# Conditions for unique (not exact) recovery from noisy compressed measurements

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?

The purpose of this contribution is to extend some recent results on sparse representations of signals in redundant bases developed in the noise-free case to the case of noisy observations. [..] We consider the case $b = Ax_0 + e$ [...] and seek conditions under which $x_0$ can be recovered from $b$ ...
• Thanks for this answer. Now I just wonder to which extent the conditions around theorem 2 + 3 are satisfied by random Gaussian matrices: - $\bar A_o$ full rank - linear independence of the active columns, I suppose. - The mutual coherence condition in (2). - The condition in (8). Any idea? May 14 '13 at 19:59