Canonical thresholding for non-sparse high-dimensional linear regression

TL;DR

This paper introduces canonical thresholding estimators for high-dimensional linear regression that do not require sparsity, relying instead on eigenvalue decay of the data covariance, with theoretical guarantees and practical performance demonstrated.

Contribution

It proposes a new family of estimators based on canonical form thresholding, linking them to LASSO and PCR, and provides theoretical analysis under eigenvalue decay conditions.

Findings

Bounds on mean squared and prediction errors established.

Eigenvalue decay conditions ensure estimator convergence.

Numerical simulations show competitive performance.

Abstract

We consider a high-dimensional linear regression problem. Unlike many papers on the topic, we do not require sparsity of the regression coefficients; instead, our main structural assumption is a decay of eigenvalues of the covariance matrix of the data. We propose a new family of estimators, called the canonical thresholding estimators, which pick largest regression coefficients in the canonical form. The estimators admit an explicit form and can be linked to LASSO and Principal Component Regression (PCR). A theoretical analysis for both fixed design and random design settings is provided. Obtained bounds on the mean squared error and the prediction error of a specific estimator from the family allow to clearly state sufficient conditions on the decay of eigenvalues to ensure convergence. In addition, we promote the use of the relative errors, strongly linked with the out-of-sample $R^2$. The study of these relative errors leads to a new concept of joint effective dimension, which incorporates the covariance of the data and the regression coefficients simultaneously, and describes the complexity of a linear regression problem. Some minimax lower bounds are established to showcase the optimality of our procedure. Numerical simulations confirm good performance of the proposed estimators compared to the previously developed methods.