On a randomized version of compressive sensing

Question

The compressive sensing theory of Candes and Tao (See http://en.wikipedia.org/wiki/Compressed_sensing) relies highly on the fact that the underlying data (such as a signal or an image) is sparse or compressible under some basis.

Now we suppose that the underlying data is probabilistic, namely the data follow some distribution. And we want to know with how much probability that the samples from the distribution would be sparse or compressive under some basis.

Is there any relevant literature? Thanks.

Answer 1

http://www-stat.stanford.edu/~candes/papers/IncoherenceCS.pdf

Answer 2

@Daniel: Can you make your question more precise? In traditional CS, the signal is $k$-sparse but their locations are typically uniformly distributed, i.e., there are $n\choose k$ possibilities. I am a bit confused as to what you mean by "data follow some distribution".

Answer 3

This paper may be related:

V. Cevher, “Learning with compressible priors,” in NIPS, Vancouver, BC, Canada, 2008, p. 7--12.

From the abstract:

We describe a set of probability distributions, dubbed compressible priors, whose independent and identically distributed (iid) realizations result in p-compressible signals. [...] We show that the membership of generalized Pareto, Student’s t, log-normal, Frechet, and log-logistic distributions to the set of compressible priors depends only on the distribution parameters and is independent of N. In contrast, we demonstrate that the membership of the generalized Gaussian dis- tribution (GGD) depends both on the signal dimension and the GGD parameters