A necessary and sufficient condition for exact sparse recovery by l1 minimization

In this Note, a new sharp sufficient condition for exact sparse recovery by l(1)-penalized minimization from linear measurements is proposed. The main contribution of this paper is to show that, for most matrices, this condition is also necessary. Moreover, when the l(1) minimizer is unique, we investigate its sensitivity to the measurements and we establish that the application associating the measurements to this minimizer is Lipschitz-continuous. (C) 2012 Published by Elsevier Masson SAS on behalf of Academie des sciences.