Noncommutative Donoho-Elad-Gribonval-Nielsen-Fuchs Sparsity Theorem

TL;DR

This paper extends the classical sparsity theorem to a noncommutative setting using frames for Hilbert C*-modules, providing a new theoretical foundation for sparse solutions in noncommutative analysis.

Contribution

It introduces a noncommutative version of the Donoho-Elad-Gribonval-Nielsen-Fuchs sparsity theorem using Hilbert C*-modules.

Findings

Established a noncommutative sparsity theorem.

Connected classical and noncommutative sparse recovery.

Provided theoretical insights into noncommutative signal analysis.

Abstract

Breakthrough Sparsity Theorem, derived independently by Donoho and Elad \textit{[Proc. Natl. Acad. Sci. USA, 2003]}, Gribonval and Nielsen \textit{[IEEE Trans. Inform. Theory, 2003]} and Fuchs \textit{[IEEE Trans. Inform. Theory, 2004]} says that unique sparse solution to NP-Hard $\ell_0$-minimization problem can be obtained using unique solution of P-Type $\ell_1$-minimization problem. In this paper, we derive noncommutative version of their result using frames for Hilbert C*-modules.