332 research outputs found
Robust one-bit compressed sensing with partial circulant matrices
We present optimal sample complexity estimates for one-bit compressed sensing
problems in a realistic scenario: the procedure uses a structured matrix (a
randomly sub-sampled circulant matrix) and is robust to analog pre-quantization
noise as well as to adversarial bit corruptions in the quantization process.
Our results imply that quantization is not a statistically expensive procedure
in the presence of nontrivial analog noise: recovery requires the same sample
size one would have needed had the measurement matrix been Gaussian and the
noisy analog measurements been given as data
Quantized Compressed Sensing for Partial Random Circulant Matrices
We provide the first analysis of a non-trivial 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 one-bit, Sigma-Delta 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 Sigma-Delta 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 non-quantization 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.Comment: 15 page
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