2,071 research outputs found
Compressive Phase Retrieval From Squared Output Measurements Via Semidefinite Programming
Given a linear system in a real or complex domain, linear regression aims to
recover the model parameters from a set of observations. Recent studies in
compressive sensing have successfully shown that under certain conditions, a
linear program, namely, l1-minimization, guarantees recovery of sparse
parameter signals even when the system is underdetermined. In this paper, we
consider a more challenging problem: when the phase of the output measurements
from a linear system is omitted. Using a lifting technique, we show that even
though the phase information is missing, the sparse signal can be recovered
exactly by solving a simple semidefinite program when the sampling rate is
sufficiently high, albeit the exact solutions to both sparse signal recovery
and phase retrieval are combinatorial. The results extend the type of
applications that compressive sensing can be applied to those where only output
magnitudes can be observed. We demonstrate the accuracy of the algorithms
through theoretical analysis, extensive simulations and a practical experiment.Comment: Parts of the derivations have submitted to the 16th IFAC Symposium on
System Identification, SYSID 2012, and parts to the 51st IEEE Conference on
Decision and Control, CDC 201
Structured random measurements in signal processing
Compressed sensing and its extensions have recently triggered interest in
randomized signal acquisition. A key finding is that random measurements
provide sparse signal reconstruction guarantees for efficient and stable
algorithms with a minimal number of samples. While this was first shown for
(unstructured) Gaussian random measurement matrices, applications require
certain structure of the measurements leading to structured random measurement
matrices. Near optimal recovery guarantees for such structured measurements
have been developed over the past years in a variety of contexts. This article
surveys the theory in three scenarios: compressed sensing (sparse recovery),
low rank matrix recovery, and phaseless estimation. The random measurement
matrices to be considered include random partial Fourier matrices, partial
random circulant matrices (subsampled convolutions), matrix completion, and
phase estimation from magnitudes of Fourier type measurements. The article
concludes with a brief discussion of the mathematical techniques for the
analysis of such structured random measurements.Comment: 22 pages, 2 figure
- …