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

TL;DR

This paper introduces an $ ext{l}_1$-norm invariant convex $k$-sparse decomposition of signals, providing theoretical insights and a simple derivation of a key RIP recovery condition relevant to compressed sensing.

Contribution

It formulates and proves a novel convex $k$-sparse decomposition invariant under $ ext{l}_1$ norm, advancing theoretical understanding in compressed sensing.

Findings

Provides a convex $k$-sparse decomposition invariant under $ ext{l}_1$ norm.

Derives the RIP recovery condition $ ext{delta}_k + heta_{k,k} < 1$.

Enhances theoretical tools for compressed sensing analysis.

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.