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About 7,124 papers for “Sparse approximation”

Quaternionic Sparse Approximation

Quentin Barthélemy, Anthony Larue, Jérôme Mars · HAL (Le Centre pour la Communication Scientifique Directe) · 2012

Cited by 3Open access

Quaternionic sparse approximation

Anthony Larue, Jérôme I. Mars · 2012

Abstract In this paper, we introduce a new processing procedure for quater-nionic signals through consideration of the well-known orthogonal matching pur-suit (OMP), which provides sparse approximation. We present a quaternionic ex-tension, the quaternionic OMP, that can be used to process a right-multiplication linear combination of quaternionic signals. As validation, this quaternionic OMP is applied to simulated data. Deconvolution is carried out and presented here with a new spikegram that is designed for visualization of quaternionic coefficients, and finally this is compared to multivariate OMP. 1

Cited by 0

Distributed Sparse Cut Approximation

Fabian Kühn, Anisur Rahaman Molla · DROPS (Schloss Dagstuhl – Leibniz Center for Informatics) · 2016

We study the problem of computing a sparse cut in an undirected network graph G=(V,E). We measure the sparsity of a cut (S,V\S) by its conductance phi(S), i.e., by the ratio of the number of edges crossing the cut and the sum of the degrees on the smaller of the two sides. We present an efficient distributed algorithm to compute a cut of low conductance. Specifically, given two parameters b and phi, if there exists a cut of balance at least b and conductance at most phi, our algorithm outputs a cut of balance at least b/2 and conductance at most ~O(sqrt{phi}), where ~O(.) hides polylogarithmic factors in the number of nodes n. Our distributed algorithm works in the \congest model, i.e., it only requires to send messages of size at most O(log(n)) bits. The time complexity of the algorithm is ~O(D + 1/b*phi), where D is the diameter of G. This is a significant improvement over a result by Das Sarma et al. [ICDCN 2015], where it is shown that a cut of the same quality can be computed in time ~O(n + 1/b*phi). The improved running time is in particular achieved by devising and applying an efficient distributed algorithm for the all-prefix-sums problem in a distributed search tree. This algorithm, which is based on the classic parallel all-prefix-sums algorithm, might be of independent interest.

Cited by 8Open access

Learning Sparse Dictionaries for Sparse Signal Approximation

Ron Rubinstein, Michael Zibulevsky, Michael Elad · 2009

An e‐cient and ∞exible dictionary structure is proposed for sparse and redundant signal representation. The structure is based on a sparsity model of the dictionary atoms over a base dictionary. The sparse dictionary provides e‐cient forward and adjoint operators, has a compact representation, and can be efiectively trained from given example data. In this, the sparse structure bridges the gap between implicit dictionaries, which have e‐cient implementations yet lack adaptability, and explicit dictionaries, which are fully adaptable but non-e‐cient and costly to deploy. In this report we discuss the advantages of sparse dictionaries, and present an e‐cient algorithm for training them. We demonstrate the advantages of the proposed structure for 3-D image denoising.

Cited by 14

Sparse approximation of complex networks

Omar De la Cruz Cabrera, Jiafeng Jin, Lothar Reichel · Applied Numerical Mathematics · 2024

This paper considers the problem of recovering a sparse approximation A ∈ R n × n of an unknown (exact) adjacency matrix A true for a network from a corrupted version of a communicability matrix C = exp ⁡ ( A true ) + N ∈ R n × n , where N denotes an unknown “noise matrix”. We consider two methods for determining an approximation A of A true : ( i ) a Newton method with soft-thresholding and line search, and ( ii ) a proximal gradient method with line search. These methods are applied to compute the solution of the minimization problem arg ⁡ min A ∈ R n × n { ‖ exp ⁡ ( A ) − C ‖ F 2 + μ ‖ vec ( A ) ‖ 1 } , where μ > 0 is a regularization parameter that controls the amount of shrinkage. We discuss the effect of μ on the computed solution, conditions for convergence, and the rate of convergence of the methods. Computed examples illustrate their performance when applied to directed and undirected networks.

Cited by 2Open access

Statistically Driven Sparse Image Approximation

Rosa Ventura, Eero P. Simoncelli · Proceedings - International Conference on Image Processing · 2007

Finding the sparsest approximation of an image as a sum of basis functions drawn from a redundant dictionary is an NP-hard problem. In the case of a dictionary whose elements form an overcomplete basis, a recently developed method, based on alternating thresholding and projection operations, provides an appealing approximate solution. When applied to images, this method produces sparser results and requires less computation than current alternative methods. Motivated by recent developments in statistical image modeling, we develop an enhancement of this method based on a locally adaptive threshold operation, and demonstrate that the enhanced algorithm is capable of finding sparser approximations with a decrease in computational complexity.

Cited by 7

Sparse Tensor Spherical Harmonics Approximation

K. Grella, C. Schwab, K. Grella et al. · 2011

The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distributed by the authors.

Cited by 0

Sparse Approximation and Its Applications

DigiNole (Florida State University) · 2011

In this thesis, we tackle the fundamental problem of how to effectively and reliably calculate sparse solutions to underdetermined systems of equations. This class of problems is found in applied mathematics, electrical engineering, statistics, geophysics, just to name a few. This dissertation concentrates on developing efficient and robust solution algorithms, and applies them in several applications in the field of signal/image processing. The first contribution concerns the development of a new Iterative Shrinkage algorithm based on Surrogate Function, ISSF-K, for finding the best K-term approximation to an image. In this problem, we seek to represent an image with K elements from an overcomplete dictionary. We present a proof that this algorithm converges to a local minimum of the NP hard sparsity constrained optimization problem. In addition, we choose curvelets as the dictionary. The approximation obtained by our approach achieves higher PSNR than that of the best K-term wavelet (Cohen-Daubechies-Fauraue 9-7) approximation. We extends ISSF to the application of Morphological Component Analysis, which leads to the second contribution, a new algorithm MCA-ISSF with an adaptive thresholding strategy. The adaptive MCA-ISSF algorithm approximates the problem from the synthesis approach, and it is the only algorithm that incorporate an adaptive strategy to update its algorithmic parameter. Compared to the existent MCA algorithms, our method is more efficient and is parameter free in the thresdholding update. The third contribution concerns the non-convex optimization problems in Compressive Sensing (CS), which is an important extension of sparse approximation. We propose two new iterative reweighted algorithms based on Alternating Direction Method of Multiplier, IR1-ADM and IR2-ADM, to solve the ell-p,0.

Cited by 0

Sparse approximation of singularity functions

P.-A. Nitsche · Repository for Publications and Research Data (ETH Zurich) · 2002

We are concerned with the sparse approximation of functions on the d-dimensional unit cube [0,1]d, which contain powers of distance functions to lower-dimensional k-faces (corners, edges, etc.). These functions arise, e.g., from corners, edges, etc., of domains in solutions to elliptic PDEs. Usually, they deteriorate the rate of convergence of numerical algorithms to approximate these solutions. We show that functions of this type can be approximated with respect to the H1 norm by sparse grid wavelet spaces VL, (VL) = NL, of biorthogonal spline wavelets of degree p essentially at the rate p: \[ \|u - P_Lu\|_{H^1([0,1]^d)} \leq CN_L^{-p}\,(\log_2 N_L)^s \|u\|, \qquad s = s(p,d), \] where || · || is a weighted Sobolev norm and PLu \in VL.

Cited by 4Open access

Optimization Methods for Sparse Approximation

Yong Zhang · Summit (Simon Fraser University) · 2014

In the last two decades, there are numerous applications in which sparse solutions are concerned. Mathematically, all these applications can be formulated into the L0 minimization problems. In this thesis, we first propose a novel augmented Lagrangian (AL) method for solving the L1-norm relaxation problems of the original L0 minimization problems and apply it to our proposed formulation of sparse principal component analysis (PCA). We next propose penalty decomposition (PD) methods for solving the original L0 minimization problems in which a sequence of penalty subproblems are solved by a block coordinate descent (BCD) method. For the AL method, we show that under some regularity assumptions, it converges to a stationary point. Additionally, we propose two nonmonotone gradient methods for solving the AL subproblems, and establish their global and local convergence. Moreover, we apply the AL method to our proposed formulation of sparse PCA and compare our approach with several existing methods on synthetic, Pitprops, and gene expression data, respectively. The computational results demonstrate that the sparse {it principal components} (PCs) produced by our approach substantially outperform those by other methods in terms of total explained variance, correlation of PCs, and orthogonality of loading vectors. For the PD methods, under some suitable assumptions, we establish some convergence results for both inner (the BCD method) and outer (the PD method) iterations, respectively. We test the performance of our PD methods by applying them to sparse logistic regression, sparse inverse covariance selection, and compressed sensing problems. The computational results demonstrate that when solutions of same cardinality are sought, our approach applied to the L0-based models generally has better solution quality and/or speed than the existing approaches that are applied to the corresponding L1-based models. Finally, we adapt the PD method to solve our proposed wavelet frame based image restoration problem. Some convergence analysis of the adapted PD method for this problem are provided. Numerical results show that the proposed model solved by the PD method can generate images with better quality than those obtained by either analysis based approach or balanced approach in terms of restoring sharp features as well as maintaining smoothness of the recovered images.

Cited by 1Open access

Sparse Interpolation and Rational Approximation

Annie Cuyt, Wen-shin Lee · Contemporary mathematics - American Mathematical Society · 2016

Sparse interpolation or exponential analysis, is widely used and in quite different applications and areas of science and engineering. Therefore researchers are often not aware of similar studies going on in another field. The current text is written as a concise tutorial, from an approximation theorist point of view. In Section 2 we summarize the mathematics involved in exponential analysis: structured matrices, generalized eigenvalue problems, singular value decomposition. The section is written with the numerical computation of the sparse interpolant in mind. In Section 3 we outline several connections of sparse interpolation with other mostly non-numeric subjects: computer algebra, number theory, linear recurrences. Some problems are only solved using exact arithmetic. In Section 4 we connect sparse interpolation to rational approximation theory. One of the major hurdles in sparse interpolation is still the correct detection of the number of components in the model. Here we show how to reliably obtain the number of terms in a numeric and noisy environment. The new insight allows to improve on existing state-of-the-art algorithms.

Cited by 9

Sparse approximation via iterative thresholding

Anna C Gilbert, See Profile, Kyle K. Herrity et al. · 2006

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Cited by 0

Sparse approximation of individual functions

Leyla Sharipovna Burusheva, Vladimir Temlyakov · arXiv (Cornell University) · 2019

Results on two different settings of asymptotic behavior of approximation characteristics of individual functions are presented. First, we discuss the following classical question for sparse approximation. Is it true that for any individual function from a given function class its sequence of errors of best sparse approximations with respect to a given dictionary decays faster than the corresponding supremum over the function class? Second, we discuss sparse approximation by greedy type algorithms. We show that for any individual function from a given class we can improve the upper bound on the rate of convergence of the error of approximation by a greedy algorithm if we use some information from the previous iterations of the algorithm. We call bounds of this type a posteriori bounds.

Cited by 0Open access