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?


"Recovery of Exact Sparse Representations in the Presence of Bounded Noise", by J. Fuchs, deals with the case you ask. From the abstract:

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 $b$ ...

I think theorem 2 and 3 are what you're looking for (at least in part).

  • $\begingroup$ 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? $\endgroup$ – Thomas Arildsen May 14 '13 at 19:59
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    $\begingroup$ I suspect the full rank and the mutual coherence conditions are true for Gaussian matrices with "very high probability". I'm not sure about condition (8). By the way, take a look at chapter 5 of Elad's book (Sparse and redundant representations). It seems that for the noisy observations case, the focus should be on stability, rather than in uniqueness. $\endgroup$ – Alejandro May 16 '13 at 12:13
  • $\begingroup$ I think you are right about stability in the noisy case. This also seems to be the theme of the paper @peter-k recommended in his comment. I will look into that. $\endgroup$ – Thomas Arildsen May 16 '13 at 13:07

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