Shannon-theoretic limits on noisy compressive sampling

In this paper, we study the number of measurements required to recover a sparse signal in M with L nonzero coefficients from compressed samples in the presence of noise. We consider a number of different recovery criteria, including the exact recovery of the support of the signal, which was previously considered in the literature, as well as new criteria for the recovery of a large fraction of the support of the signal, and the recovery of a large fraction of the energy of the signal. For these recovery criteria, we prove that O(L) (an asymptotically linear multiple of L) measurements are necessary and sufficient for signal recovery, whenever L grows linearly as a function of M. This improves on the existing literature that is mostly focused on variants of a specific recovery algorithm based on convex programming, for which O(Llog(M 0L)) measurements are required. In contrast, the implementation of our proof method would have a higher complexity. We also show that O(Llog(M0L)) measurem...

• Type: Journal
• Year: 2010
• Where: TIT
• Authors: Mehmet Akçakaya, Vahid Tarokh
• Keywords: Education, Large Fraction, Measurements, Recovery Criteria, TIT 2010