Adaptive spectral regularizations of high dimensional linear models

TL;DR

This paper develops data-driven spectral regularization methods for high-dimensional linear models to accurately recover unknown vectors despite noise, providing theoretical guarantees even when noise levels are unknown.

Contribution

It introduces adaptive spectral regularization techniques based on ordered smoothers with oracle inequalities for unknown noise levels.

Findings

Effective regularization parameter selection for unknown noise levels.

Theoretical oracle inequalities established for the proposed methods.

Improved recovery guarantees in high-dimensional settings.

Abstract

This paper focuses on recovering an unknown vector $\beta$ from the noisy data $Y=X\beta +\sigma\xi$, where $X$ is a known $n\times p$-matrix, $\xi $ is a standard white Gaussian noise, and $\sigma$ is an unknown noise level. In order to estimate $\beta$, a spectral regularization method is used, and our goal is to choose its regularization parameter with the help of the data $Y$. In this paper, we deal solely with regularization methods based on the so-called ordered smoothers and provide some oracle inequalities in the case, where the noise level is unknown.