A Simple Proof for Recoverability of `1-Minimization: Go Over or Under?

It is well-known by now that `1 minimization can help recover sparse solutions to under-determined linear equations or sparsely corrupted solutions to over-determined equations, and the two problems are equivalent under appropriate conditions. So far almost all theoretic results have been obtained through studying the “under-determined side” of the problem. In this note, we take a different approach from the “overdetermined side” and show that a recoverability result (with the best available order) follows almost immediately from an inequality of Garnaev and Gluskin. We also connect dots with recoverability conditions obtained from different spaces.