On statistical applications of l1-recovery
Résumé
We consider the problem of recovery of a sparse signal $x\in R^M$ from noisy observation $y=Ax+\sigma\xi$, where $A\in R^{n\times M}$, $\xi\sim N(0,I_n)$ and $n$ may be (much) smaller than $M$. We propose new methods of for recovery of $x$ based on $\ell_1$-minimization. Though they are intimately related to well-known techniques such as Lasso and Dantzig Selector, they often possess better statistical properties. These new procedures are based on veri able su cient conditions of exact recovery in Compressive Sensing and come with effi ciently veri fiable guaranties of performance. We also discuss fast implementation of the estimation routines, based on Non-Euclidean Basis Pursuit algorithm.