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.

Add some comments for the noise case

References in corpus (1)

• Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-rank Matrices