7,856 research outputs found
Mismatched Estimation in Large Linear Systems
We study the excess mean square error (EMSE) above the minimum mean square
error (MMSE) in large linear systems where the posterior mean estimator (PME)
is evaluated with a postulated prior that differs from the true prior of the
input signal. We focus on large linear systems where the measurements are
acquired via an independent and identically distributed random matrix, and are
corrupted by additive white Gaussian noise (AWGN). The relationship between the
EMSE in large linear systems and EMSE in scalar channels is derived, and closed
form approximations are provided. Our analysis is based on the decoupling
principle, which links scalar channels to large linear system analyses.
Numerical examples demonstrate that our closed form approximations are
accurate.Comment: 5 pages, 2 figure
Compressive Imaging via Approximate Message Passing with Image Denoising
We consider compressive imaging problems, where images are reconstructed from
a reduced number of linear measurements. Our objective is to improve over
existing compressive imaging algorithms in terms of both reconstruction error
and runtime. To pursue our objective, we propose compressive imaging algorithms
that employ the approximate message passing (AMP) framework. AMP is an
iterative signal reconstruction algorithm that performs scalar denoising at
each iteration; in order for AMP to reconstruct the original input signal well,
a good denoiser must be used. We apply two wavelet based image denoisers within
AMP. The first denoiser is the "amplitude-scaleinvariant Bayes estimator"
(ABE), and the second is an adaptive Wiener filter; we call our AMP based
algorithms for compressive imaging AMP-ABE and AMP-Wiener. Numerical results
show that both AMP-ABE and AMP-Wiener significantly improve over the state of
the art in terms of runtime. In terms of reconstruction quality, AMP-Wiener
offers lower mean square error (MSE) than existing compressive imaging
algorithms. In contrast, AMP-ABE has higher MSE, because ABE does not denoise
as well as the adaptive Wiener filter.Comment: 15 pages; 2 tables; 7 figures; to appear in IEEE Trans. Signal
Proces
Analysis of Approximate Message Passing with a Class of Non-Separable Denoisers
Approximate message passing (AMP) is a class of efficient algorithms for
solving high-dimensional linear regression tasks where one wishes to recover an
unknown signal \beta_0 from noisy, linear measurements y = A \beta_0 + w. When
applying a separable denoiser at each iteration, the performance of AMP (for
example, the mean squared error of its estimates) can be accurately tracked by
a simple, scalar iteration referred to as state evolution. Although separable
denoisers are sufficient if the unknown signal has independent and identically
distributed entries, in many real-world applications, like image or audio
signal reconstruction, the unknown signal contains dependencies between
entries. In these cases, a coordinate-wise independence structure is not a good
approximation to the true prior of the unknown signal. In this paper we assume
the unknown signal has dependent entries, and using a class of non-separable
sliding-window denoisers, we prove that a new form of state evolution still
accurately predicts AMP performance. This is an early step in understanding the
role of non-separable denoisers within AMP, and will lead to a characterization
of more general denoisers in problems including compressive image
reconstruction.Comment: 37 pages, 1 figure. A shorter version of this paper to appear in the
proceedings of ISIT 201
Multiprocessor Approximate Message Passing with Column-Wise Partitioning
Solving a large-scale regularized linear inverse problem using multiple
processors is important in various real-world applications due to the
limitations of individual processors and constraints on data sharing policies.
This paper focuses on the setting where the matrix is partitioned column-wise.
We extend the algorithmic framework and the theoretical analysis of approximate
message passing (AMP), an iterative algorithm for solving linear inverse
problems, whose asymptotic dynamics are characterized by state evolution (SE).
In particular, we show that column-wise multiprocessor AMP (C-MP-AMP) obeys an
SE under the same assumptions when the SE for AMP holds. The SE results imply
that (i) the SE of C-MP-AMP converges to a state that is no worse than that of
AMP and (ii) the asymptotic dynamics of C-MP-AMP and AMP can be identical.
Moreover, for a setting that is not covered by SE, numerical results show that
damping can improve the convergence performance of C-MP-AMP.Comment: This document contains complete details of the previous version
(i.e., arXiv:1701.02578v1), which was accepted for publication in ICASSP 201
Empirical Bayes and Full Bayes for Signal Estimation
We consider signals that follow a parametric distribution where the parameter
values are unknown. To estimate such signals from noisy measurements in scalar
channels, we study the empirical performance of an empirical Bayes (EB)
approach and a full Bayes (FB) approach. We then apply EB and FB to solve
compressed sensing (CS) signal estimation problems by successively denoising a
scalar Gaussian channel within an approximate message passing (AMP) framework.
Our numerical results show that FB achieves better performance than EB in
scalar channel denoising problems when the signal dimension is small. In the CS
setting, the signal dimension must be large enough for AMP to work well; for
large signal dimensions, AMP has similar performance with FB and EB.Comment: This work was presented at the Information Theory and Application
workshop (ITA), San Diego, CA, Feb. 201
Approximate Message Passing in Coded Aperture Snapshot Spectral Imaging
We consider a compressive hyperspectral imaging reconstruction problem, where
three-dimensional spatio-spectral information about a scene is sensed by a
coded aperture snapshot spectral imager (CASSI). The approximate message
passing (AMP) framework is utilized to reconstruct hyperspectral images from
CASSI measurements, and an adaptive Wiener filter is employed as a
three-dimensional image denoiser within AMP. We call our algorithm
"AMP-3D-Wiener." The simulation results show that AMP-3D-Wiener outperforms
existing widely-used algorithms such as gradient projection for sparse
reconstruction (GPSR) and two-step iterative shrinkage/thresholding (TwIST)
given the same amount of runtime. Moreover, in contrast to GPSR and TwIST,
AMP-3D-Wiener need not tune any parameters, which simplifies the reconstruction
process.Comment: to appear in Globalsip 201
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