Tuned Regularized Estimators for Linear Regression via Covariance Fitting

TL;DR

This paper introduces a data-driven approach to tuning regularized linear regression estimators using covariance fitting, unifying and improving methods like ridge, LASSO, and LAD through a common framework.

Contribution

It proposes a novel covariance fitting technique to adaptively tune regularized estimators, unifying several known methods under a single theoretical framework.

Findings

The covariance fitting approach effectively tunes estimators in practice.

Unified framework encompasses ridge, LASSO, and LAD estimators.

Numerical examples demonstrate practical relevance and performance.

Abstract

We consider the problem of finding tuned regularized parameter estimators for linear models. We start by showing that three known optimal linear estimators belong to a wider class of estimators that can be formulated as a solution to a weighted and constrained minimization problem. The optimal weights, however, are typically unknown in many applications. This begs the question, how should we choose the weights using only the data? We propose using the covariance fitting SPICE-methodology to obtain data-adaptive weights and show that the resulting class of estimators yields tuned versions of known regularized estimators - such as ridge regression, LASSO, and regularized least absolute deviation. These theoretical results unify several important estimators under a common umbrella. The resulting tuned estimators are also shown to be practically relevant by means of a number of numerical examples.