing results. We recover sharp results for the global minimaxity as a consequence of our study. Fixing the scale of the signal-to-noise ratio, we prove that the estimation error can be much smaller than the global minimax error. We construct a new optimal estimator for the scaled minimax sparse estimation. An optimal adaptive procedure is also described.