The Experts below are selected from a list of 3582 Experts worldwide ranked by ideXlab platform
Holger Rauhut - One of the best experts on this subject based on the ideXlab platform.
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Random Sampling of Sparse Trigonometric Polynomials, II. Orthogonal Matching Pursuit versus Basis Pursuit
Foundations of Computational Mathematics, 2008Co-Authors: Stefan Kunis, Holger RauhutAbstract:We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Thresholding. While recovery by Basis Pursuit has recently been studied by several authors, we provide theoretical results on the success probability of reconstruction via Thresholding and OMP for both a continuous and a discrete probability model for the sampling points. We present numerical experiments, which indicate that usually Basis Pursuit is significantly slower than greedy algorithms, while the recovery rates are very similar.
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random sampling of sparse trigonometric polynomials ii orthogonal matching Pursuit versus Basis Pursuit
arXiv: Classical Analysis and ODEs, 2006Co-Authors: Stefan Kunis, Holger RauhutAbstract:We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Thresholding. While recovery by Basis Pursuit has recently been studied by several authors, we provide theoretical results on the success probability of reconstruction via Thresholding and OMP for both a continuous and a discrete probability model for the sampling points. We present numerical experiments, which indicate that usually Basis Pursuit is significantly slower than greedy algorithms, while the recovery rates are very similar.
Stefan Kunis - One of the best experts on this subject based on the ideXlab platform.
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Random Sampling of Sparse Trigonometric Polynomials, II. Orthogonal Matching Pursuit versus Basis Pursuit
Foundations of Computational Mathematics, 2008Co-Authors: Stefan Kunis, Holger RauhutAbstract:We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Thresholding. While recovery by Basis Pursuit has recently been studied by several authors, we provide theoretical results on the success probability of reconstruction via Thresholding and OMP for both a continuous and a discrete probability model for the sampling points. We present numerical experiments, which indicate that usually Basis Pursuit is significantly slower than greedy algorithms, while the recovery rates are very similar.
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random sampling of sparse trigonometric polynomials ii orthogonal matching Pursuit versus Basis Pursuit
arXiv: Classical Analysis and ODEs, 2006Co-Authors: Stefan Kunis, Holger RauhutAbstract:We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Thresholding. While recovery by Basis Pursuit has recently been studied by several authors, we provide theoretical results on the success probability of reconstruction via Thresholding and OMP for both a continuous and a discrete probability model for the sampling points. We present numerical experiments, which indicate that usually Basis Pursuit is significantly slower than greedy algorithms, while the recovery rates are very similar.
Alex Bronstein - One of the best experts on this subject based on the ideXlab platform.
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Sparse null space Basis Pursuit and analysis dictionary learning for high-dimensional data analysis
2015 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2015Co-Authors: Xiao Bian, Hamid Krim, Alex BronsteinAbstract:Sparse models in dictionary learning have been successfully applied in a wide variety of machine learning and computer vision problems, and have also recently been of increasing research interest. Another interesting related problem based on a linear equality constraint, namely the sparse null space problem (SNS), first appeared in 1986, and has since inspired results on sparse Basis Pursuit. In this paper, we investigate the relation between the SNS problem and the analysis dictionary learning problem, and show that the SNS problem plays a central role, and may be utilized to solve dictionary learning problems. Moreover, we propose an efficient algorithm of sparse null space Basis Pursuit, and extend it to a solution of analysis dictionary learning. Experimental results on numerical synthetic data and real-world data are further presented to validate the performance of our method.
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ICASSP - Sparse null space Basis Pursuit and analysis dictionary learning for high-dimensional data analysis
2015 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2015Co-Authors: Xiao Bian, Hamid Krim, Alex BronsteinAbstract:Sparse models in dictionary learning have been successfully applied in a wide variety of machine learning and computer vision problems, and have also recently been of increasing research interest. Another interesting related problem based on a linear equality constraint, namely the sparse null space problem (SNS), first appeared in 1986, and has since inspired results on sparse Basis Pursuit. In this paper, we investigate the relation between the SNS problem and the analysis dictionary learning problem, and show that the SNS problem plays a central role, and may be utilized to solve dictionary learning problems. Moreover, we propose an efficient algorithm of sparse null space Basis Pursuit, and extend it to a solution of analysis dictionary learning. Experimental results on numerical synthetic data and real-world data are further presented to validate the performance of our method.
Eero P. Simoncelli - One of the best experts on this subject based on the ideXlab platform.
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continuous Basis Pursuit and its applications
2012Co-Authors: Eero P. Simoncelli, Daniel Tranchina, Chaitanya EkanadhamAbstract:Transformation-invariance is a major source of nonlinear structure in many real signal ensembles. To model this structure, we develop a methodology for decomposing a signal into a sparse linear combination of continuously transformed features. The central idea is to approximate the manifold(s) of transformed features(s) by linearly combining interpolation functions using constrained coefficients that can be recovered via convex programming. The advantage of this approach over traditional sparse coding methods is threefold: (1) it is built upon a more accurate probabilistic source model for transformation-invariant ensembles, (2) it uses a more efficient dictionary, and (3) both structural and transformational information can be extracted separately from the representation via well-defined mappings, providing transformation-invariant and -equivariant information, respectively. The method can be used with any linear interpolator, and includes Basis Pursuit denoising as a special case corresponding to nearest-neighbor interpolation. We propose a novel polar interpolation method with which our method significantly outperforms Basis Pursuit on a sparse deconvolution task. In addition, our method outperforms the state-of-the-art in identifying neural action potentials from voltage recordings on multiple simulated and real data sets. The advantage of our method is primarily due to its superior handling of near-synchronous action potentials, which overlap in the trace and are not recoverable by standard spike sorting methods. Finally, we develop a hierarchical formulation in which successive layers encode more complex features and their associated transformation parameters. A two-layer time- and frequency-shiftable representation is learned from speech data. The second layer encoding compactly represents sounds in terms of acoustic features such as harmonic stacks, sweeps, and ramps in time-frequency space. Despite its compactness, synthesis reveals that it is a faithful representation of the original sound and yields significant improvement over wavelet thresholding techniques on an acoustic denoising task. These two applications demonstrate the advantage of representations which separate content and transformation, and our proposed methodology provides an effective tool for computing such a representation.
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Sparse decomposition of transformation-invariant signals with continuous Basis Pursuit
2011 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2011Co-Authors: Chaitanya Ekanadham, Daniel Tranchina, Eero P. SimoncelliAbstract:Consider the decomposition of a signal into features that undergo transformations drawn from a continuous family. Current methods discretely sample the transformations and apply sparse recovery methods to the resulting finite dictionary. These methods do not exploit the underlying continuous structure, thereby limiting the ability to produce sparse solutions. Instead, we employ interpolation functions which linearly approximate the manifold of scaled and transformed features. Coefficients are interpreted as interpolation weights, and we formulate a convex optimization problem for obtaining them, enforcing both reconstruction accuracy and sparsity. We compare our method, which we call continuous Basis Pursuit (CBP) with the standard Basis Pursuit approach on a sparse deconvolution task. CBP yields substantially sparser solutions without sacrificing accuracy, and does so with a smaller dictionary. We conclude that for signals generated by transformation-invariant processes, a representation that explicitly accommodates the transformation(s) can yield sparser and more interpretable decompositions.
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ICASSP - Sparse decomposition of transformation-invariant signals with continuous Basis Pursuit
2011 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2011Co-Authors: Chaitanya Ekanadham, Daniel Tranchina, Eero P. SimoncelliAbstract:Consider the decomposition of a signal into features that undergo transformations drawn from a continuous family. Current methods discretely sample the transformations and apply sparse recovery methods to the resulting finite dictionary. These methods do not exploit the underlying continuous structure, thereby limiting the ability to produce sparse solutions. Instead, we employ interpolation functions which linearly approximate the manifold of scaled and transformed features. Coefficients are interpreted as interpolation weights, and we formulate a convex optimization problem for obtaining them, enforcing both reconstruction accuracy and sparsity. We compare our method, which we call continuous Basis Pursuit (CBP) with the standard Basis Pursuit approach on a sparse deconvolution task. CBP yields substantially sparser solutions without sacrificing accuracy, and does so with a smaller dictionary. We conclude that for signals generated by transformation-invariant processes, a representation that explicitly accommodates the transformation(s) can yield sparser and more interpretable decompositions.
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Recovery of Sparse Translation-Invariant Signals With Continuous Basis Pursuit
IEEE Transactions on Signal Processing, 2011Co-Authors: Chaitanya Ekanadham, Daniel Tranchina, Eero P. SimoncelliAbstract:We consider the problem of decomposing a signal into a linear combination of features, each a continuously translated version of one of a small set of elementary features. Although these constituents are drawn from a continuous family, most current signal decomposition methods rely on a finite dictionary of discrete examples selected from this family (e.g., shifted copies of a set of basic waveforms), and apply sparse optimization methods to select and solve for the relevant coefficients. Here, we generate a dictionary that includes auxiliary interpolation functions that approximate translates of features via adjustment of their coefficients. We formulate a constrained convex optimization problem, in which the full set of dictionary coefficients represents a linear approximation of the signal, the auxiliary coefficients are constrained so as to only represent translated features, and sparsity is imposed on the primary coefficients using an L1 penalty. The Basis Pursuit denoising (BP) method may be seen as a special case, in which the auxiliary interpolation functions are omitted, and we thus refer to our methodology as continuous Basis Pursuit (CBP). We develop two implementations of CBP for a one-dimensional translation-invariant source, one using a first-order Taylor approximation, and another using a form of trigonometric spline. We examine the tradeoff between sparsity and signal reconstruction accuracy in these methods, demonstrating empirically that trigonometric CBP substantially outperforms Taylor CBP, which, in turn, offers substantial gains over ordinary BP. In addition, the CBP bases can generally achieve equally good or better approximations with much coarser sampling than BP, leading to a reduction in dictionary dimensionality.
Rui Zhang - One of the best experts on this subject based on the ideXlab platform.
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multi trace Basis Pursuit inversion with spatial regularization
Journal of Geophysics and Engineering, 2013Co-Authors: Rui Zhang, Sanjay SrinivasanAbstract:Basis Pursuit inversion (BPI) was developed to derive spiky reflection coefficients, which was applied on post-stack seismic data trace-by-trace for subsurface reflectivity. Such a trace-by-trace operation has no control on lateral continuity. To address this, we extend the former BPI to a multi-trace procedure with spatial regularization?a spatial derivative operator?to stabilize the lateral variation. The original objective function of BPI is reformed with a ?Z? shape spatial derivative imposed, which can balance the lateral continuity and vertical resolution. Synthetic tests show improved lateral continuity by spatial regularized BPI in the case of noise contamination. Real data applications demonstrate its effectiveness to enhance the lateral continuity of inverted spiky reflectivity.
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a prestack Basis Pursuit seismic inversion
Geophysics, 2013Co-Authors: Rui Zhang, Sanjay SrinivasanAbstract:ABSTRACTResolving thin layers and clearly delineating layer boundaries in inverted seismic sections are very important goals for exploration and production. Many seismic inversion methods based on a least-squares optimization approach with Tikhonov-type regularization can lead to unfocused transitions between adjacent layers. A Basis Pursuit inversion (BPI) algorithm based on the L1 norm optimization method can, however, resolve sharp boundaries between layers. We have formulated a BPI algorithm for amplitude-versus-angle inversion and investigated its potential to improve contrasts between layers. Like the BPI for poststack case, the sparse layer constraint, rather than the sparse spike constraint, is used to construct the model space as a wedge dictionary. All the elements of the dictionary are bed reflectivities, which include solutions consisting of thin beds as well. With this dictionary, we use an L1 norm optimization framework to derive three reflectivities, namely, Rp, Rs, and Rρ. Although BPI doe...
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a pre stack Basis Pursuit seismic inversion
74th European Association of Geoscientists and Engineers Conference and Exhibition 2012 Incorporating SPE EUROPEC 2012: Responsibly Securing Natural R, 2012Co-Authors: Rui Zhang, Sanjay SrinivasanAbstract:Zhang and Castagna (2011) formulated Basis Pursuit inversion for post-stack seismic data. Here we extend it to the pre-stack angle gather domain. The major contribution of this work is the introduction and feasibility of application of Basis Pursuit inversion on pre-stack seismic data. The BPI inversion follows a L1 norm optimization framework. The resulting inversion enforces sharpness of the layer interfaces, thereby achieving better focus of layer boundaries. Unlike other AVA inversion methods which output smooth Vp, Vs and density, our method can generate spiky reflectivities of Rp, Rs and Rρ and blocky velocities and density, which can resolve structural boundaries and lithology information better for interpretation.
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seismic sparse layer reflectivity inversion using Basis Pursuit decomposition
Geophysics, 2011Co-Authors: Rui Zhang, John P CastagnaAbstract:A Basis Pursuit inversion of seismic reflection data for reflection coefficients is introduced as an alternative method of incorporating a priori information in the seismic inversion process. The inversion is accomplished by building a dictionary of functions representing reflectivity patterns and constituting the seismic trace as a superposition of these patterns. Basis Pursuit decomposition finds a sparse number of reflection responses that sum to form the seismic trace. When the dictionary of functions is chosen to be a wedge-model of reflection coefficient pairs convolved with the seismic wavelet, the resulting reflectivity inversion is a sparse-layer inversion, rather than a sparse-spike inversion. Synthetic tests suggest that a sparse-layer inversion using Basis Pursuit can better resolve thin beds than a comparable sparse-spike inversion. Application to field data indicates that sparse-layer inversion results in the potentially improved detectability and resolution of some thin layers and reveals a...
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sparse layer inversion with Basis Pursuit decomposition
Seg Technical Program Expanded Abstracts, 2011Co-Authors: Rui Zhang, John P CastagnaAbstract:Inversion of seismic reflection data for reflection coefficients can be formulated using predefined reflectivity patterns as Basis functions. When Basis Pursuit decomposition with a wavelet dictionary of seismic layer responses is employed, the process is a sparse-layer, rather than a sparse-spike, inversion and consequently has improved resolution of thin layers. This is demonstrated on synthetic and real seismic data.
