## "Compressed" Compressed Sensing

The field of compressed sensing has shown that a sparse but otherwise arbitrary vector can be recovered exactly from a small number of randomly constructed linear projections (or samples). The question addressed in this paper is whether an even smaller number of samples is sufficient when there exists prior knowledge about the distribution of the unknown vector, or when only partial recovery is needed. An information-theoretic lower bound with connections to free probability theory and an upper bound corresponding to a computationally simple thresholding estimator are derived. It is shown that in certain cases (e.g. discrete valued vectors or large distortions) the number of samples can be decreased. Interestingly though, it is also shown that in many cases no reduction is possible.

Published in:
2010 Ieee International Symposium On Information Theory, 1548-1552
Presented at:
2010 IEEE International Symposium on Information Theory, Austin, TX, Jul 13, 2010
Year:
2010
Publisher:
Ieee Service Center, 445 Hoes Lane, Po Box 1331, Piscataway, Nj 08855-1331 Usa
ISBN:
978-1-4244-6960-4
Laboratories: