19,876 research outputs found
Masking Strategies for Image Manifolds
We consider the problem of selecting an optimal mask for an image manifold,
i.e., choosing a subset of the pixels of the image that preserves the
manifold's geometric structure present in the original data. Such masking
implements a form of compressive sensing through emerging imaging sensor
platforms for which the power expense grows with the number of pixels acquired.
Our goal is for the manifold learned from masked images to resemble its full
image counterpart as closely as possible. More precisely, we show that one can
indeed accurately learn an image manifold without having to consider a large
majority of the image pixels. In doing so, we consider two masking methods that
preserve the local and global geometric structure of the manifold,
respectively. In each case, the process of finding the optimal masking pattern
can be cast as a binary integer program, which is computationally expensive but
can be approximated by a fast greedy algorithm. Numerical experiments show that
the relevant manifold structure is preserved through the data-dependent masking
process, even for modest mask sizes
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
Compressive Parameter Estimation for Sparse Translation-Invariant Signals Using Polar Interpolation
We propose new compressive parameter estimation algorithms that make use of
polar interpolation to improve the estimator precision. Our work extends
previous approaches involving polar interpolation for compressive parameter
estimation in two aspects: (i) we extend the formulation from real non-negative
amplitude parameters to arbitrary complex ones, and (ii) we allow for mismatch
between the manifold described by the parameters and its polar approximation.
To quantify the improvements afforded by the proposed extensions, we evaluate
six algorithms for estimation of parameters in sparse translation-invariant
signals, exemplified with the time delay estimation problem. The evaluation is
based on three performance metrics: estimator precision, sampling rate and
computational complexity. We use compressive sensing with all the algorithms to
lower the necessary sampling rate and show that it is still possible to attain
good estimation precision and keep the computational complexity low. Our
numerical experiments show that the proposed algorithms outperform existing
approaches that either leverage polynomial interpolation or are based on a
conversion to a frequency-estimation problem followed by a super-resolution
algorithm. The algorithms studied here provide various tradeoffs between
computational complexity, estimation precision, and necessary sampling rate.
The work shows that compressive sensing for the class of sparse
translation-invariant signals allows for a decrease in sampling rate and that
the use of polar interpolation increases the estimation precision.Comment: 13 pages, 5 figures, to appear in IEEE Transactions on Signal
Processing; minor edits and correction
Conditioning of Random Block Subdictionaries with Applications to Block-Sparse Recovery and Regression
The linear model, in which a set of observations is assumed to be given by a
linear combination of columns of a matrix, has long been the mainstay of the
statistics and signal processing literature. One particular challenge for
inference under linear models is understanding the conditions on the dictionary
under which reliable inference is possible. This challenge has attracted
renewed attention in recent years since many modern inference problems deal
with the "underdetermined" setting, in which the number of observations is much
smaller than the number of columns in the dictionary. This paper makes several
contributions for this setting when the set of observations is given by a
linear combination of a small number of groups of columns of the dictionary,
termed the "block-sparse" case. First, it specifies conditions on the
dictionary under which most block subdictionaries are well conditioned. This
result is fundamentally different from prior work on block-sparse inference
because (i) it provides conditions that can be explicitly computed in
polynomial time, (ii) the given conditions translate into near-optimal scaling
of the number of columns of the block subdictionaries as a function of the
number of observations for a large class of dictionaries, and (iii) it suggests
that the spectral norm and the quadratic-mean block coherence of the dictionary
(rather than the worst-case coherences) fundamentally limit the scaling of
dimensions of the well-conditioned block subdictionaries. Second, this paper
investigates the problems of block-sparse recovery and block-sparse regression
in underdetermined settings. Near-optimal block-sparse recovery and regression
are possible for certain dictionaries as long as the dictionary satisfies
easily computable conditions and the coefficients describing the linear
combination of groups of columns can be modeled through a mild statistical
prior.Comment: 39 pages, 3 figures. A revised and expanded version of the paper
published in IEEE Transactions on Information Theory (DOI:
10.1109/TIT.2015.2429632); this revision includes corrections in the proofs
of some of the result
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