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The Dantzig selector: Statistical estimation when $p$ is much larger than $n$. (With discussions and rejoinder). (English) Zbl 1139.62019

Summary: In many important statistical applications, the number of variables or parameters $p$ is much larger than the number of observations $n$. Suppose then that we have observations $y=X\beta +z$, where $\beta \in {ℝ}^{p}$ is a parameter vector of interest, $X$ is a data matrix with possibly far fewer rows than columns, $n\ll p$, and the ${z}_{i}$’s are i.i.d. $N\left(0,{\sigma }^{2}\right)$. Is it possible to estimate $\beta$ reliably based on the noisy data $y$? To estimate $\beta$, we introduce a new estimator – we call it the Dantzig selector – which is a solution to the ${\ell }_{1}$-regularization problem

$\underset{\stackrel{˜}{\beta }\in {ℝ}^{p}}{min}\parallel \stackrel{˜}{\beta }{\parallel }_{{\ell }_{1}}\phantom{\rule{1.em}{0ex}}\text{subject}\phantom{\rule{4.pt}{0ex}}\text{to}\phantom{\rule{1.em}{0ex}}{\parallel {X}^{*}r\parallel }_{{\ell }_{\infty }}\le \left(1+{t}^{-1}\right)\sqrt{2logp}·\sigma ,$

where $r$ is the residual vector $y-X\stackrel{˜}{\beta }$ and $t$ is a positive scalar. We show that if $X$ obeys a uniform uncertainty principle (with unit-normed columns) and if the true parameter vector $\beta$ is sufficiently sparse (which here roughly guarantees that the model is identifiable), then with very large probability,

$\parallel \stackrel{^}{\beta }{-\beta \parallel }_{{\ell }_{2}}^{2}\le {C}^{2}·2logp·\left({\sigma }^{2}+\sum _{i}min\left({\beta }_{i}^{2},{\sigma }^{2}\right)\right)·$

Our results are nonasymptotic and we give values for the constant $C$. Even though $n$ may be much smaller than $p$, our estimator achieves a loss within a logarithmic factor of the ideal mean squared error one would achieve with an oracle which would supply perfect information about which coordinates are nonzero, and which were above the noise level.

In multivariate regression and from a model selection viewpoint, our result says that it is possible nearly to select the best subset of variables by solving a very simple convex program, which, in fact, can easily be recast as a convenient linear program.

##### MSC:
 62G08 Nonparametric regression 62G05 Nonparametric estimation 94A08 Image processing (compression, reconstruction, etc.) 94A12 Signal theory (characterization, reconstruction, filtering, etc.) 90C05 Linear programming 62C05 General considerations in statistical decision theory
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