44 research outputs found
Construction of a Large Class of Deterministic Sensing Matrices that Satisfy a Statistical Isometry Property
Compressed Sensing aims to capture attributes of -sparse signals using
very few measurements. In the standard Compressed Sensing paradigm, the
\m\times \n measurement matrix \A is required to act as a near isometry on
the set of all -sparse signals (Restricted Isometry Property or RIP).
Although it is known that certain probabilistic processes generate \m \times
\n matrices that satisfy RIP with high probability, there is no practical
algorithm for verifying whether a given sensing matrix \A has this property,
crucial for the feasibility of the standard recovery algorithms. In contrast
this paper provides simple criteria that guarantee that a deterministic sensing
matrix satisfying these criteria acts as a near isometry on an overwhelming
majority of -sparse signals; in particular, most such signals have a unique
representation in the measurement domain. Probability still plays a critical
role, but it enters the signal model rather than the construction of the
sensing matrix. We require the columns of the sensing matrix to form a group
under pointwise multiplication. The construction allows recovery methods for
which the expected performance is sub-linear in \n, and only quadratic in
\m; the focus on expected performance is more typical of mainstream signal
processing than the worst-case analysis that prevails in standard Compressed
Sensing. Our framework encompasses many families of deterministic sensing
matrices, including those formed from discrete chirps, Delsarte-Goethals codes,
and extended BCH codes.Comment: 16 Pages, 2 figures, to appear in IEEE Journal of Selected Topics in
Signal Processing, the special issue on Compressed Sensin
Model Selection: Two Fundamental Measures of Coherence and Their Algorithmic Significance
The problem of model selection arises in a number of contexts, such as
compressed sensing, subset selection in linear regression, estimation of
structures in graphical models, and signal denoising. This paper generalizes
the notion of \emph{incoherence} in the existing literature on model selection
and introduces two fundamental measures of coherence---termed as the worst-case
coherence and the average coherence---among the columns of a design matrix. In
particular, it utilizes these two measures of coherence to provide an in-depth
analysis of a simple one-step thresholding (OST) algorithm for model selection.
One of the key insights offered by the ensuing analysis is that OST is feasible
for model selection as long as the design matrix obeys an easily verifiable
property. In addition, the paper also characterizes the model-selection
performance of OST in terms of the worst-case coherence, \mu, and establishes
that OST performs near-optimally in the low signal-to-noise ratio regime for N
x C design matrices with \mu = O(N^{-1/2}). Finally, in contrast to some of the
existing literature on model selection, the analysis in the paper is
nonasymptotic in nature, it does not require knowledge of the true model order,
it is applicable to generic (random or deterministic) design matrices, and it
neither requires submatrices of the design matrix to have full rank, nor does
it assume a statistical prior on the values of the nonzero entries of the data
vector.Comment: 5 pages; Accepted for Proc. 2010 IEEE International Symposium on
Information Theory (ISIT 2010
Efficient and Robust Compressed Sensing Using Optimized Expander Graphs
Expander graphs have been recently proposed to construct efficient compressed sensing algorithms. In particular, it has been shown that any n-dimensional vector that is k-sparse can be fully recovered using O(klog n) measurements and only O(klog n) simple recovery iterations. In this paper, we improve upon this result by considering expander graphs with expansion coefficient beyond 3/4 and show that, with the same number of measurements, only O(k) recovery iterations are required, which is a significant improvement when n is large. In fact, full recovery can be accomplished by at most 2k very simple iterations. The number of iterations can be reduced arbitrarily close to k, and the recovery algorithm can be implemented very efficiently using a simple priority queue with total recovery time O(nlog(n/k))). We also show that by tolerating a small penal- ty on the number of measurements, and not on the number of recovery iterations, one can use the efficient construction of a family of expander graphs to come up with explicit measurement matrices for this method. We compare our result with other recently developed expander-graph-based methods and argue that it compares favorably both in terms of the number of required measurements and in terms of the time complexity and the simplicity of recovery. Finally, we will show how our analysis extends to give a robust algorithm that finds the position and sign of the k significant elements of an almost k-sparse signal and then, using very simple optimization techniques, finds a k-sparse signal which is close to the best k-term approximation of the original signal
Uncovering elements of style
This paper relates the style of 16th century Flemish paintings by Goossen van der Weyden (GvdW) to the style of preliminary sketches or underpaintings made prior to executing the painting. Van der Weyden made underpaintings in markedly different styles for reasons as yet not understood by art historians. The analysis presented here starts from a classification of the underpaintings into four distinct styles by experts in art history. Analysis of the painted surfaces by a combination of wavelet analysis, hidden Markov trees and boosting algorithms can distinguish the four underpainting styles with greater than 90% cross-validation accuracy. On a subsequent blind test this classifier provided insight into the hypothesis by art historians that different patches of the finished painting were executed by different hands
Performance bounds for expander-based compressed sensing in Poisson noise
This paper provides performance bounds for compressed sensing in the presence
of Poisson noise using expander graphs. The Poisson noise model is appropriate
for a variety of applications, including low-light imaging and digital
streaming, where the signal-independent and/or bounded noise models used in the
compressed sensing literature are no longer applicable. In this paper, we
develop a novel sensing paradigm based on expander graphs and propose a MAP
algorithm for recovering sparse or compressible signals from Poisson
observations. The geometry of the expander graphs and the positivity of the
corresponding sensing matrices play a crucial role in establishing the bounds
on the signal reconstruction error of the proposed algorithm. We support our
results with experimental demonstrations of reconstructing average packet
arrival rates and instantaneous packet counts at a router in a communication
network, where the arrivals of packets in each flow follow a Poisson process.Comment: revised version; accepted to IEEE Transactions on Signal Processin
Efficient and Robust Compressed Sensing using High-Quality Expander Graphs
Expander graphs have been recently proposed to construct efficient compressed
sensing algorithms. In particular, it has been shown that any -dimensional
vector that is -sparse (with ) can be fully recovered using
measurements and only simple recovery
iterations. In this paper we improve upon this result by considering expander
graphs with expansion coefficient beyond 3/4 and show that, with the same
number of measurements, only recovery iterations are required, which is
a significant improvement when is large. In fact, full recovery can be
accomplished by at most very simple iterations. The number of iterations
can be made arbitrarily close to , and the recovery algorithm can be
implemented very efficiently using a simple binary search tree. We also show
that by tolerating a small penalty on the number of measurements, and not on
the number of recovery iterations, one can use the efficient construction of a
family of expander graphs to come up with explicit measurement matrices for
this method. We compare our result with other recently developed
expander-graph-based methods and argue that it compares favorably both in terms
of the number of required measurements and in terms of the recovery time
complexity. Finally we will show how our analysis extends to give a robust
algorithm that finds the position and sign of the significant elements of
an almost -sparse signal and then, using very simple optimization
techniques, finds in sublinear time a -sparse signal which approximates the
original signal with very high precision