393 research outputs found
Sparse recovery in bounded Riesz systems with applications to numerical methods for PDEs
We study sparse recovery with structured random measurement matrices having
independent, identically distributed, and uniformly bounded rows and with a
nontrivial covariance structure. This class of matrices arises from random
sampling of bounded Riesz systems and generalizes random partial Fourier
matrices. Our main result improves the currently available results for the null
space and restricted isometry properties of such random matrices. The main
novelty of our analysis is a new upper bound for the expectation of the
supremum of a Bernoulli process associated with a restricted isometry constant.
We apply our result to prove new performance guarantees for the CORSING method,
a recently introduced numerical approximation technique for partial
differential equations (PDEs) based on compressive sensing
A Simplified Approach to Recovery Conditions for Low Rank Matrices
Recovering sparse vectors and low-rank matrices from noisy linear
measurements has been the focus of much recent research. Various reconstruction
algorithms have been studied, including and nuclear norm minimization
as well as minimization with . These algorithms are known to
succeed if certain conditions on the measurement map are satisfied. Proofs of
robust recovery for matrices have so far been much more involved than in the
vector case.
In this paper, we show how several robust classes of recovery conditions can
be extended from vectors to matrices in a simple and transparent way, leading
to the best known restricted isometry and nullspace conditions for matrix
recovery. Our results rely on the ability to "vectorize" matrices through the
use of a key singular value inequality.Comment: 6 pages, This is a modified version of a paper submitted to ISIT
2011; Proc. Intl. Symp. Info. Theory (ISIT), Aug 201
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