45 research outputs found
Asymptotic Analysis of Complex LASSO via Complex Approximate Message Passing (CAMP)
Recovering a sparse signal from an undersampled set of random linear
measurements is the main problem of interest in compressed sensing. In this
paper, we consider the case where both the signal and the measurements are
complex. We study the popular reconstruction method of -regularized
least squares or LASSO. While several studies have shown that the LASSO
algorithm offers desirable solutions under certain conditions, the precise
asymptotic performance of this algorithm in the complex setting is not yet
known. In this paper, we extend the approximate message passing (AMP) algorithm
to the complex signals and measurements and obtain the complex approximate
message passing algorithm (CAMP). We then generalize the state evolution
framework recently introduced for the analysis of AMP, to the complex setting.
Using the state evolution, we derive accurate formulas for the phase transition
and noise sensitivity of both LASSO and CAMP
Maximin Analysis of Message Passing Algorithms for Recovering Block Sparse Signals
We consider the problem of recovering a block (or group) sparse signal from
an underdetermined set of random linear measurements, which appear in
compressed sensing applications such as radar and imaging. Recent results of
Donoho, Johnstone, and Montanari have shown that approximate message passing
(AMP) in combination with Stein's shrinkage outperforms group LASSO for large
block sizes. In this paper, we prove that, for a fixed block size and in the
strong undersampling regime (i.e., having very few measurements compared to the
ambient dimension), AMP cannot improve upon group LASSO, thereby complementing
the results of Donoho et al
On Phase Transition of Compressed Sensing in the Complex Domain
The phase transition is a performance measure of the sparsity-undersampling
tradeoff in compressed sensing (CS). This letter reports our first observation
and evaluation of an empirical phase transition of the minimization
approach to the complex valued CS (CVCS), which is positioned well above the
known phase transition of the real valued CS in the phase plane. This result
can be considered as an extension of the existing phase transition theory of
the block-sparse CS (BSCS) based on the universality argument, since the CVCS
problem does not meet the condition required by the phase transition theory of
BSCS but its observed phase transition coincides with that of BSCS. Our result
is obtained by applying the recently developed ONE-L1 algorithms to the
empirical evaluation of the phase transition of CVCS.Comment: 4 pages, 3 figure