Consistent selection via the Lasso for high dimensional approximating regression models

TL;DR

This paper studies the consistency of the Lasso method for selecting the most parsimonious approximation of a regression function from a large dictionary in high-dimensional settings, extending traditional linear models.

Contribution

It introduces a framework for consistent selection of function indices using Lasso in high-dimensional approximation models with data-dependent penalties.

Findings

Lasso achieves consistent selection under certain conditions.

The method works for a number of functions up to polynomial in sample size.

Provides theoretical guarantees for high-dimensional approximation scenarios.

Abstract

In this article we investigate consistency of selection in regression models via the popular Lasso method. Here we depart from the traditional linear regression assumption and consider approximations of the regression function $f$ with elements of a given dictionary of $M$ functions. The target for consistency is the index set of those functions from this dictionary that realize the most parsimonious approximation to $f$ among all linear combinations belonging to an $L_2$ ball centered at $f$ and of radius $r_{n,M}^2$. In this framework we show that a consistent estimate of this index set can be derived via $\ell_1$ penalized least squares, with a data dependent penalty and with tuning sequence $r_{n,M}>\sqrt{\log(Mn)/n}$, where $n$ is the sample size. Our results hold for any $1\leq M\leq n^{\gamma}$, for any $\gamma>0$.