Restricted Isometries for Partial Random Circulant Matrices

Details Restricted Isometries for Partial Random Circulant Matrices Restricted Isometries for Partial Random Circulant Matrices

2010-10-09

arxiv.org/abs/1010.1847v1

Computer Science

Information Theory

Scientific paper

In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to succeed when the restricted isometry constants of the sampling matrix are small. Many potential applications of compressed sensing involve a data-acquisition process that proceeds by convolution with a random pulse followed by (nonrandom) subsampling. At present, the theoretical analysis of this measurement technique is lacking. This paper demonstrates that the $s$th order restricted isometry constant is small when the number $m$ of samples satisfies $m \gtrsim (s \log n)^{3/2}$, where $n$ is the length of the pulse. This bound improves on previous estimates, which exhibit quadratic scaling.

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