Compressed sensing and optimal denoising of monotone signals

Eftychios A. Pnevmatikakis

ICASSP 2017accepted0 citations

Compressed sensing and optimal denoising of monotone signals

Abstract

We consider the problems of compressed sensing and optimal denoising for signals x 0 ∈ ℝ N that are monotone, i.e., x 0 (i + 1) ≥ x 0 (i), and sparsely varying, i.e., x 0 (i + 1) > x 0 (i) only for a small number k of indices i. We approach the compressed sensing problem by minimizing the total variation norm restricted to the class of monotone signals subject to equality constraints obtained from a number of measurements Ax 0 . For random Gaussian sensing matrices A ∈ ℝ m×N we derive a closed form expression for the number of measurements m required for successful reconstruction with high probability. We show that the probability undergoes a phase transition as m varies, and depends not only on the number of change points, but also on their location. For denoising we regularize with the same norm and derive a formula for the optimal regularizer weight that depends only mildly on x 0 . We obtain our results using the statistical dimension tool.

BibTeX

@inproceedings{icassp2017_compressedsensin,
  title = {Compressed sensing and optimal denoising of monotone signals},
  author = {Eftychios A. Pnevmatikakis},
  booktitle = {ICASSP 2017},
  year = {2017}
}