A Remark on the Lasso and the Dantzig Selector

Numerous authors have established a connection between the Compressed Sensing problem without noise and the estimation of the Gelfand widths. This article shows that this connection is still true in the noisy case. Indeed, we investigate the lasso and the Dantzig selector in terms of the distortion of the design. This latter measures how far is the intersection between the kernel of the design matrix and the unit l1-ball from an l2-ball. In particular, we exhibit the weakest condition to get oracle inequalities in terms of the s-best term approximation.