4,547 research outputs found
Replica Symmetry Breaking in Compressive Sensing
For noisy compressive sensing systems, the asymptotic distortion with respect
to an arbitrary distortion function is determined when a general class of
least-square based reconstruction schemes is employed. The sampling matrix is
considered to belong to a large ensemble of random matrices including i.i.d.
and projector matrices, and the source vector is assumed to be i.i.d. with a
desired distribution. We take a statistical mechanical approach by representing
the asymptotic distortion as a macroscopic parameter of a spin glass and
employing the replica method for the large-system analysis. In contrast to
earlier studies, we evaluate the general replica ansatz which includes the RS
ansatz as well as RSB. The generality of the solution enables us to study the
impact of symmetry breaking. Our numerical investigations depict that for the
reconstruction scheme with the "zero-norm" penalty function, the RS fails to
predict the asymptotic distortion for relatively large compression rates;
however, the one-step RSB ansatz gives a valid prediction of the performance
within a larger regime of compression rates.Comment: 7 pages, 3 figures, presented at ITA 201
Distributed Quantization for Compressed Sensing
We study distributed coding of compressed sensing (CS) measurements using
vector quantizer (VQ). We develop a distributed framework for realizing
optimized quantizer that enables encoding CS measurements of correlated sparse
sources followed by joint decoding at a fusion center. The optimality of VQ
encoder-decoder pairs is addressed by minimizing the sum of mean-square errors
between the sparse sources and their reconstruction vectors at the fusion
center. We derive a lower-bound on the end-to-end performance of the studied
distributed system, and propose a practical encoder-decoder design through an
iterative algorithm.Comment: 5 Pages, Accepted for presentation in ICASSP 201
Recovery from Linear Measurements with Complexity-Matching Universal Signal Estimation
We study the compressed sensing (CS) signal estimation problem where an input
signal is measured via a linear matrix multiplication under additive noise.
While this setup usually assumes sparsity or compressibility in the input
signal during recovery, the signal structure that can be leveraged is often not
known a priori. In this paper, we consider universal CS recovery, where the
statistics of a stationary ergodic signal source are estimated simultaneously
with the signal itself. Inspired by Kolmogorov complexity and minimum
description length, we focus on a maximum a posteriori (MAP) estimation
framework that leverages universal priors to match the complexity of the
source. Our framework can also be applied to general linear inverse problems
where more measurements than in CS might be needed. We provide theoretical
results that support the algorithmic feasibility of universal MAP estimation
using a Markov chain Monte Carlo implementation, which is computationally
challenging. We incorporate some techniques to accelerate the algorithm while
providing comparable and in many cases better reconstruction quality than
existing algorithms. Experimental results show the promise of universality in
CS, particularly for low-complexity sources that do not exhibit standard
sparsity or compressibility.Comment: 29 pages, 8 figure
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