Handling correlated and repeated measurements with the smoothed multivariate square-root Lasso

TL;DR

This paper introduces a novel convex estimator for high-dimensional regression that effectively handles complex noise structures and correlated measurements without relying on data averaging, improving robustness and applicability.

Contribution

It proposes a new smoothed multivariate square-root Lasso estimator that jointly estimates noise levels and regression coefficients using non-averaged measurements, enhancing modeling of complex noise.

Findings

Demonstrates improved noise modeling on simulated data

Shows practical benefits on neuroimaging datasets

Provides a convex optimization framework leveraging sparsity

Abstract

Sparsity promoting norms are frequently used in high dimensional regression. A limitation of such Lasso-type estimators is that the optimal regularization parameter depends on the unknown noise level. Estimators such as the concomitant Lasso address this dependence by jointly estimating the noise level and the regression coefficients. Additionally, in many applications, the data is obtained by averaging multiple measurements: this reduces the noise variance, but it dramatically reduces sample sizes and prevents refined noise modeling. In this work, we propose a concomitant estimator that can cope with complex noise structure by using non-averaged measurements. The resulting optimization problem is convex and amenable, thanks to smoothing theory, to state-of-the-art optimization techniques that leverage the sparsity of the solutions. Practical benefits are demonstrated on toy datasets, realistic simulated data and real neuroimaging data.