3,870 research outputs found
Compression-Based Compressed Sensing
Modern compression algorithms exploit complex structures that are present in
signals to describe them very efficiently. On the other hand, the field of
compressed sensing is built upon the observation that "structured" signals can
be recovered from their under-determined set of linear projections. Currently,
there is a large gap between the complexity of the structures studied in the
area of compressed sensing and those employed by the state-of-the-art
compression codes. Recent results in the literature on deterministic signals
aim at bridging this gap through devising compressed sensing decoders that
employ compression codes. This paper focuses on structured stochastic processes
and studies the application of rate-distortion codes to compressed sensing of
such signals. The performance of the formerly-proposed compressible signal
pursuit (CSP) algorithm is studied in this stochastic setting. It is proved
that in the very low distortion regime, as the blocklength grows to infinity,
the CSP algorithm reliably and robustly recovers instances of a stationary
process from random linear projections as long as their count is slightly more
than times the rate-distortion dimension (RDD) of the source. It is also
shown that under some regularity conditions, the RDD of a stationary process is
equal to its information dimension (ID). This connection establishes the
optimality of the CSP algorithm at least for memoryless stationary sources, for
which the fundamental limits are known. Finally, it is shown that the CSP
algorithm combined by a family of universal variable-length fixed-distortion
compression codes yields a family of universal compressed sensing recovery
algorithms
Universal Compressed Sensing
In this paper, the problem of developing universal algorithms for compressed
sensing of stochastic processes is studied. First, R\'enyi's notion of
information dimension (ID) is generalized to analog stationary processes. This
provides a measure of complexity for such processes and is connected to the
number of measurements required for their accurate recovery. Then a minimum
entropy pursuit (MEP) optimization approach is proposed, and it is proven that
it can reliably recover any stationary process satisfying some mixing
constraints from sufficient number of randomized linear measurements, without
having any prior information about the distribution of the process. It is
proved that a Lagrangian-type approximation of the MEP optimization problem,
referred to as Lagrangian-MEP problem, is identical to a heuristic
implementable algorithm proposed by Baron et al. It is shown that for the right
choice of parameters the Lagrangian-MEP algorithm, in addition to having the
same asymptotic performance as MEP optimization, is also robust to the
measurement noise. For memoryless sources with a discrete-continuous mixture
distribution, the fundamental limits of the minimum number of required
measurements by a non-universal compressed sensing decoder is characterized by
Wu et al. For such sources, it is proved that there is no loss in universal
coding, and both the MEP and the Lagrangian-MEP asymptotically achieve the
optimal performance
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
Approximate Sparsity Pattern Recovery: Information-Theoretic Lower Bounds
Recovery of the sparsity pattern (or support) of an unknown sparse vector
from a small number of noisy linear measurements is an important problem in
compressed sensing. In this paper, the high-dimensional setting is considered.
It is shown that if the measurement rate and per-sample signal-to-noise ratio
(SNR) are finite constants independent of the length of the vector, then the
optimal sparsity pattern estimate will have a constant fraction of errors.
Lower bounds on the measurement rate needed to attain a desired fraction of
errors are given in terms of the SNR and various key parameters of the unknown
vector. The tightness of the bounds in a scaling sense, as a function of the
SNR and the fraction of errors, is established by comparison with existing
achievable bounds. Near optimality is shown for a wide variety of practically
motivated signal models
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
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