Smoothing $\ell_1$-penalized estimators for high-dimensional time-course data

TL;DR

This paper proposes a method to improve the convergence rates of $ ext{L}_1$-penalized estimators like Lasso for high-dimensional time-course data by combining information across time points, while maintaining oracle properties.

Contribution

It introduces a smoothing approach for $ ext{L}_1$-penalized estimators in high-dimensional time-series, enhancing convergence rates without sacrificing variable selection consistency.

Findings

Improved convergence rates for Lasso and Adaptive Lasso in time-course data.

Adaptive Lasso retains oracle properties and consistent variable selection.

Validated methods on simulated data and DNA motif finding.

Abstract

When a series of (related) linear models has to be estimated it is often appropriate to combine the different data-sets to construct more efficient estimators. We use $\ell_1$-penalized estimators like the Lasso or the Adaptive Lasso which can simultaneously do parameter estimation and model selection. We show that for a time-course of high-dimensional linear models the convergence rates of the Lasso and of the Adaptive Lasso can be improved by combining the different time-points in a suitable way. Moreover, the Adaptive Lasso still enjoys oracle properties and consistent variable selection. The finite sample properties of the proposed methods are illustrated on simulated data and on a real problem of motif finding in DNA sequences.