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Published October 2009 | Published
Journal Article Open

Near-ideal model selection by ℓ1 minimization

Abstract

We consider the fundamental problem of estimating the mean of a vector y=Xβ+z, where X is an n×p design matrix in which one can have far more variables than observations, and z is a stochastic error term -— the so-called "p>n" setup. When β is sparse, or, more generally, when there is a sparse subset of covariates providing a close approximation to the unknown mean vector, we ask whether or not it is possible to accurately estimate Xβ using a computationally tractable algorithm. We show that, in a surprisingly wide range of situations, the lasso happens to nearly select the best subset of variables. Quantitatively speaking, we prove that solving a simple quadratic program achieves a squared error within a logarithmic factor of the ideal mean squared error that one would achieve with an oracle supplying perfect information about which variables should and should not be included in the model. Interestingly, our results describe the average performance of the lasso; that is, the performance one can expect in an vast majority of cases where Xβ is a sparse or nearly sparse superposition of variables, but not in all cases. Our results are nonasymptotic and widely applicable, since they simply require that pairs of predictor variables are not too collinear.

Additional Information

© Institute of Mathematical Statistics, 2009. Received June 2008. Supported in part by a National Science Foundation Grant CCF-515362 and by the 2006 Waterman Award (NSF). E. Candès would like to thank Chiara Sabatti for fruitful discussions and for offering some insightful comments about an early draft of this paper. E. Candès. also acknowledges inspiring conversations with Terence Tao and Joel Tropp about parts of this paper. We thank the anonymous referees for their constructive comments.

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August 21, 2023
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