79,610 research outputs found
Coherence-based Partial Exact Recovery Condition for OMP/OLS
We address the exact recovery of the support of a k-sparse vector with
Orthogonal Matching Pursuit (OMP) and Orthogonal Least Squares (OLS) in a
noiseless setting. We consider the scenario where OMP/OLS have selected good
atoms during the first l iterations (l<k) and derive a new sufficient and
worst-case necessary condition for their success in k steps. Our result is
based on the coherence \mu of the dictionary and relaxes Tropp's well-known
condition \mu<1/(2k-1) to the case where OMP/OLS have a partial knowledge of
the support
A Compressed Sampling and Dictionary Learning Framework for WDM-Based Distributed Fiber Sensing
We propose a compressed sampling and dictionary learning framework for
fiber-optic sensing using wavelength-tunable lasers. A redundant dictionary is
generated from a model for the reflected sensor signal. Imperfect prior
knowledge is considered in terms of uncertain local and global parameters. To
estimate a sparse representation and the dictionary parameters, we present an
alternating minimization algorithm that is equipped with a pre-processing
routine to handle dictionary coherence. The support of the obtained sparse
signal indicates the reflection delays, which can be used to measure
impairments along the sensing fiber. The performance is evaluated by
simulations and experimental data for a fiber sensor system with common core
architecture.Comment: Accepted for publication in Journal of the Optical Society of America
A [ \copyright\ 2017 Optical Society of America.]. One print or electronic
copy may be made for personal use only. Systematic reproduction and
distribution, duplication of any material in this paper for a fee or for
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Exact Recovery Conditions for Sparse Representations with Partial Support Information
We address the exact recovery of a k-sparse vector in the noiseless setting
when some partial information on the support is available. This partial
information takes the form of either a subset of the true support or an
approximate subset including wrong atoms as well. We derive a new sufficient
and worst-case necessary (in some sense) condition for the success of some
procedures based on lp-relaxation, Orthogonal Matching Pursuit (OMP) and
Orthogonal Least Squares (OLS). Our result is based on the coherence "mu" of
the dictionary and relaxes the well-known condition mu<1/(2k-1) ensuring the
recovery of any k-sparse vector in the non-informed setup. It reads
mu<1/(2k-g+b-1) when the informed support is composed of g good atoms and b
wrong atoms. We emphasize that our condition is complementary to some
restricted-isometry based conditions by showing that none of them implies the
other.
Because this mutual coherence condition is common to all procedures, we carry
out a finer analysis based on the Null Space Property (NSP) and the Exact
Recovery Condition (ERC). Connections are established regarding the
characterization of lp-relaxation procedures and OMP in the informed setup.
First, we emphasize that the truncated NSP enjoys an ordering property when p
is decreased. Second, the partial ERC for OMP (ERC-OMP) implies in turn the
truncated NSP for the informed l1 problem, and the truncated NSP for p<1.Comment: arXiv admin note: substantial text overlap with arXiv:1211.728
Approximation errors of online sparsification criteria
Many machine learning frameworks, such as resource-allocating networks,
kernel-based methods, Gaussian processes, and radial-basis-function networks,
require a sparsification scheme in order to address the online learning
paradigm. For this purpose, several online sparsification criteria have been
proposed to restrict the model definition on a subset of samples. The most
known criterion is the (linear) approximation criterion, which discards any
sample that can be well represented by the already contributing samples, an
operation with excessive computational complexity. Several computationally
efficient sparsification criteria have been introduced in the literature, such
as the distance, the coherence and the Babel criteria. In this paper, we
provide a framework that connects these sparsification criteria to the issue of
approximating samples, by deriving theoretical bounds on the approximation
errors. Moreover, we investigate the error of approximating any feature, by
proposing upper-bounds on the approximation error for each of the
aforementioned sparsification criteria. Two classes of features are described
in detail, the empirical mean and the principal axes in the kernel principal
component analysis.Comment: 10 page
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