2,629 research outputs found
Perturbed Orthogonal Matching Pursuit
Cataloged from PDF version of article.Compressive Sensing theory details how a sparsely
represented signal in a known basis can be reconstructed with
an underdetermined linear measurement model. However, in reality
there is a mismatch between the assumed and the actual
bases due to factors such as discretization of the parameter
space defining basis components, sampling jitter in A/D conversion,
and model errors. Due to this mismatch, a signal may
not be sparse in the assumed basis, which causes significant performance
degradation in sparse reconstruction algorithms. To
eliminate the mismatch problem, this paper presents a novel
perturbed orthogonal matching pursuit (POMP) algorithm that
performs controlled perturbation of selected support vectors to
decrease the orthogonal residual at each iteration. Based on detailed
mathematical analysis, conditions for successful reconstruction
are derived. Simulations show that robust results with much
smaller reconstruction errors in the case of perturbed bases can
be obtained as compared to standard sparse reconstruction techniques
Greed is Fine: on Finding Sparse Zeros of Hilbert Operators
We propose an generalization of the classical Orthogonal Matching Pursuit (OMP) algorithm for finding sparse zeros of Hilbert operator. First we introduce a new condition called the restricted diagonal deviation property which allow us to analysis of the consistency of the estimated support and vector. Secondly when using a perturbed version of the operator, we show that a partial recovery of the support is possible and remain possible even if some of the steps of the algorithm are inexact. Finally we discuss about the links between recent works on other version of OMP
Oracle-order Recovery Performance of Greedy Pursuits with Replacement against General Perturbations
Applying the theory of compressive sensing in practice always takes different
kinds of perturbations into consideration. In this paper, the recovery
performance of greedy pursuits with replacement for sparse recovery is analyzed
when both the measurement vector and the sensing matrix are contaminated with
additive perturbations. Specifically, greedy pursuits with replacement include
three algorithms, compressive sampling matching pursuit (CoSaMP), subspace
pursuit (SP), and iterative hard thresholding (IHT), where the support
estimation is evaluated and updated in each iteration. Based on restricted
isometry property, a unified form of the error bounds of these recovery
algorithms is derived under general perturbations for compressible signals. The
results reveal that the recovery performance is stable against both
perturbations. In addition, these bounds are compared with that of oracle
recovery--- least squares solution with the locations of some largest entries
in magnitude known a priori. The comparison shows that the error bounds of
these algorithms only differ in coefficients from the lower bound of oracle
recovery for some certain signal and perturbations, as reveals that
oracle-order recovery performance of greedy pursuits with replacement is
guaranteed. Numerical simulations are performed to verify the conclusions.Comment: 27 pages, 4 figures, 5 table
Learning Active Basis Models by EM-Type Algorithms
EM algorithm is a convenient tool for maximum likelihood model fitting when
the data are incomplete or when there are latent variables or hidden states. In
this review article we explain that EM algorithm is a natural computational
scheme for learning image templates of object categories where the learning is
not fully supervised. We represent an image template by an active basis model,
which is a linear composition of a selected set of localized, elongated and
oriented wavelet elements that are allowed to slightly perturb their locations
and orientations to account for the deformations of object shapes. The model
can be easily learned when the objects in the training images are of the same
pose, and appear at the same location and scale. This is often called
supervised learning. In the situation where the objects may appear at different
unknown locations, orientations and scales in the training images, we have to
incorporate the unknown locations, orientations and scales as latent variables
into the image generation process, and learn the template by EM-type
algorithms. The E-step imputes the unknown locations, orientations and scales
based on the currently learned template. This step can be considered
self-supervision, which involves using the current template to recognize the
objects in the training images. The M-step then relearns the template based on
the imputed locations, orientations and scales, and this is essentially the
same as supervised learning. So the EM learning process iterates between
recognition and supervised learning. We illustrate this scheme by several
experiments.Comment: Published in at http://dx.doi.org/10.1214/09-STS281 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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