Journal
INFORMATION AND INFERENCE-A JOURNAL OF THE IMA
Volume 3, Issue 1, Pages 40-58Publisher
OXFORD UNIV PRESS
DOI: 10.1093/imaiai/iat007
Keywords
compressed sensing; quantization; random frames; root-exponential accuracy; Sigma-Delta; sub-Gaussian matrices
Categories
Funding
- Banting Postdoctoral Fellowship
- Natural Sciences and Engineering Research Counsel of Canada (NSERC)
- NSERC Discovery Accelerator Supplement Award
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Suppose that the collection {ei}(i-1)(m) forms a frame for R-k, where each entry of the vector ei is a sub-Gaussian random variable. We consider expansions in such a frame, which are then quantized using a Sigma-Delta scheme. We show that an arbitrary signal in Rk can be recovered from its quantized frame coefficients up to an error which decays root-exponentially in the oversampling rate m/k. Here the quantization scheme is assumed to be chosen appropriately depending on the oversampling rate and the quantization alphabet can be coarse. The result holds with high probability on the draw of the frame uniformly for all signals. The crux of the argument is a bound on the extreme singular values of the product of a deterministic matrix and a sub-Gaussian frame. For fine quantization alphabets, we leverage this bound to show polynomial error decay in the context of compressed sensing. Our results extend previous results for structured deterministic frame expansions and Gaussian compressed sensing measurements.
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