Trace Lasso: a trace norm regularization for correlated designs

TL;DR

The paper introduces the trace Lasso, a new regularization norm that accounts for covariate correlations in linear models, improving stability over traditional methods like Lasso and Elastic Net.

Contribution

It proposes the trace Lasso norm, a novel convex penalty that incorporates correlation structure into regularization, with analysis and an optimization algorithm.

Findings

Trace Lasso outperforms Elastic Net on correlated data

It provides more stable estimates in highly correlated settings

The method is validated on synthetic datasets

Abstract

Using the $\ell_1$-norm to regularize the estimation of the parameter vector of a linear model leads to an unstable estimator when covariates are highly correlated. In this paper, we introduce a new penalty function which takes into account the correlation of the design matrix to stabilize the estimation. This norm, called the trace Lasso, uses the trace norm, which is a convex surrogate of the rank, of the selected covariates as the criterion of model complexity. We analyze the properties of our norm, describe an optimization algorithm based on reweighted least-squares, and illustrate the behavior of this norm on synthetic data, showing that it is more adapted to strong correlations than competing methods such as the elastic net.