Consistent selection via the Lasso for high dimensional approximating regression models

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 $1\le M\le n^{\gamma }$, for any $\gamma >0$.

Published in at http://dx.doi.org/10.1214/074921708000000101 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Regularization with the Smooth-Lasso procedure
• Randomized maximum-contrast selection: subagging for large-scale regression
• Variable Selection with Exponential Weights and $l_0$-Penalization
• High-dimensional stochastic optimization with the generalized Dantzig estimator
• The Smooth-Lasso and other ${\ell }_1+{\ell }_2$-penalized methods