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COMPUTER SCIENCE (1157 journals)                  1 2 3 4 5 6 | Last

Showing 1 - 200 of 872 Journals sorted alphabetically
3D Printing and Additive Manufacturing     Full-text available via subscription   (Followers: 13)
Abakós     Open Access   (Followers: 3)
Academy of Information and Management Sciences Journal     Full-text available via subscription   (Followers: 69)
ACM Computing Surveys     Hybrid Journal   (Followers: 22)
ACM Journal on Computing and Cultural Heritage     Hybrid Journal   (Followers: 9)
ACM Journal on Emerging Technologies in Computing Systems     Hybrid Journal   (Followers: 13)
ACM Transactions on Accessible Computing (TACCESS)     Hybrid Journal   (Followers: 3)
ACM Transactions on Algorithms (TALG)     Hybrid Journal   (Followers: 16)
ACM Transactions on Applied Perception (TAP)     Hybrid Journal   (Followers: 6)
ACM Transactions on Architecture and Code Optimization (TACO)     Hybrid Journal   (Followers: 9)
ACM Transactions on Autonomous and Adaptive Systems (TAAS)     Hybrid Journal   (Followers: 7)
ACM Transactions on Computation Theory (TOCT)     Hybrid Journal   (Followers: 11)
ACM Transactions on Computational Logic (TOCL)     Hybrid Journal   (Followers: 4)
ACM Transactions on Computer Systems (TOCS)     Hybrid Journal   (Followers: 18)
ACM Transactions on Computer-Human Interaction     Hybrid Journal   (Followers: 13)
ACM Transactions on Computing Education (TOCE)     Hybrid Journal   (Followers: 3)
ACM Transactions on Design Automation of Electronic Systems (TODAES)     Hybrid Journal   (Followers: 1)
ACM Transactions on Economics and Computation     Hybrid Journal  
ACM Transactions on Embedded Computing Systems (TECS)     Hybrid Journal   (Followers: 4)
ACM Transactions on Information Systems (TOIS)     Hybrid Journal   (Followers: 20)
ACM Transactions on Intelligent Systems and Technology (TIST)     Hybrid Journal   (Followers: 8)
ACM Transactions on Interactive Intelligent Systems (TiiS)     Hybrid Journal   (Followers: 3)
ACM Transactions on Multimedia Computing, Communications, and Applications (TOMCCAP)     Hybrid Journal   (Followers: 10)
ACM Transactions on Reconfigurable Technology and Systems (TRETS)     Hybrid Journal   (Followers: 7)
ACM Transactions on Sensor Networks (TOSN)     Hybrid Journal   (Followers: 8)
ACM Transactions on Speech and Language Processing (TSLP)     Hybrid Journal   (Followers: 11)
ACM Transactions on Storage     Hybrid Journal  
ACS Applied Materials & Interfaces     Full-text available via subscription   (Followers: 21)
Acta Automatica Sinica     Full-text available via subscription   (Followers: 3)
Acta Universitatis Cibiniensis. Technical Series     Open Access  
Ad Hoc Networks     Hybrid Journal   (Followers: 11)
Adaptive Behavior     Hybrid Journal   (Followers: 11)
Advanced Engineering Materials     Hybrid Journal   (Followers: 26)
Advanced Science Letters     Full-text available via subscription   (Followers: 7)
Advances in Adaptive Data Analysis     Hybrid Journal   (Followers: 8)
Advances in Artificial Intelligence     Open Access   (Followers: 16)
Advances in Calculus of Variations     Hybrid Journal   (Followers: 2)
Advances in Catalysis     Full-text available via subscription   (Followers: 5)
Advances in Computational Mathematics     Hybrid Journal   (Followers: 15)
Advances in Computer Science : an International Journal     Open Access   (Followers: 13)
Advances in Computing     Open Access   (Followers: 2)
Advances in Data Analysis and Classification     Hybrid Journal   (Followers: 53)
Advances in Engineering Software     Hybrid Journal   (Followers: 25)
Advances in Geosciences (ADGEO)     Open Access   (Followers: 10)
Advances in Human-Computer Interaction     Open Access   (Followers: 20)
Advances in Materials Sciences     Open Access   (Followers: 16)
Advances in Operations Research     Open Access   (Followers: 11)
Advances in Parallel Computing     Full-text available via subscription   (Followers: 7)
Advances in Porous Media     Full-text available via subscription   (Followers: 4)
Advances in Remote Sensing     Open Access   (Followers: 37)
Advances in Science and Research (ASR)     Open Access   (Followers: 6)
Advances in Technology Innovation     Open Access   (Followers: 1)
AEU - International Journal of Electronics and Communications     Hybrid Journal   (Followers: 8)
African Journal of Information and Communication     Open Access   (Followers: 6)
African Journal of Mathematics and Computer Science Research     Open Access   (Followers: 4)
Air, Soil & Water Research     Open Access   (Followers: 7)
AIS Transactions on Human-Computer Interaction     Open Access   (Followers: 6)
Algebras and Representation Theory     Hybrid Journal   (Followers: 1)
Algorithms     Open Access   (Followers: 11)
American Journal of Computational and Applied Mathematics     Open Access   (Followers: 4)
American Journal of Computational Mathematics     Open Access   (Followers: 4)
American Journal of Information Systems     Open Access   (Followers: 7)
American Journal of Sensor Technology     Open Access   (Followers: 4)
Anais da Academia Brasileira de Ciências     Open Access   (Followers: 2)
Analog Integrated Circuits and Signal Processing     Hybrid Journal   (Followers: 7)
Analysis in Theory and Applications     Hybrid Journal   (Followers: 1)
Animation Practice, Process & Production     Hybrid Journal   (Followers: 5)
Annals of Combinatorics     Hybrid Journal   (Followers: 3)
Annals of Data Science     Hybrid Journal   (Followers: 9)
Annals of Mathematics and Artificial Intelligence     Hybrid Journal   (Followers: 6)
Annals of Pure and Applied Logic     Open Access   (Followers: 2)
Annals of Software Engineering     Hybrid Journal   (Followers: 12)
Annual Reviews in Control     Hybrid Journal   (Followers: 6)
Anuario Americanista Europeo     Open Access  
Applicable Algebra in Engineering, Communication and Computing     Hybrid Journal   (Followers: 2)
Applied and Computational Harmonic Analysis     Full-text available via subscription   (Followers: 2)
Applied Artificial Intelligence: An International Journal     Hybrid Journal   (Followers: 14)
Applied Categorical Structures     Hybrid Journal   (Followers: 2)
Applied Clinical Informatics     Hybrid Journal   (Followers: 2)
Applied Computational Intelligence and Soft Computing     Open Access   (Followers: 12)
Applied Computer Systems     Open Access   (Followers: 1)
Applied Informatics     Open Access  
Applied Mathematics and Computation     Hybrid Journal   (Followers: 32)
Applied Medical Informatics     Open Access   (Followers: 10)
Applied Numerical Mathematics     Hybrid Journal   (Followers: 5)
Applied Soft Computing     Hybrid Journal   (Followers: 16)
Applied Spatial Analysis and Policy     Hybrid Journal   (Followers: 4)
Architectural Theory Review     Hybrid Journal   (Followers: 3)
Archive of Applied Mechanics     Hybrid Journal   (Followers: 5)
Archive of Numerical Software     Open Access  
Archives and Museum Informatics     Hybrid Journal   (Followers: 125)
Archives of Computational Methods in Engineering     Hybrid Journal   (Followers: 4)
Artifact     Hybrid Journal   (Followers: 2)
Artificial Life     Hybrid Journal   (Followers: 6)
Asia Pacific Journal on Computational Engineering     Open Access  
Asia-Pacific Journal of Information Technology and Multimedia     Open Access   (Followers: 1)
Asian Journal of Computer Science and Information Technology     Open Access  
Asian Journal of Control     Hybrid Journal  
Assembly Automation     Hybrid Journal   (Followers: 2)
at - Automatisierungstechnik     Hybrid Journal   (Followers: 1)
Australian Educational Computing     Open Access  
Automatic Control and Computer Sciences     Hybrid Journal   (Followers: 3)
Automatic Documentation and Mathematical Linguistics     Hybrid Journal   (Followers: 5)
Automatica     Hybrid Journal   (Followers: 9)
Automation in Construction     Hybrid Journal   (Followers: 6)
Autonomous Mental Development, IEEE Transactions on     Hybrid Journal   (Followers: 8)
Basin Research     Hybrid Journal   (Followers: 5)
Behaviour & Information Technology     Hybrid Journal   (Followers: 52)
Bioinformatics     Hybrid Journal   (Followers: 308)
Biomedical Engineering     Hybrid Journal   (Followers: 16)
Biomedical Engineering and Computational Biology     Open Access   (Followers: 13)
Biomedical Engineering, IEEE Reviews in     Full-text available via subscription   (Followers: 17)
Biomedical Engineering, IEEE Transactions on     Hybrid Journal   (Followers: 31)
Briefings in Bioinformatics     Hybrid Journal   (Followers: 46)
British Journal of Educational Technology     Hybrid Journal   (Followers: 123)
Broadcasting, IEEE Transactions on     Hybrid Journal   (Followers: 10)
c't Magazin fuer Computertechnik     Full-text available via subscription   (Followers: 2)
CALCOLO     Hybrid Journal  
Calphad     Hybrid Journal  
Canadian Journal of Electrical and Computer Engineering     Full-text available via subscription   (Followers: 14)
Catalysis in Industry     Hybrid Journal   (Followers: 1)
CEAS Space Journal     Hybrid Journal  
Cell Communication and Signaling     Open Access   (Followers: 1)
Central European Journal of Computer Science     Hybrid Journal   (Followers: 5)
CERN IdeaSquare Journal of Experimental Innovation     Open Access  
Chaos, Solitons & Fractals     Hybrid Journal   (Followers: 3)
Chemometrics and Intelligent Laboratory Systems     Hybrid Journal   (Followers: 15)
ChemSusChem     Hybrid Journal   (Followers: 7)
China Communications     Full-text available via subscription   (Followers: 7)
Chinese Journal of Catalysis     Full-text available via subscription   (Followers: 2)
CIN Computers Informatics Nursing     Full-text available via subscription   (Followers: 12)
Circuits and Systems     Open Access   (Followers: 16)
Clean Air Journal     Full-text available via subscription   (Followers: 2)
CLEI Electronic Journal     Open Access  
Clin-Alert     Hybrid Journal   (Followers: 1)
Cluster Computing     Hybrid Journal   (Followers: 1)
Cognitive Computation     Hybrid Journal   (Followers: 4)
COMBINATORICA     Hybrid Journal  
Combustion Theory and Modelling     Hybrid Journal   (Followers: 13)
Communication Methods and Measures     Hybrid Journal   (Followers: 11)
Communication Theory     Hybrid Journal   (Followers: 20)
Communications Engineer     Hybrid Journal   (Followers: 1)
Communications in Algebra     Hybrid Journal   (Followers: 3)
Communications in Partial Differential Equations     Hybrid Journal   (Followers: 3)
Communications of the ACM     Full-text available via subscription   (Followers: 53)
Communications of the Association for Information Systems     Open Access   (Followers: 18)
COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering     Hybrid Journal   (Followers: 3)
Complex & Intelligent Systems     Open Access  
Complex Adaptive Systems Modeling     Open Access  
Complex Analysis and Operator Theory     Hybrid Journal   (Followers: 2)
Complexity     Hybrid Journal   (Followers: 6)
Complexus     Full-text available via subscription  
Composite Materials Series     Full-text available via subscription   (Followers: 9)
Computación y Sistemas     Open Access  
Computation     Open Access  
Computational and Applied Mathematics     Hybrid Journal   (Followers: 2)
Computational and Mathematical Methods in Medicine     Open Access   (Followers: 2)
Computational and Mathematical Organization Theory     Hybrid Journal   (Followers: 2)
Computational and Structural Biotechnology Journal     Open Access   (Followers: 2)
Computational and Theoretical Chemistry     Hybrid Journal   (Followers: 9)
Computational Astrophysics and Cosmology     Open Access   (Followers: 1)
Computational Biology and Chemistry     Hybrid Journal   (Followers: 12)
Computational Chemistry     Open Access   (Followers: 2)
Computational Cognitive Science     Open Access   (Followers: 2)
Computational Complexity     Hybrid Journal   (Followers: 4)
Computational Condensed Matter     Open Access  
Computational Ecology and Software     Open Access   (Followers: 9)
Computational Economics     Hybrid Journal   (Followers: 9)
Computational Geosciences     Hybrid Journal   (Followers: 14)
Computational Linguistics     Open Access   (Followers: 23)
Computational Management Science     Hybrid Journal  
Computational Mathematics and Modeling     Hybrid Journal   (Followers: 8)
Computational Mechanics     Hybrid Journal   (Followers: 4)
Computational Methods and Function Theory     Hybrid Journal  
Computational Molecular Bioscience     Open Access   (Followers: 2)
Computational Optimization and Applications     Hybrid Journal   (Followers: 7)
Computational Particle Mechanics     Hybrid Journal   (Followers: 1)
Computational Research     Open Access   (Followers: 1)
Computational Science and Discovery     Full-text available via subscription   (Followers: 2)
Computational Science and Techniques     Open Access  
Computational Statistics     Hybrid Journal   (Followers: 13)
Computational Statistics & Data Analysis     Hybrid Journal   (Followers: 31)
Computer     Full-text available via subscription   (Followers: 84)
Computer Aided Surgery     Hybrid Journal   (Followers: 3)
Computer Applications in Engineering Education     Hybrid Journal   (Followers: 6)
Computer Communications     Hybrid Journal   (Followers: 10)
Computer Engineering and Applications Journal     Open Access   (Followers: 5)
Computer Journal     Hybrid Journal   (Followers: 7)
Computer Methods in Applied Mechanics and Engineering     Hybrid Journal   (Followers: 22)
Computer Methods in Biomechanics and Biomedical Engineering     Hybrid Journal   (Followers: 10)
Computer Methods in the Geosciences     Full-text available via subscription   (Followers: 1)
Computer Music Journal     Hybrid Journal   (Followers: 16)
Computer Physics Communications     Hybrid Journal   (Followers: 6)
Computer Science - Research and Development     Hybrid Journal   (Followers: 7)
Computer Science and Engineering     Open Access   (Followers: 17)
Computer Science and Information Technology     Open Access   (Followers: 11)
Computer Science Education     Hybrid Journal   (Followers: 12)
Computer Science Journal     Open Access   (Followers: 20)
Computer Science Master Research     Open Access   (Followers: 10)
Computer Science Review     Hybrid Journal   (Followers: 10)

        1 2 3 4 5 6 | Last

Journal Cover Applied and Computational Harmonic Analysis
  [SJR: 1.589]   [H-I: 65]   [2 followers]  Follow
   Full-text available via subscription Subscription journal
   ISSN (Print) 1063-5203 - ISSN (Online) 1096-603X
   Published by Elsevier Homepage  [3044 journals]
  • Approximation scheme for essentially bandlimited and space-concentrated
           functions on a disk
    • Authors: Boris Landa; Yoel Shkolnisky
      Pages: 381 - 403
      Abstract: Publication date: November 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 3
      Author(s): Boris Landa, Yoel Shkolnisky
      We introduce an approximation scheme for almost bandlimited functions which are sufficiently concentrated in a disk, based on their equally spaced samples on a Cartesian grid. The scheme is based on expanding the function into a series of two-dimensional prolate spheroidal wavefunctions, and estimating the expansion coefficients using the available samples. We prove that the approximate expansion coefficients have particularly simple formulas, in the form of a dot product of the available samples with samples of the basis functions. We also derive error bounds for the error incurred by approximating the expansion coefficients as well as by truncating the expansion. In particular, we derive a bound on the approximation error in terms of the assumed space/frequency concentration, and provide a simple truncation rule to control the length of the expansion and the resulting approximation error.

      PubDate: 2017-09-02T16:12:43Z
      DOI: 10.1016/j.acha.2016.01.006
  • Finite-length and asymptotic analysis of averaged correlogram for
           undersampled data
    • Authors: Mahdi Shaghaghi; Sergiy A. Vorobyov
      Pages: 404 - 423
      Abstract: Publication date: November 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 3
      Author(s): Mahdi Shaghaghi, Sergiy A. Vorobyov
      This paper gives the finite-length analysis of a spectrum estimation method for the case that the samples are obtained at a rate lower than the Nyquist rate. The method is referred to as the averaged correlogram for undersampled data. It is based on partitioning the spectrum into a number of segments and estimating the average power within each spectral segment. This method is able to estimate the power spectrum density of a signal from undersampled data without essentially requiring the signal to be sparse. We derive the bias and the variance of the spectrum estimator, and show that there is a tradeoff between the accuracy of the estimation, the frequency resolution, and the complexity of the estimator. A closed-form approximation of the estimation variance is derived, which clearly shows how the variance is related to different parameters. The asymptotic behavior of the estimator is also investigated, and it is proved that in the case of a white Gaussian process, this spectrum estimator is consistent. Moreover, the estimation made for different spectral segments becomes uncorrelated as the signal length tends to infinity. Finally, numerical examples and simulation results are provided, which approve the theoretical conclusions.

      PubDate: 2017-09-02T16:12:43Z
      DOI: 10.1016/j.acha.2016.02.001
  • Computation of 2D Fourier transforms and diffraction integrals using
           Gaussian radial basis functions
    • Authors: A. Martínez-Finkelshtein; D. Ramos-López; D.R. Iskander
      Pages: 424 - 448
      Abstract: Publication date: November 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 3
      Author(s): A. Martínez-Finkelshtein, D. Ramos-López, D.R. Iskander
      We implement an efficient method of computation of two dimensional Fourier-type integrals based on approximation of the integrand by Gaussian radial basis functions, which constitute a standard tool in approximation theory. As a result, we obtain a rapidly converging series expansion for the integrals, allowing for their accurate calculation. We apply this idea to the evaluation of diffraction integrals, used for the computation of the through-focus characteristics of an optical system. We implement this method and compare its performance in terms of complexity, accuracy and execution time with several alternative approaches, especially with the extended Nijboer–Zernike theory, which is also outlined in the text for the reader's convenience. The proposed method yields a reliable and fast scheme for simultaneous evaluation of such kind of integrals for several values of the defocus parameter, as required in the characterization of the through-focus optics.

      PubDate: 2017-09-02T16:12:43Z
      DOI: 10.1016/j.acha.2016.01.007
  • Simplified vanishing moment criteria for wavelets over general dilation
           groups, with applications to abelian and shearlet dilation groups
    • Authors: Hartmut Führ; Reihaneh Raisi Tousi
      Pages: 449 - 481
      Abstract: Publication date: November 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 3
      Author(s): Hartmut Führ, Reihaneh Raisi Tousi
      We consider the coorbit theory associated to a square-integrable, irreducible quasi-regular representation of a semidirect product group G = R d ⋊ H . The existence of coorbit spaces for this very general setting has been recently established, together with concrete vanishing moment criteria for analyzing vectors and atoms that can be used in the coorbit scheme. These criteria depend on fairly technical assumptions on the dual action of the dilation group, and it is one of the chief purposes of this paper to considerably simplify these assumptions. We then proceed to verify the assumptions for large classes of dilation groups, in particular for all abelian dilation groups in arbitrary dimensions, as well as a class called generalized shearlet dilation groups, containing and extending all known examples of shearlet dilation groups employed in dimensions two and higher. We explain how these groups can be systematically constructed from certain commutative associative algebras of the same dimension, and give a full list, up to conjugacy, of shearing groups in dimensions three and four. In the latter case, three previously unknown groups are found. As a result, the existence of Banach frames consisting of compactly supported wavelets, with simultaneous convergence in a whole range of coorbit spaces, is established for all groups involved.

      PubDate: 2017-09-02T16:12:43Z
      DOI: 10.1016/j.acha.2016.03.003
  • Algorithms and error bounds for noisy phase retrieval with low-redundancy
    • Authors: Bernhard G. Bodmann; Nathaniel Hammen
      Pages: 482 - 503
      Abstract: Publication date: November 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 3
      Author(s): Bernhard G. Bodmann, Nathaniel Hammen
      The main objective of this paper is to find algorithms accompanied by explicit error bounds for phase retrieval from noisy magnitudes of frame coefficients when the underlying frame has a low redundancy. We achieve these goals with frames consisting of N = 6 d − 3 vectors spanning a d-dimensional complex Hilbert space. The two algorithms we use, phase propagation or the kernel method, are polynomial time in the dimension d. To ensure a successful approximate recovery, we assume that the noise is sufficiently small compared to the squared norm of the vector to be recovered. In this regime, we derive an explicit error bound that is inverse proportional to the signal-to-noise ratio, with a constant of proportionality that depends only on the dimension d. Properties of the reproducing kernel space of complex polynomials and of trigonometric polynomials are central in our error estimates.

      PubDate: 2017-09-02T16:12:43Z
      DOI: 10.1016/j.acha.2016.03.005
  • Multiscale geometric methods for data sets I: Multiscale SVD, noise and
    • Authors: Anna V. Little; Mauro Maggioni; Lorenzo Rosasco
      Pages: 504 - 567
      Abstract: Publication date: November 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 3
      Author(s): Anna V. Little, Mauro Maggioni, Lorenzo Rosasco
      Large data sets are often modeled as being noisy samples from probability distributions μ in R D , with D large. It has been noticed that oftentimes the support M of these probability distributions seems to be well-approximated by low-dimensional sets, perhaps even by manifolds. We shall consider sets that are locally well-approximated by k-dimensional planes, with k ≪ D , with k-dimensional manifolds isometrically embedded in R D being a special case. Samples from μ are furthermore corrupted by D-dimensional noise. Certain tools from multiscale geometric measure theory and harmonic analysis seem well-suited to be adapted to the study of samples from such probability distributions, in order to yield quantitative geometric information about them. In this paper we introduce and study multiscale covariance matrices, i.e. covariances corresponding to the distribution restricted to a ball of radius r, with a fixed center and varying r, and under rather general geometric assumptions we study how their empirical, noisy counterparts behave. We prove that in the range of scales where these covariance matrices are most informative, the empirical, noisy covariances are close to their expected, noiseless counterparts. In fact, this is true as soon as the number of samples in the balls where the covariance matrices are computed is linear in the intrinsic dimension of M . As an application, we present an algorithm for estimating the intrinsic dimension of M .

      PubDate: 2017-09-02T16:12:43Z
      DOI: 10.1016/j.acha.2015.09.009
  • On the number of iterations for convergence of CoSaMP and Subspace Pursuit
    • Authors: Siddhartha Satpathi; Mrityunjoy Chakraborty
      Pages: 568 - 576
      Abstract: Publication date: November 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 3
      Author(s): Siddhartha Satpathi, Mrityunjoy Chakraborty
      In compressive sensing, one important parameter that characterizes the various greedy recovery algorithms is the iteration bound which provides the maximum number of iterations by which the algorithm is guaranteed to converge. In this letter, we present a new iteration bound for the compressive sampling matching pursuit (CoSaMP) algorithm by certain mathematical manipulations including formulation of appropriate sufficient conditions that ensure passage of a chosen support through the two selection stages of CoSaMP, “Augment” and “Update”. Subsequently, we extend the treatment to the subspace pursuit (SP) algorithm. The proposed iteration bounds for both CoSaMP and SP algorithms are seen to be improvements over their existing counterparts, revealing that both CoSaMP and SP algorithms converge in fewer iterations than suggested by results available in literature.

      PubDate: 2017-09-02T16:12:43Z
      DOI: 10.1016/j.acha.2016.10.001
  • Spectral echolocation via the wave embedding
    • Authors: Alexander Cloninger; Stefan Steinerberger
      Pages: 577 - 590
      Abstract: Publication date: November 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 3
      Author(s): Alexander Cloninger, Stefan Steinerberger
      Spectral embedding uses eigenfunctions of the discrete Laplacian on a weighted graph to obtain coordinates for an embedding of an abstract data set into Euclidean space. We propose a new pre-processing step of first using the eigenfunctions to simulate a low-frequency wave moving over the data and using both position as well as change in time of the wave to obtain a refined metric to which classical methods of dimensionality reduction can then applied. This is motivated by the behavior of waves, symmetries of the wave equation and the hunting technique of bats. It is shown to be effective in practice and also works for other partial differential equations – the method yields improved results even for the classical heat equation.

      PubDate: 2017-09-02T16:12:43Z
      DOI: 10.1016/j.acha.2017.01.002
  • Decomposition matrices for the special case of data on the triangular
           lattice of SU(3)
    • Authors: M. Bodner; J. Patera; M. Szajewska
      Abstract: Publication date: September 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 2
      Author(s): M. Bodner, J. Patera, M. Szajewska
      A method for the decomposition of data functions sampled on a finite fragment of triangular lattices is described for the lattice corresponding to the simple Lie group S U ( 3 ) . The basic tile (fundamental region) of S U ( 3 ) is an equilateral triangle. The decomposition matrices refer to lattices that carry data of any density. This main result is summarized in Section 4 Theorem 2.

      PubDate: 2017-07-12T06:45:03Z
      DOI: 10.1016/j.acha.2017.02.003
  • Evaluation of small elements of the eigenvectors of certain symmetric
           tridiagonal matrices with high relative accuracy
    • Authors: Andrei Osipov
      Pages: 173 - 211
      Abstract: Publication date: September 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 2
      Author(s): Andrei Osipov
      Evaluation of the eigenvectors of symmetric tridiagonal matrices is one of the most basic tasks in numerical linear algebra. It is a widely known fact that, in the case of well separated eigenvalues, the eigenvectors can be evaluated with high relative accuracy. Nevertheless, in general, each coordinate of the eigenvector is evaluated with only high absolute accuracy. In particular, those coordinates whose magnitude is below the machine precision are not expected to be evaluated with any accuracy whatsoever. It turns out that, under certain conditions, frequently encountered in applications, small (e.g. 10 − 50 ) coordinates of eigenvectors of symmetric tridiagonal matrices can be evaluated with high relative accuracy. In this paper, we investigate such conditions, carry out the analysis, and describe the resulting numerical schemes. While our schemes can be viewed as a modification of already existing (and well known) numerical algorithms, the related error analysis appears to be new. Our results are illustrated via several numerical examples.

      PubDate: 2017-07-12T06:45:03Z
      DOI: 10.1016/j.acha.2015.12.002
  • Error bounds for compressed sensing algorithms with group sparsity: A
           unified approach
    • Authors: M. Eren Ahsen; M. Vidyasagar
      Pages: 212 - 232
      Abstract: Publication date: September 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 2
      Author(s): M. Eren Ahsen, M. Vidyasagar
      In compressed sensing, in order to recover a sparse or nearly sparse vector from possibly noisy measurements, the most popular approach is ℓ 1 -norm minimization. Upper bounds for the ℓ 2 -norm of the error between the true and estimated vectors are given in [1] and reviewed in [2], while bounds for the ℓ 1 -norm are given in [3]. When the unknown vector is not conventionally sparse but is “group sparse” instead, a variety of alternatives to the ℓ 1 -norm have been proposed in the literature, including the group LASSO, sparse group LASSO, and group LASSO with tree structured overlapping groups. However, no error bounds are available for any of these modified objective functions. In the present paper, a unified approach is presented for deriving upper bounds on the error between the true vector and its approximation, based on the notion of decomposable and γ-decomposable norms. The bounds presented cover all of the norms mentioned above, and also provide a guideline for choosing norms in future to accommodate alternate forms of sparsity.

      PubDate: 2017-07-12T06:45:03Z
      DOI: 10.1016/j.acha.2015.11.006
  • Neural network with unbounded activation functions is universal
    • Authors: Sho Sonoda; Noboru Murata
      Pages: 233 - 268
      Abstract: Publication date: September 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 2
      Author(s): Sho Sonoda, Noboru Murata
      This paper presents an investigation of the approximation property of neural networks with unbounded activation functions, such as the rectified linear unit (ReLU), which is the new de-facto standard of deep learning. The ReLU network can be analyzed by the ridgelet transform with respect to Lizorkin distributions. By showing three reconstruction formulas by using the Fourier slice theorem, the Radon transform, and Parseval's relation, it is shown that a neural network with unbounded activation functions still satisfies the universal approximation property. As an additional consequence, the ridgelet transform, or the backprojection filter in the Radon domain, is what the network learns after backpropagation. Subject to a constructive admissibility condition, the trained network can be obtained by simply discretizing the ridgelet transform, without backpropagation. Numerical examples not only support the consistency of the admissibility condition but also imply that some non-admissible cases result in low-pass filtering.

      PubDate: 2017-07-12T06:45:03Z
      DOI: 10.1016/j.acha.2015.12.005
  • A multifractal formalism for non-concave and non-increasing spectra: The
           leaders profile method
    • Authors: Céline Esser; Thomas Kleyntssens; Samuel Nicolay
      Pages: 269 - 291
      Abstract: Publication date: September 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 2
      Author(s): Céline Esser, Thomas Kleyntssens, Samuel Nicolay
      We present an implementation of a multifractal formalism based on the types of histogram of wavelet leaders. This method yields non-concave spectra and is not limited to their increasing part. We show both from the theoretical and from the applied points of view that this approach is more efficient than the wavelet-based multifractal formalisms previously introduced.

      PubDate: 2017-07-12T06:45:03Z
      DOI: 10.1016/j.acha.2015.12.006
  • Fully discrete needlet approximation on the sphere
    • Authors: Yu Guang Wang; Quoc T. Le Gia; Ian H. Sloan; Robert S. Womersley
      Pages: 292 - 316
      Abstract: Publication date: September 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 2
      Author(s): Yu Guang Wang, Quoc T. Le Gia, Ian H. Sloan, Robert S. Womersley
      Spherical needlets are highly localized radial polynomials on the sphere S d ⊂ R d + 1 , d ≥ 2 , with centers at the nodes of a suitable cubature rule. The original semidiscrete spherical needlet approximation of Narcowich, Petrushev and Ward is not computable, in that the needlet coefficients depend on inner product integrals. In this work we approximate these integrals by a second quadrature rule with an appropriate degree of precision, to construct a fully discrete needlet approximation. We prove that the resulting approximation is equivalent to filtered hyperinterpolation, that is to a filtered Fourier–Laplace series partial sum with inner products replaced by appropriate cubature sums. It follows that the L p -error of discrete needlet approximation of order J for 1 ≤ p ≤ ∞ and s > d / p has for a function f in the Sobolev space W p s ( S d ) the optimal rate of convergence in the sense of optimal recovery, namely O ( 2 − J s ) . Moreover, this is achieved with a filter function that is of smoothness class C ⌊ d + 3 2 ⌋ , in contrast to the usually assumed C ∞ . A numerical experiment for a class of functions in known Sobolev smoothness classes gives L 2 errors for the fully discrete needlet approximation that are almost identical to those for the original semidiscrete needlet approximation. Another experiment uses needlets over the whole sphere for the lower levels together with high-level needlets with centers restricted to a local region. The resulting errors are reduced in the local region away from the boundary, indicating that local refinement in special regions is a promising strategy.

      PubDate: 2017-07-12T06:45:03Z
      DOI: 10.1016/j.acha.2016.01.003
  • Appell sequences, continuous wavelet transforms and series expansions
    • Authors: Say Song Goh; Tim N.T. Goodman; S.L. Lee
      Pages: 317 - 345
      Abstract: Publication date: September 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 2
      Author(s): Say Song Goh, Tim N.T. Goodman, S.L. Lee
      A series expansion with remainder for functions in a Sobolev space is derived in terms of the classical Bernoulli polynomials, the B-spline scale-space and the continuous wavelet transforms with the derivatives of the standardized B-splines as mother wavelets. In the limit as their orders tend to infinity, the B-splines and their derivatives converge to the Gaussian function and its derivatives respectively, the associated Bernoulli polynomials converge to the Hermite polynomials, and the corresponding series expansion is an expansion in terms of the Hermite polynomials, the Gaussian scale-space and the continuous wavelet transforms with the derivatives of the Gaussian function as mother wavelets. A similar expansion is also derived in terms of continuous wavelet transforms in which the mother wavelets are the spline framelets that approximate the derivatives of the standardized B-splines.

      PubDate: 2017-07-12T06:45:03Z
      DOI: 10.1016/j.acha.2016.01.005
  • Explicit universal sampling sets in finite vector spaces
    • Authors: Lucia Morotti
      Pages: 354 - 369
      Abstract: Publication date: September 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 2
      Author(s): Lucia Morotti
      In this paper we construct explicit sampling sets and present reconstruction algorithms for Fourier signals on finite vector spaces G, with G = p r for a suitable prime p. The two sampling sets have sizes of order O ( p t 2 r 2 ) and O ( p t 2 r 3 log ⁡ ( p ) ) respectively, where t is the number of large coefficients in the Fourier transform. The algorithms approximate the function up to a small constant of the best possible approximation with t non-zero Fourier coefficients. The fastest of the algorithms has complexity O ( p 2 t 2 r 3 log ⁡ ( p ) ) .

      PubDate: 2017-07-12T06:45:03Z
      DOI: 10.1016/j.acha.2016.06.001
  • A note on Markov normalized magnetic eigenmaps
    • Authors: Alexander Cloninger
      Pages: 370 - 380
      Abstract: Publication date: September 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 2
      Author(s): Alexander Cloninger
      We note that building a magnetic Laplacian from the Markov transition matrix, rather than the graph adjacency matrix, yields several benefits for the magnetic eigenmaps algorithm. The two largest benefits are that the embedding becomes more stable as a function of the rotation parameter g, and the principal eigenvector of the magnetic Laplacian now converges to the page rank of the network as a function of diffusion time. We show empirically that this normalization improves the phase and real/imaginary embeddings of the low-frequency eigenvectors of the magnetic Laplacian.

      PubDate: 2017-07-12T06:45:03Z
      DOI: 10.1016/j.acha.2016.11.002
  • Recovery analysis for weighted ℓ1-minimization using the null space
    • Authors: Hassan Mansour; Rayan Saab
      Pages: 23 - 38
      Abstract: Publication date: July 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 1
      Author(s): Hassan Mansour, Rayan Saab
      We study the recovery of sparse signals from underdetermined linear measurements when a potentially erroneous support estimate is available. Our results are twofold. First, we derive necessary and sufficient conditions for signal recovery from compressively sampled measurements using weighted ℓ 1 -norm minimization. These conditions, which depend on the choice of weights as well as the size and accuracy of the support estimate, are on the null space of the measurement matrix. They can guarantee recovery even when standard ℓ 1 minimization fails. Second, we derive bounds on the number of Gaussian measurements for these conditions to be satisfied, i.e., for weighted ℓ 1 minimization to successfully recover all sparse signals whose support has been estimated sufficiently accurately. Our bounds show that weighted ℓ 1 minimization requires significantly fewer measurements than standard ℓ 1 minimization when the support estimate is relatively accurate.

      PubDate: 2017-05-17T15:00:34Z
      DOI: 10.1016/j.acha.2015.10.005
  • Far-field compression for fast kernel summation methods in high dimensions
    • Authors: William B. March; George Biros
      Pages: 39 - 75
      Abstract: Publication date: July 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 1
      Author(s): William B. March, George Biros
      We consider fast kernel summations in high dimensions: given a large set of points in d dimensions (with d ≫ 3 ) and a pair-potential function (the kernel function), we compute a weighted sum of all pairwise kernel interactions for each point in the set. Direct summation is equivalent to a (dense) matrix–vector multiplication and scales quadratically with the number of points. Fast kernel summation algorithms reduce this cost to log-linear or linear complexity. Treecodes and Fast Multipole Methods (FMMs) deliver tremendous speedups by constructing approximate representations of interactions of points that are far from each other. In algebraic terms, these representations correspond to low-rank approximations of blocks of the overall interaction matrix. Existing approaches require an excessive number of kernel evaluations with increasing d and number of points in the dataset. To address this issue, we use a randomized algebraic approach in which we first sample the rows of a block and then construct its approximate, low-rank interpolative decomposition. We examine the feasibility of this approach theoretically and experimentally. We provide a new theoretical result showing a tighter bound on the reconstruction error from uniformly sampling rows than the existing state-of-the-art. We demonstrate that our sampling approach is competitive with existing (but prohibitively expensive) methods from the literature. We also construct kernel matrices for the Laplacian, Gaussian, and polynomial kernels—all commonly used in physics and data analysis. We explore the numerical properties of blocks of these matrices, and show that they are amenable to our approach. Depending on the data set, our randomized algorithm can successfully compute low rank approximations in high dimensions. We report results for data sets with ambient dimensions from four to 1,000.

      PubDate: 2017-05-17T15:00:34Z
      DOI: 10.1016/j.acha.2015.09.007
  • A sampling theory for non-decaying signals
    • Authors: Ha Q. Nguyen; Michael Unser
      Pages: 76 - 93
      Abstract: Publication date: July 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 1
      Author(s): Ha Q. Nguyen, Michael Unser
      The classical assumption in sampling and spline theories is that the input signal is square-integrable, which prevents us from applying such techniques to signals that do not decay or even grow at infinity. In this paper, we develop a sampling theory for multidimensional non-decaying signals living in weighted L p spaces. The sampling and reconstruction of an analog signal can be done by a projection onto a shift-invariant subspace generated by an interpolating kernel. We show that, if this kernel and its biorthogonal counterpart are elements of appropriate hybrid-norm spaces, then both the sampling and the reconstruction are stable. This is an extension of earlier results by Aldroubi and Gröchenig. The extension is required because it allows us to develop the theory for the ideal sampling of non-decaying signals in weighted Sobolev spaces. When the d-dimensional signal and its d / p + ε derivatives, for arbitrarily small ε > 0 , grow no faster than a polynomial in the L p sense, the sampling operator is shown to be bounded even without a sampling kernel. As a consequence, the signal can also be interpolated from its samples with a nicely behaved interpolating kernel.

      PubDate: 2017-05-17T15:00:34Z
      DOI: 10.1016/j.acha.2015.10.006
  • Gabor frames of Gaussian beams for the Schrödinger equation
    • Authors: Michele Berra; Iulia Martina Bulai; Elena Cordero; Fabio Nicola
      Pages: 94 - 121
      Abstract: Publication date: July 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 1
      Author(s): Michele Berra, Iulia Martina Bulai, Elena Cordero, Fabio Nicola
      The present paper is devoted to the semiclassical analysis of linear Schrödinger equations from a Gabor frame perspective. We consider (time-dependent) smooth Hamiltonians with at most quadratic growth. Then we construct higher order parametrices for the corresponding Schrödinger equations by means of ħ-Gabor frames, as recently defined by M. de Gosson, and we provide precise L 2 -estimates of their accuracy, in terms of the Planck constant ħ. Nonlinear parametrices, in the spirit of the nonlinear approximation, are also presented. Numerical experiments are exhibited to compare our results with the early literature.

      PubDate: 2017-05-17T15:00:34Z
      DOI: 10.1016/j.acha.2015.11.001
  • High-dimensional change-point estimation: Combining filtering with convex
    • Authors: Yong Sheng Soh; Venkat Chandrasekaran
      Pages: 122 - 147
      Abstract: Publication date: July 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 1
      Author(s): Yong Sheng Soh, Venkat Chandrasekaran
      We consider change-point estimation in a sequence of high-dimensional signals given noisy observations. Classical approaches to this problem such as the filtered derivative method are useful for sequences of scalar-valued signals, but they have undesirable scaling behavior in the high-dimensional setting. However, many high-dimensional signals encountered in practice frequently possess latent low-dimensional structure. Motivated by this observation, we propose a technique for high-dimensional change-point estimation that combines the filtered derivative approach from previous work with convex optimization methods based on atomic norm regularization, which are useful for exploiting structure in high-dimensional data. Our algorithm is applicable in online settings as it operates on small portions of the sequence of observations at a time, and it is well-suited to the high-dimensional setting both in terms of computational scalability and of statistical efficiency. The main result of this paper shows that our method performs change-point estimation reliably as long as the product of the smallest-sized change (the Euclidean-norm-squared of the difference between signals at a change-point) and the smallest distance between change-points (number of time instances) is larger than a Gaussian width parameter that characterizes the low-dimensional complexity of the underlying signal sequence.

      PubDate: 2017-05-17T15:00:34Z
      DOI: 10.1016/j.acha.2015.11.003
  • Frames of directional wavelets on n-dimensional spheres
    • Authors: I. Iglewska-Nowak
      Pages: 148 - 161
      Abstract: Publication date: July 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 1
      Author(s): I. Iglewska-Nowak
      The major goal of the paper is to prove that discrete frames of (directional) wavelets derived from an approximate identity exist. Additionally, a kind of energy conservation property is shown to hold in the case when a wavelet family is not its own reconstruction family. Although an additional constraint on the spectrum of the wavelet family must be satisfied, it is shown that all the wavelets so far defined in the literature possess this property.

      PubDate: 2017-05-17T15:00:34Z
      DOI: 10.1016/j.acha.2016.01.004
  • Indefinite kernels in least squares support vector machines and principal
           component analysis
    • Authors: Xiaolin Huang; Andreas Maier; Joachim Hornegger; Johan A.K. Suykens
      Pages: 162 - 172
      Abstract: Publication date: July 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 1
      Author(s): Xiaolin Huang, Andreas Maier, Joachim Hornegger, Johan A.K. Suykens
      Because of several successful applications, indefinite kernels have attracted many research interests in recent years. This paper addresses indefinite learning in the framework of least squares support vector machines (LS-SVM). Unlike existing indefinite kernel learning methods, which usually involve non-convex problems, the indefinite LS-SVM is still easy to solve, but the kernel trick and primal-dual relationship for LS-SVM with a Mercer kernel is no longer valid. In this paper, we give a feature space interpretation for indefinite LS-SVM. In the same framework, kernel principal component analysis with an infinite kernel is discussed as well. In numerical experiments, LS-SVM with indefinite kernels for classification and kernel principal component analysis is evaluated. Its good performance together with the feature space interpretation given in this paper imply the potential use of indefinite LS-SVM in real applications.

      PubDate: 2017-05-17T15:00:34Z
      DOI: 10.1016/j.acha.2016.09.001
  • On computing distributions of products of random variables via Gaussian
           multiresolution analysis
    • Authors: Gregory Beylkin; Lucas Monzón; Ignas Satkauskas
      Abstract: Publication date: Available online 5 September 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Gregory Beylkin, Lucas Monzón, Ignas Satkauskas
      We introduce a new approximate multiresolution analysis (MRA) using a single Gaussian as the scaling function, which we call Gaussian MRA (GMRA). As an initial application, we employ this new tool to accurately and efficiently compute the probability density function (PDF) of the product of independent random variables. In contrast with Monte-Carlo (MC) type methods (the only other universal approach known to address this problem), our method not only achieves accuracies beyond the reach of MC but also produces a PDF expressed as a Gaussian mixture, thus allowing for further efficient computations. We also show that an exact MRA corresponding to our GMRA can be constructed for a matching user-selected accuracy.

      PubDate: 2017-09-08T02:02:44Z
      DOI: 10.1016/j.acha.2017.08.008
  • Stability of operator expansions under discretization
    • Authors: Michael Wilson
      Abstract: Publication date: Available online 4 September 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Michael Wilson
      We show that, when wavelets in almost-orthogonal expansions of linear operators are replaced by fine dyadic discretizations, the resulting approximations are (in the L 2 → L 2 sense) close to the original operators and that these discretizations are stable with respect to small errors in translation and dilation.

      PubDate: 2017-09-08T02:02:44Z
      DOI: 10.1016/j.acha.2017.08.009
  • Similarity matrix framework for data from union of subspaces
    • Authors: A. Aldroubi; A. Sekmen; A.B. Koku; A.F. Cakmak
      Abstract: Publication date: Available online 4 September 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): A. Aldroubi, A. Sekmen, A.B. Koku, A.F. Cakmak
      This paper presents a framework for finding similarity matrices for the segmentation of data W = [ w 1 ⋯ w N ] ⊂ R D drawn from a union U = ⋃ i = 1 M S i of independent subspaces { S i } i = 1 M of dimensions { d i } i = 1 M . It is shown that any factorization of W = B P , where columns of B form a basis for data W and they also come from U , can be used to produce a similarity matrix Ξ W . In other words, Ξ W ( i , j ) ≠ 0 , when the columns w i and w j of W come from the same subspace, and Ξ W ( i , j ) = 0 , when the columns w i and w j of W come from different subspaces. Furthermore, Ξ W = Q d m a x , where d m a x = max ⁡ { d i } i = 1 M and Q ∈ R N × N with Q ( i , j ) = P T P ( i , j ) . It is shown that a similarity matrix obtained from the reduced row echelon form of W is a special case of the theory. It is also proven that the Shape Interaction Matrix defined as V V T , where W = U Σ V T is the skinny singular value decomposition of W, is not necessarily a similarity matrix. But, taking powers of its absolute value always generates a similarity matrix. An interesting finding of this research is that a similarity matrix can be obtained using a skeleton decomposition of W. First, a square sub-matrix A ∈ R r ×
      PubDate: 2017-09-08T02:02:44Z
      DOI: 10.1016/j.acha.2017.08.006
  • Diffusion Nets
    • Authors: Gal Mishne; Uri Shaham; Alexander Cloninger; Israel Cohen
      Abstract: Publication date: Available online 31 August 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Gal Mishne, Uri Shaham, Alexander Cloninger, Israel Cohen
      Non-linear manifold learning enables high-dimensional data analysis, but requires out-of-sample-extension methods to process new data points. In this paper, we propose a manifold learning algorithm based on deep learning to create an encoder, which maps a high-dimensional dataset to its low-dimensional embedding, and a decoder, which takes the embedded data back to the high-dimensional space. Stacking the encoder and decoder together constructs an autoencoder, which we term a diffusion net, that performs out-of-sample-extension as well as outlier detection. We introduce new neural net constraints for the encoder, which preserve the local geometry of the points, and we prove rates of convergence for the encoder. Also, our approach is efficient in both computational complexity and memory requirements, as opposed to previous methods that require storage of all training points in both the high-dimensional and the low-dimensional spaces to calculate the out-of-sample-extension and the pre-image of new points.

      PubDate: 2017-09-02T16:12:43Z
      DOI: 10.1016/j.acha.2017.08.007
  • Landmark diffusion maps (L-dMaps): Accelerated manifold learning
           out-of-sample extension
    • Authors: Andrew W. Long; Andrew L. Ferguson
      Abstract: Publication date: Available online 31 August 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Andrew W. Long, Andrew L. Ferguson
      Diffusion maps are a nonlinear manifold learning technique based on harmonic analysis of a diffusion process over the data. Out-of-sample extensions with computational complexity O ( N ) , where N is the number of points comprising the manifold, frustrate applications to online learning applications requiring rapid embedding of high-dimensional data streams. We propose landmark diffusion maps (L-dMaps) to reduce the complexity to O ( M ) , where M ≪ N is the number of landmark points selected using pruned spanning trees or k-medoids. Offering ( N / M ) speedups in out-of-sample extension, L-dMaps enable the application of diffusion maps to high-volume and/or high-velocity streaming data. We illustrate our approach on three datasets: the Swiss roll, molecular simulations of a C24H50 polymer chain, and biomolecular simulations of alanine dipeptide. We demonstrate up to 50-fold speedups in out-of-sample extension for the molecular systems with less than 4% errors in manifold reconstruction fidelity relative to calculations over the full dataset.

      PubDate: 2017-09-02T16:12:43Z
      DOI: 10.1016/j.acha.2017.08.004
  • Analysis of time-frequency scattering transforms
    • Authors: Wojciech Czaja; Weilin Li
      Abstract: Publication date: Available online 23 August 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Wojciech Czaja, Weilin Li
      In this paper we address the problem of constructing a feature extractor which combines Mallat's scattering transform framework with time-frequency (Gabor) representations. To do this, we introduce a class of frames, called uniform covering frames, which includes a variety of semi-discrete Gabor systems. Incorporating a uniform covering frame with a neural network structure yields the Fourier scattering transform S F and the truncated Fourier scattering transform. We prove that S F propagates energy along frequency decreasing paths and its energy decays exponentially as a function of the depth. These quantitative estimates are fundamental in showing that S F satisfies the typical scattering transform properties, and in controlling the information loss due to width and depth truncation. We introduce the fast Fourier scattering transform algorithm, and illustrate the algorithm's performance. The time-frequency covering techniques developed in this paper are flexible and give insight into the analysis of scattering transforms.

      PubDate: 2017-09-02T16:12:43Z
      DOI: 10.1016/j.acha.2017.08.005
  • On sets of large Fourier transform under changes in domain
    • Authors: Joel Laity; Barak Shani
      Abstract: Publication date: Available online 23 August 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Joel Laity, Barak Shani
      A function f : Z n → C can be represented as a linear combination f ( x ) = ∑ α ∈ Z n f ˆ ( α ) χ α , n ( x ) where f ˆ is the (discrete) Fourier transform of f. Clearly, the basis { χ α , n ( x ) : = exp ⁡ ( 2 π i α x / n ) } depends on the value n. We show that if f has “large” Fourier coefficients, then the function f ˜ : Z m → C , given by f ˜ ( x ) = { f ( x ) when  0 ≤ x < min ⁡ ( n , m ) , 0 otherwise , also has “large” coefficients. Moreover, they are all contained in a “small” interval around ⌊ m n α ⌉ for each α ∈ Z n such that f ˆ ( α ) is large. One can use this result to recover the large Fourier coefficients of a function f by redefining it on a convenient domain. One can also use this result to reprove a result by Morillo and Ràfols: single-bit functions, defined over any domain, are Fourier concentrated.

      PubDate: 2017-09-02T16:12:43Z
      DOI: 10.1016/j.acha.2017.08.002
  • The spectrogram expansion of Wigner functions
    • Authors: Johannes Keller
      Abstract: Publication date: Available online 18 August 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Johannes Keller
      Wigner functions generically attain negative values and hence are not probability densities. We prove an asymptotic expansion of Wigner functions in terms of Hermite spectrograms, which are probability densities. The expansion provides exact formulas for the quantum expectations of polynomial observables. In the high frequency regime it allows to approximate quantum expectation values up to any order of accuracy in the high frequency parameter. We present a Markov Chain Monte Carlo method to sample from the new densities and illustrate our findings by numerical experiments.

      PubDate: 2017-09-02T16:12:43Z
      DOI: 10.1016/j.acha.2017.08.003
  • Spatially distributed sampling and reconstruction
    • Authors: Cheng Cheng; Yingchun Jiang; Qiyu Sun
      Abstract: Publication date: Available online 14 August 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Cheng Cheng, Yingchun Jiang, Qiyu Sun
      A spatially distributed network contains a large amount of agents with limited sensing, data processing, and communication capabilities. Recent technological advances have opened up possibilities to deploy spatially distributed networks for signal sampling and reconstruction. In this paper, we introduce a graph structure for a distributed sampling and reconstruction system by coupling agents in a spatially distributed network with innovative positions of signals. A fundamental problem in sampling theory is the robustness of signal reconstruction in the presence of sampling noises. For a distributed sampling and reconstruction system, the robustness could be reduced to the stability of its sensing matrix. In this paper, we split a distributed sampling and reconstruction system into a family of overlapping smaller subsystems, and we show that the stability of the sensing matrix holds if and only if its quasi-restrictions to those subsystems have uniform stability. This new stability criterion could be pivotal for the design of a robust distributed sampling and reconstruction system against supplement, replacement and impairment of agents, as we only need to check the uniform stability of affected subsystems. In this paper, we also propose an exponentially convergent distributed algorithm for signal reconstruction, that provides a suboptimal approximation to the original signal in the presence of bounded sampling noises.

      PubDate: 2017-09-02T16:12:43Z
      DOI: 10.1016/j.acha.2017.07.007
  • Digital Gabor filters do generate MRA-based wavelet tight frames
    • Authors: Hui Ji; Zuowei Shen; Yufei Zhao
      Abstract: Publication date: Available online 10 August 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Hui Ji, Zuowei Shen, Yufei Zhao
      Gabor frames, especially digital Gabor filters, have long been known as indispensable tools for local time–frequency analysis of discrete signals. With strong orientation selectivity, tensor products discrete (tight) Gabor frames also see their applications in image analysis and restoration. However, the lack of multi-scale structures existing in MRA-based wavelet (tight) frames makes discrete Gabor frames less effective on modeling local structures of signals with varying sizes. Indeed, historically speaking, it was the motivation of studying wavelet systems. By applying the unitary extension principle on some most often seen digital Gabor filters (e.g. local discrete Fourier transform and discrete Cosine transform), we are surprised to find out that these digital filter banks generate MRA-based wavelet tight frames in square integrable function space, and the corresponding refinable functions and wavelets can be explicitly given. In other words, the discrete tight frames associated with these digital Gabor filters can be used as the filter banks of MRA wavelet tight frames, which introduce both multi-scale structures and fast cascade implementation of discrete signal decomposition/reconstruction. Discrete tight frames generated by such filters with both wavelet and Gabor structures can see their potential applications in image processing and recovery.

      PubDate: 2017-09-02T16:12:43Z
      DOI: 10.1016/j.acha.2017.08.001
  • Vandermonde matrices with nodes in the unit disk and the large sieve
    • Authors: Céline Aubel; Helmut Bölcskei
      Abstract: Publication date: Available online 1 August 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Céline Aubel, Helmut Bölcskei
      We derive bounds on the extremal singular values and the condition number of N × K , with N ⩾ K , Vandermonde matrices with nodes in the unit disk. The mathematical techniques we develop to prove our main results are inspired by a link—first established by Selberg [1] and later extended by Moitra [2]—between the extremal singular values of Vandermonde matrices with nodes on the unit circle and large sieve inequalities. Our main conceptual contribution lies in establishing a connection between the extremal singular values of Vandermonde matrices with nodes in the unit disk and a novel large sieve inequality involving polynomials in z ∈ C with z ⩽ 1 . Compared to Bazán's upper bound on the condition number [3], which, to the best of our knowledge, constitutes the only analytical result—available in the literature—on the condition number of Vandermonde matrices with nodes in the unit disk, our bound not only takes a much simpler form, but is also sharper for certain node configurations. Moreover, the bound we obtain can be evaluated consistently in a numerically stable fashion, whereas the evaluation of Bazán's bound requires the solution of a linear system of equations which has the same condition number as the Vandermonde matrix under consideration and can therefore lead to numerical instability in practice. As a byproduct, our result—when particularized to the case of nodes on the unit circle—slightly improves upon the Selberg–Moitra bound.

      PubDate: 2017-08-03T08:25:09Z
      DOI: 10.1016/j.acha.2017.07.006
  • Spark-level sparsity and the ℓ1 tail minimization
    • Authors: Chun-Kit Lai; Shidong Li; Daniel Mondo
      Abstract: Publication date: Available online 27 July 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Chun-Kit Lai, Shidong Li, Daniel Mondo
      Solving compressed sensing problems relies on the properties of sparse signals. It is commonly assumed that the sparsity s needs to be less than one half of the spark of the sensing matrix A, and then the unique sparsest solution exists, and is recoverable by ℓ 1 -minimization or related procedures. We discover, however, a measure theoretical uniqueness exists for nearly spark-level sparsity from compressed measurements A x = b . Specifically, suppose A is of full spark with m rows, and suppose m 2 < s < m . Then the solution to A x = b is unique for x with ‖ x ‖ 0 ≤ s up to a set of measure 0 in every s-sparse plane. This phenomenon is observed and confirmed by an ℓ 1 -tail minimization procedure, which recovers sparse signals uniquely with s > m 2 in thousands and thousands of random tests. We further show instead that the mere ℓ 1 -minimization would actually fail if s > m 2 even from the same measure theoretical point of view.

      PubDate: 2017-08-03T08:25:09Z
      DOI: 10.1016/j.acha.2017.07.001
  • Shift–Invariant and Sampling Spaces Associated with the Special
           Affine Fourier Transform
    • Authors: Ayush Bhandari; Ahmed I. Zayed
      Abstract: Publication date: Available online 25 July 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Ayush Bhandari, Ahmed I. Zayed
      The Special Affine Fourier Transformation or the SAFT generalizes a number of well known unitary transformations as well as signal processing and optics related mathematical operations. Shift-invariant spaces also play an important role in sampling theory, multiresolution analysis, and many other areas of signal and image processing. Shannon's sampling theorem, which is at the heart of modern digital communications, is a special case of sampling in shift-invariant spaces. Furthermore, it is well known that the Poisson summation formula is equivalent to the sampling theorem and that the Zak transform is closely connected to the sampling theorem and the Poisson summation formula. These results have been known to hold in the Fourier transform domain for decades and were recently shown to hold in the Fractional Fourier transform domain by A. Bhandari and A. Zayed. The main goal of this article is to show that these results also hold true in the SAFT domain. We provide a short, self–contained proof of Shannon's theorem for functions bandlimited in the SAFT domain and then show that sampling in the SAFT domain is equivalent to orthogonal projection of functions onto a subspace of bandlimited basis associated with the SAFT domain. This interpretation of sampling leads to least–squares optimal sampling theorem. Furthermore, we show that this approximation procedure is linked with convolution and semi–discrete convolution operators that are associated with the SAFT domain. We conclude the article with an application of fractional delay filtering of SAFT bandlimited functions.

      PubDate: 2017-08-03T08:25:09Z
      DOI: 10.1016/j.acha.2017.07.002
  • The fast Slepian transform
    • Authors: Santhosh Karnik; Zhihui Zhu; Michael B. Wakin; Justin Romberg; Mark A. Davenport
      Abstract: Publication date: Available online 19 July 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Santhosh Karnik, Zhihui Zhu, Michael B. Wakin, Justin Romberg, Mark A. Davenport
      The discrete prolate spheroidal sequences (DPSS's) provide an efficient representation for discrete signals that are perfectly timelimited and nearly bandlimited. Due to the high computational complexity of projecting onto the DPSS basis – also known as the Slepian basis – this representation is often overlooked in favor of the fast Fourier transform (FFT). We show that there exist fast constructions for computing approximate projections onto the leading Slepian basis elements. The complexity of the resulting algorithms is comparable to the FFT, and scales favorably as the quality of the desired approximation is increased. In the process of bounding the complexity of these algorithms, we also establish new nonasymptotic results on the eigenvalue distribution of discrete time-frequency localization operators. We then demonstrate how these algorithms allow us to efficiently compute the solution to certain least-squares problems that arise in signal processing. We also provide simulations comparing these fast, approximate Slepian methods to exact Slepian methods as well as the traditional FFT based methods.

      PubDate: 2017-07-24T07:52:31Z
      DOI: 10.1016/j.acha.2017.07.005
  • Functional Reproducing Kernel Hilbert Spaces for Non-Point-Evaluation
           Functional Data
    • Authors: Rui Wang; Yuesheng Xu
      Abstract: Publication date: Available online 18 July 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Rui Wang, Yuesheng Xu
      Motivated by the need of processing non-point-evaluation functional data, we introduce the notion of functional reproducing kernel Hilbert spaces (FRKHSs). This space admits a unique functional reproducing kernel which reproduces a family of continuous linear functionals on the space. The theory of FRKHSs and the associated functional reproducing kernels are established. A special class of FRKHSs, which we call the perfect FRKHSs, are studied, which reproduce the family of the standard point-evaluation functionals and at the same time another different family of continuous linear (non-point-evaluation) functionals. The perfect FRKHSs are characterized in terms of features, especially for those with respect to integral functionals. In particular, several specific examples of the perfect FRKHSs are presented. We apply the theory of FRKHSs to sampling and regularized learning, where non-point-evaluation functional data are used. Specifically, a general complete reconstruction formula from linear functional values is established in the framework of FRKHSs. The average sampling and the reconstruction of vector-valued functions are considered in specific FRKHSs. We also investigate in the FRKHS setting the regularized learning schemes, which learn a target element from non-point-evaluation functional data. The desired representer theorems of the learning problems are established to demonstrate the key roles played by the FRKHSs and the functional reproducing kernels in machine learning from non-point-evaluation functional data. We finally illustrate that the continuity of linear functionals, used to obtain the non-point-evaluation functional data, on an FRKHS is necessary for the stability of the numerical reconstruction algorithm using the data.

      PubDate: 2017-07-24T07:52:31Z
      DOI: 10.1016/j.acha.2017.07.003
  • The unitary extension principle on locally compact abelian groups
    • Authors: Ole Christensen; Say Song Goh
      Abstract: Publication date: Available online 18 July 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Ole Christensen, Say Song Goh
      The unitary extension principle (UEP) by Ron and Shen yields conditions for the construction of a multi-generated tight wavelet frame for L 2 ( R s ) based on a given refinable function. In this paper we show that the UEP can be generalized to locally compact abelian groups. In the general setting, the resulting frames are generated by modulates of a collection of functions; via the Fourier transform this corresponds to a generalized shift-invariant system. Both the stationary and the nonstationary case are covered. We provide general constructions, based on B-splines on the group itself as well as on characteristic functions on the dual group. Finally, we consider a number of concrete groups and derive explicit constructions of the resulting frames.

      PubDate: 2017-07-24T07:52:31Z
      DOI: 10.1016/j.acha.2017.07.004
  • Toward the classification of biangular harmonic frames
    • Authors: Peter G. Casazza; Amineh Farzannia; John I. Haas; Tin T. Tran
      Abstract: Publication date: Available online 28 June 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Peter G. Casazza, Amineh Farzannia, John I. Haas, Tin T. Tran
      Equiangular tight frames (ETFs) and biangular tight frames (BTFs) - sets of unit vectors with basis-like properties whose pairwise absolute inner products admit exactly one or two values, respectively - are useful for many applications. A well-understood class of ETFs are those which manifest as harmonic frames – vector sets defined in terms of the characters of finite abelian groups – because they are characterized by combinatorial objects called difference sets. This work is dedicated to the study of the underlying combinatorial structures of harmonic BTFs. We show that if a harmonic frame is generated by a divisible difference set, a partial difference set or by a special structure with certain Gauss summing properties – all three of which are generalizations of difference sets that fall under the umbrella term “bidifference set” – then it is either a BTF or an ETF. However, we also show that the relationship between harmonic BTFs and bidifference sets is not as straightforward as the correspondence between harmonic ETFs and difference sets, as there are examples of bidifference sets that do not generate harmonic BTFs. In addition, we study another class of combinatorial structures, the nested divisible difference sets, which yields an example of a harmonic BTF that is not generated by a bidifference set.

      PubDate: 2017-07-03T03:57:49Z
      DOI: 10.1016/j.acha.2017.06.004
  • A classification of continuous wavelet transforms in dimension three
    • Authors: Bradley Currey; Hartmut Führ; Vignon Oussa
      Abstract: Publication date: Available online 23 June 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Bradley Currey, Hartmut Führ, Vignon Oussa
      This paper presents a full catalogue, up to conjugacy and subgroups of finite index, of all matrix groups H < GL ( 3 , R ) that give rise to a continuous wavelet transform with associated irreducible quasi-regular representation. For each group in this class, coorbit theory allows to consistently define spaces of sparse signals, and to construct atomic decompositions converging simultaneously in a whole range of these spaces. As an application of the classification, we investigate the existence of compactly supported admissible vectors and atoms for the groups.

      PubDate: 2017-07-03T03:57:49Z
      DOI: 10.1016/j.acha.2017.06.003
  • Distributed learning with multi-penalty regularization
    • Authors: Zheng-Chu Guo; Shao-Bo Lin; Lei Shi
      Abstract: Publication date: Available online 21 June 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Zheng-Chu Guo, Shao-Bo Lin, Lei Shi
      In this paper, we study distributed learning with multi-penalty regularization based on a divide-and-conquer approach. Using Neumann expansion and a second order decomposition for difference of operator inverses approach, we derive optimal learning rates for distributed multi-penalty regularization in expectation. As a byproduct, we also deduce optimal learning rates for multi-penalty regularization, which was not given in the literature. These results are applied to the distributed manifold regularization and optimal learning rates are given.

      PubDate: 2017-06-21T17:19:46Z
      DOI: 10.1016/j.acha.2017.06.001
  • Bendlets: A second-order shearlet transform with bent elements
    • Authors: Christian Lessig; Philipp Petersen; Martin Schäfer
      Abstract: Publication date: Available online 16 June 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Christian Lessig, Philipp Petersen, Martin Schäfer
      We introduce bendlets, a shearlet-like system that is based on anisotropic scaling, translation, shearing, and bending of a compactly supported generator. With shearing being linear and bending quadratic in spatial coordinates, bendlets provide what we term a second-order shearlet system. As we show in this article, the decay rates of the associated transform enable the precise characterization of location, orientation and curvature of discontinuities in piecewise constant images. These results yield an improvement over existing directional representation systems where curvature only controls the constant of the decay rate of the transform. We also detail the construction of shearlet systems of arbitrary order. A practical implementation of bendlets is provided as an extension of the ShearLab toolbox, which we use to verify our theoretical classification results.

      PubDate: 2017-06-21T17:19:46Z
      DOI: 10.1016/j.acha.2017.03.005
  • Compressed sensing with local structure: Uniform recovery guarantees for
           the sparsity in levels class
    • Authors: Chen Li; Ben Adcock
      Abstract: Publication date: Available online 6 June 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Chen Li, Ben Adcock
      In compressed sensing, it is often desirable to consider signals possessing additional structure beyond sparsity. One such structured signal model – which forms the focus of this paper – is the local sparsity in levels class. This class has recently found applications in problems such as compressive imaging, multi-sensor acquisition systems and sparse regularization in inverse problems. In this paper we present uniform recovery guarantees for this class when the measurement matrix corresponds to a subsampled isometry. We do this by establishing a variant of the standard restricted isometry property for sparse in levels vectors, known as the restricted isometry property in levels. Interestingly, besides the usual log factors, our uniform recovery guarantees are simpler and less stringent than existing nonuniform recovery guarantees. For the particular case of discrete Fourier sampling with Haar wavelet sparsity, a corollary of our main theorem yields a new recovery guarantee which improves over the current state-of-the-art.

      PubDate: 2017-06-21T17:19:46Z
      DOI: 10.1016/j.acha.2017.05.006
  • Time-frequency analysis of bivariate signals
    • Authors: Julien Flamant; Nicolas Le Bihan; Pierre Chainais
      Abstract: Publication date: Available online 1 June 2017
      Source:Applied and Computational Harmonic Analysis
      Author(s): Julien Flamant, Nicolas Le Bihan, Pierre Chainais
      Many phenomena are described by bivariate signals or bidimensional vectors in applications ranging from radar to EEG, optics and oceanography. We show that an adequate quaternion Fourier transform permits to build relevant time-frequency representations of bivariate signals that naturally identify geometrical or polarization properties. First, a bivariate counterpart of the usual analytic signal of real signals is introduced, called the quaternion embedding of bivariate signals. Then two fundamental theorems ensure that a quaternion short term Fourier transform and a quaternion continuous wavelet transform are well defined and obey desirable properties such as conservation laws and reconstruction formulas. The resulting spectrograms and scalograms provide meaningful representations of both the time-frequency and geometrical/polarization content of the signal. Moreover the numerical implementation remains simply based on the use of FFT. A toolbox is available for reproducibility. Synthetic and real-world examples illustrate the relevance and efficiency of the proposed approach.

      PubDate: 2017-06-06T16:26:09Z
      DOI: 10.1016/j.acha.2017.05.007
  • Time-frequency and time-scale analysis of deformed stationary processes,
           with application to non-stationary sound modeling
    • Authors: Omer
      Abstract: Publication date: July 2017
      Source:Applied and Computational Harmonic Analysis, Volume 43, Issue 1
      Author(s): H. Omer, B. Torrésani
      A class of random non-stationary signals termed timbre×dynamics is introduced and studied. These signals are obtained by non-linear transformations of stationary random Gaussian signals, in such a way that the transformation can be approximated by translations in an appropriate representation domain. In such situations, approximate maximum likelihood estimation techniques can be derived, which yield simultaneous estimation of the transformation and the power spectrum of the underlying stationary signal. This paper focuses on the case of modulation and time warping of stationary signals, and proposes and studies estimation algorithms (based on time-frequency and time-scale representations respectively) for these quantities of interest. The proposed approach is validated on numerical simulations on synthetic signals, and examples on real life car engine sounds.

      PubDate: 2017-05-17T15:00:34Z
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