The Annals of Statistics
- Ann. Statist.
- Volume 37, Number 5A (2009), 2145-2177.
Near-ideal model selection by ℓ1 minimization
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.
Ann. Statist., Volume 37, Number 5A (2009), 2145-2177.
First available in Project Euclid: 15 July 2009
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 62C05: General considerations 62G05: Estimation
Secondary: 94A08: Image processing (compression, reconstruction, etc.) [See also 68U10] 94A12: Signal theory (characterization, reconstruction, filtering, etc.)
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