NEAR-IDEAL MODEL SELECTION BY l1 MINIMIZATION

• 成果类型: Article
• 署名作者: Candes, Emmanuel J.; Plan, Yaniv
• 署名单位: California Institute of Technology
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/08-AOS653
• 发表日期: 2009
• 页码: 2145-2177
• 关键词: dantzig selector; Lasso; regression; REPRESENTATIONS; persistence

摘要

We consider the fundamental problem of estimating the mean of a vector y = X beta + 7, where X is an n x 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 beta 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 X beta 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 X beta 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.