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

Breakthrough Sparsity Theorem, derived independently by Donoho and Elad $\mathit{\text{[Proc.}}\mathit{\text{Natl.}}\mathit{\text{Acad.}}\mathit{\text{Sci.}}\mathit{\text{USA,}}\mathit{\text{2003]}}$, Gribonval and Nielsen $\mathit{\text{[IEEE}}\mathit{\text{Trans.}}\mathit{\text{Inform.}}\mathit{\text{Theory,}}\mathit{\text{2003]}}$ and Fuchs $\mathit{\text{[IEEE}}\mathit{\text{Trans.}}\mathit{\text{Inform.}}\mathit{\text{Theory,}}\mathit{\text{2004]}}$ says that unique sparse solution to NP-Hard ${\mathrm{\ell }}_0$-minimization problem can be obtained using unique solution of P-Type ${\mathrm{\ell }}_1$-minimization problem. In this paper, we derive noncommutative version of their result using frames for Hilbert C*-modules.