960 research outputs found
Information Theoretic Principles of Universal Discrete Denoising
Today, the internet makes tremendous amounts of data widely available. Often,
the same information is behind multiple different available data sets. This
lends growing importance to latent variable models that try to learn the hidden
information from the available imperfect versions. For example, social media
platforms can contain an abundance of pictures of the same person or object,
yet all of which are taken from different perspectives. In a simplified
scenario, one may consider pictures taken from the same perspective, which are
distorted by noise. This latter application allows for a rigorous mathematical
treatment, which is the content of this contribution. We apply a recently
developed method of dependent component analysis to image denoising when
multiple distorted copies of one and the same image are available, each being
corrupted by a different and unknown noise process. In a simplified scenario,
we assume that the distorted image is corrupted by noise that acts
independently on each pixel. We answer completely the question of how to
perform optimal denoising, when at least three distorted copies are available:
First we define optimality of an algorithm in the presented scenario, and then
we describe an aymptotically optimal universal discrete denoising algorithm
(UDDA). In the case of binary data and binary symmetric noise, we develop a
simplified variant of the algorithm, dubbed BUDDA, which we prove to attain
universal denoising uniformly.Comment: 10 pages, 6 figure
An Overview of Multi-Processor Approximate Message Passing
Approximate message passing (AMP) is an algorithmic framework for solving
linear inverse problems from noisy measurements, with exciting applications
such as reconstructing images, audio, hyper spectral images, and various other
signals, including those acquired in compressive signal acquisiton systems. The
growing prevalence of big data systems has increased interest in large-scale
problems, which may involve huge measurement matrices that are unsuitable for
conventional computing systems. To address the challenge of large-scale
processing, multiprocessor (MP) versions of AMP have been developed. We provide
an overview of two such MP-AMP variants. In row-MP-AMP, each computing node
stores a subset of the rows of the matrix and processes corresponding
measurements. In column- MP-AMP, each node stores a subset of columns, and is
solely responsible for reconstructing a portion of the signal. We will discuss
pros and cons of both approaches, summarize recent research results for each,
and explain when each one may be a viable approach. Aspects that are
highlighted include some recent results on state evolution for both MP-AMP
algorithms, and the use of data compression to reduce communication in the MP
network
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
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
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