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We provide the first analysis of a non-trivial quantization scheme forcompressed sensing measurements arising from structured measurements.Specifically, our analysis studies compressed sensing matrices consisting ofrows selected at random, without replacement, from a circulant matrix generatedby a random subgaussian vector. We quantize the measurements using stable,possibly one-bit, Sigma-Delta schemes, and use a reconstruction method based onconvex optimization. We show that the part of the reconstruction error due toquantization decays polynomially in the number of measurements. This is in linewith analogous results on Sigma-Delta quantization associated with randomGaussian or subgaussian matrices, and significantly better than resultsassociated with the widely assumed memoryless scalar quantization. Moreover, weprove that our approach is stable and robust; i.e., the reconstruction errordegrades gracefully in the presence of non-quantization noise and when theunderlying signal is not strictly sparse. The analysis relies on resultsconcerning subgaussian chaos processes as well as a variation of McDiarmid'sinequality.

Quantized Compressed Sensing for Partial Random Circulant Matrices