I am looking for theory on whether compressed sensing reconstruction via ℓ1-minimization is unique and under which conditions.
I have looked through:
Tropp, J. A., "Just relax: convex programming methods for identifying sparse signals in noise," IEEE Transactions on Information Theory, 2006, 52, 1030-1051.
This paper states that the minimizer is unique but I can't quite boil down exactly what is required for this uniqueness (Eq. (ℓ1-Error) and Theorem 14) to hold.
I also looked at:
Candès, E. J.; Romberg, J. & Tao, T., "Stable signal recovery from incomplete and inaccurate measurements," Communications on Pure and Applied Mathematics, 2006, 59, 1207-1223.
However, they do not seem to claim that the minimizer is unique.
Since both of these papers are from the earlier days of compressed sensing, I suspect that there may be newer results on this uniqueness of the solution. Do any of you have some hints?