Optimal Choice of Weights for Sparse Recovery With Prior Information

Compressed sensing deals with the recovery of sparse signals from linear measurements. Without any additional information, it is possible to recover an $s$-sparse signal using $m \gtrsim s \log(d/s)$ measurements in a robust and stable way. Some applications provide additional information, such as on the location of the support of the signal. Using this information, it is conceivable the threshold amount of measurements can be lowered. A proposed algorithm for this task is \emph{weighted $\ell_1$-minimization}. Put shortly, one modifies standard $\ell_1$-minimization by assigning different weights to different parts of the index set $[1, \dots d]$. The task of choosing the weights is however non-trivial. This paper provides a complete answer to the question of an optimal choice of the weights. In fact, it is shown that it is possible to directly calculate unique weights that are optimal in the sense that the threshold amount of measurements needed for exact recovery is minimized. The proof uses recent results about the connection between convex geometry and compressed sensing-type algorithms.