30 research outputs found
Noisy Signal Recovery via Iterative Reweighted L1-Minimization
Compressed sensing has shown that it is possible to reconstruct sparse high
dimensional signals from few linear measurements. In many cases, the solution
can be obtained by solving an L1-minimization problem, and this method is
accurate even in the presence of noise. Recent a modified version of this
method, reweighted L1-minimization, has been suggested. Although no provable
results have yet been attained, empirical studies have suggested the reweighted
version outperforms the standard method. Here we analyze the reweighted
L1-minimization method in the noisy case, and provide provable results showing
an improvement in the error bound over the standard bounds
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
Improved sparse recovery thresholds with two-step reweighted â„“_1 minimization
It is well known that ℓ_1 minimization can be used to recover sufficiently sparse unknown signals from compressed linear measurements. In fact, exact thresholds on the sparsity, as a function of the ratio between the system dimensions, so that with high probability almost all sparse signals can be recovered from iid Gaussian measurements, have been computed and are referred to as “weak thresholds” [4]. In this paper, we introduce a reweighted ℓ_1 recovery algorithm composed of two steps: a standard ℓ_1 minimization step to identify a set of entries where the signal is likely to reside, and a weighted ℓ_1 minimization step where entries outside this set are penalized. For signals where the non-sparse component has iid Gaussian entries, we prove a “strict” improvement in the weak recovery threshold. Simulations suggest that the improvement can be quite impressive—over 20% in the example we consider