The Dantzig selector : statistical estimation when p is much larger than

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 = Ax+ z, where x ∈ R is a parameter vector of interest, A is a data matrix with possibly far fewer rows than columns, n p, and the zi’s are i.i.d. N(0, σ). Is it possible to estimate x reliably based on the noisy data y? To estimate x, we introduce a new estimator—we call the Dantzig selector—which is solution to the `1-regularization problem min x̃∈Rp ‖x̃‖`1 subject to ‖A r‖`∞ ≤ (1 + t−1) √ 2 log p · σ, where r is the residual vector y − Ax̃ and t is a positive scalar. We show that if A obeys a uniform uncertainty principle (with unit-normed columns) and if the true parameter vector x is sufficiently sparse (which here roughly guarantees that the model is identifiable), then with very large probability ‖x̂− x‖2`2 ≤ C 2 · 2 log p · ( σ + ∑