Quantized Compressed Sensing for Partial Random Circulant Matrices
Abstract
We provide the first analysis of a nontrivial quantization scheme for compressed sensing measurements arising from structured measurements. Specifically, our analysis studies compressed sensing matrices consisting of rows selected at random, without replacement, from a circulant matrix generated by a random subgaussian vector. We quantize the measurements using stable, possibly onebit, SigmaDelta schemes, and use a reconstruction method based on convex optimization. We show that the part of the reconstruction error due to quantization decays polynomially in the number of measurements. This is in line with analogous results on SigmaDelta quantization associated with random Gaussian or subgaussian matrices, and significantly better than results associated with the widely assumed memoryless scalar quantization. Moreover, we prove that our approach is stable and robust; i.e., the reconstruction error degrades gracefully in the presence of nonquantization noise and when the underlying signal is not strictly sparse. The analysis relies on results concerning subgaussian chaos processes as well as a variation of McDiarmid's inequality.
 Publication:

arXiv eprints
 Pub Date:
 February 2017
 arXiv:
 arXiv:1702.04711
 Bibcode:
 2017arXiv170204711F
 Keywords:

 Computer Science  Information Theory;
 Mathematics  Numerical Analysis;
 94A12;
 94A20;
 41A25;
 15B05;
 15B52
 EPrint:
 15 pages