The Dantzig selector:: Statistical estimation when p is much larger than n

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 epsilon R-p is a parameter vector of interest, X is a data matrix with possibly far fewer rows than columns, n << p, and the z(i)'s are i.i.d. N(0, sigma(2)). 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 l(1)-regularization problem (min) ((beta) over tilde epsilon Rp) parallel to(beta) over tilde parallel to l(1) subject to parallel to X*r parallel to l(infinity) <= (1 + t(-1)) root 2 log p.sigma, where r is the residual vector y - X (beta) over tilde 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 to(beta) over cap-beta parallel to(2)(l2) <= C-2 . 2 log p . (sigma(2) + Sigma(i) min(beta(2)(i), sigma(2))). 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 (LP).