On the $\ell_1$-Norm Invariant Convex k-Sparse Decomposition of Signals

Abstract

Inspired by an interesting idea of Cai and Zhang, we formulate and prove the convex $k$-sparse decomposition of vectors which is invariant with respect to $\ell_1$ norm. This result fits well in discussing compressed sensing problems under RIP, but we believe it also has independent interest. As an application, a simple derivation of the RIP recovery condition $\delta_k+\theta_{k,k} < 1$ is presented.