7,607 research outputs found
Proof of Convergence and Performance Analysis for Sparse Recovery via Zero-point Attracting Projection
A recursive algorithm named Zero-point Attracting Projection (ZAP) is
proposed recently for sparse signal reconstruction. Compared with the reference
algorithms, ZAP demonstrates rather good performance in recovery precision and
robustness. However, any theoretical analysis about the mentioned algorithm,
even a proof on its convergence, is not available. In this work, a strict proof
on the convergence of ZAP is provided and the condition of convergence is put
forward. Based on the theoretical analysis, it is further proved that ZAP is
non-biased and can approach the sparse solution to any extent, with the proper
choice of step-size. Furthermore, the case of inaccurate measurements in noisy
scenario is also discussed. It is proved that disturbance power linearly
reduces the recovery precision, which is predictable but not preventable. The
reconstruction deviation of -compressible signal is also provided. Finally,
numerical simulations are performed to verify the theoretical analysis.Comment: 29 pages, 6 figure
The Convergence Guarantees of a Non-convex Approach for Sparse Recovery
In the area of sparse recovery, numerous researches hint that non-convex
penalties might induce better sparsity than convex ones, but up until now those
corresponding non-convex algorithms lack convergence guarantees from the
initial solution to the global optimum. This paper aims to provide performance
guarantees of a non-convex approach for sparse recovery. Specifically, the
concept of weak convexity is incorporated into a class of sparsity-inducing
penalties to characterize the non-convexity. Borrowing the idea of the
projected subgradient method, an algorithm is proposed to solve the non-convex
optimization problem. In addition, a uniform approximate projection is adopted
in the projection step to make this algorithm computationally tractable for
large scale problems. The convergence analysis is provided in the noisy
scenario. It is shown that if the non-convexity of the penalty is below a
threshold (which is in inverse proportion to the distance between the initial
solution and the sparse signal), the recovered solution has recovery error
linear in both the step size and the noise term. Numerical simulations are
implemented to test the performance of the proposed approach and verify the
theoretical analysis.Comment: 33 pages, 7 figure
Signal Recovery in Perturbed Fourier Compressed Sensing
In many applications in compressed sensing, the measurement matrix is a
Fourier matrix, i.e., it measures the Fourier transform of the underlying
signal at some specified `base' frequencies , where is the
number of measurements. However due to system calibration errors, the system
may measure the Fourier transform at frequencies
that are different from the base frequencies and where
are unknown. Ignoring perturbations of this nature can lead to major errors in
signal recovery. In this paper, we present a simple but effective alternating
minimization algorithm to recover the perturbations in the frequencies \emph{in
situ} with the signal, which we assume is sparse or compressible in some known
basis. In many cases, the perturbations can be expressed
in terms of a small number of unique parameters . We demonstrate that
in such cases, the method leads to excellent quality results that are several
times better than baseline algorithms (which are based on existing off-grid
methods in the recent literature on direction of arrival (DOA) estimation,
modified to suit the computational problem in this paper). Our results are also
robust to noise in the measurement values. We also provide theoretical results
for (1) the convergence of our algorithm, and (2) the uniqueness of its
solution under some restrictions.Comment: New theortical results about uniqueness and convergence now included.
More challenging experiments now include
Recent Progress in Image Deblurring
This paper comprehensively reviews the recent development of image
deblurring, including non-blind/blind, spatially invariant/variant deblurring
techniques. Indeed, these techniques share the same objective of inferring a
latent sharp image from one or several corresponding blurry images, while the
blind deblurring techniques are also required to derive an accurate blur
kernel. Considering the critical role of image restoration in modern imaging
systems to provide high-quality images under complex environments such as
motion, undesirable lighting conditions, and imperfect system components, image
deblurring has attracted growing attention in recent years. From the viewpoint
of how to handle the ill-posedness which is a crucial issue in deblurring
tasks, existing methods can be grouped into five categories: Bayesian inference
framework, variational methods, sparse representation-based methods,
homography-based modeling, and region-based methods. In spite of achieving a
certain level of development, image deblurring, especially the blind case, is
limited in its success by complex application conditions which make the blur
kernel hard to obtain and be spatially variant. We provide a holistic
understanding and deep insight into image deblurring in this review. An
analysis of the empirical evidence for representative methods, practical
issues, as well as a discussion of promising future directions are also
presented.Comment: 53 pages, 17 figure
Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)
The implicit objective of the biennial "international - Traveling Workshop on
Interactions between Sparse models and Technology" (iTWIST) is to foster
collaboration between international scientific teams by disseminating ideas
through both specific oral/poster presentations and free discussions. For its
second edition, the iTWIST workshop took place in the medieval and picturesque
town of Namur in Belgium, from Wednesday August 27th till Friday August 29th,
2014. The workshop was conveniently located in "The Arsenal" building within
walking distance of both hotels and town center. iTWIST'14 has gathered about
70 international participants and has featured 9 invited talks, 10 oral
presentations, and 14 posters on the following themes, all related to the
theory, application and generalization of the "sparsity paradigm":
Sparsity-driven data sensing and processing; Union of low dimensional
subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph
sensing/processing; Blind inverse problems and dictionary learning; Sparsity
and computational neuroscience; Information theory, geometry and randomness;
Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?;
Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website:
http://sites.google.com/site/itwist1
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