The Annals of Statistics

Near-ideal model selection by ℓ1 minimization

Emmanuel J. Candès and Yaniv Plan

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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.

Article information

Source
Ann. Statist., Volume 37, Number 5A (2009), 2145-2177.

Dates
First available in Project Euclid: 15 July 2009

Permanent link to this document
https://projecteuclid.org/euclid.aos/1247663751

Digital Object Identifier
doi:10.1214/08-AOS653

Mathematical Reviews number (MathSciNet)
MR2543688

Zentralblatt MATH identifier
1173.62053

Subjects
Primary: 62C05: General considerations 62G05: Estimation
Secondary: 94A08: Image processing (compression, reconstruction, etc.) [See also 68U10] 94A12: Signal theory (characterization, reconstruction, filtering, etc.)

Keywords
Model selection oracle inequalities the lasso compressed sensing incoherence eigenvalues of random matrices

Citation

Candès, Emmanuel J.; Plan, Yaniv. Near-ideal model selection by ℓ 1 minimization. Ann. Statist. 37 (2009), no. 5A, 2145--2177. doi:10.1214/08-AOS653. https://projecteuclid.org/euclid.aos/1247663751


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