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

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 $δ_k+θ_{k,k} < 1$ is presented.