Authors:Xing Fu; Dachun Yang Pages: 1 - 37 Abstract: Publication date: January 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 1 Author(s): Xing Fu, Dachun Yang Let ( X , d , μ ) be a metric measure space of homogeneous type in the sense of R.R. Coifman and G. Weiss and H at 1 ( X ) be the atomic Hardy space. Via orthonormal bases of regular wavelets and spline functions recently constructed by P. Auscher and T. Hytönen, together with obtaining some crucial lower bounds for regular wavelets, the authors give an unconditional basis of H at 1 ( X ) and several equivalent characterizations of H at 1 ( X ) in terms of wavelets, which are proved useful.

Authors:Deguang Han; Fusheng Lv; Wenchang Sun Pages: 38 - 58 Abstract: Publication date: January 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 1 Author(s): Deguang Han, Fusheng Lv, Wenchang Sun In this paper, we study the feasibility and stability of recovering signals in finite-dimensional spaces from unordered partial frame coefficients. We prove that with an almost self-located robust frame, any signal except from a Lebesgue measure zero subset can be recovered from its unordered partial frame coefficients. However, the recovery is not necessarily stable with almost self-located robust frames. We propose a new class of frames, namely self-located robust frames, that ensures stable recovery for any input signal with unordered partial frame coefficients. In particular, the recovery is exact whenever the received unordered partial frame coefficients are noise-free. We also present some characterizations and constructions for (almost) self-located robust frames. Based on these characterizations and construction algorithms, we prove that any randomly generated frame is almost surely self-located robust. Moreover, frames generated with cube roots of different prime numbers are also self-located robust.

Authors:Jason D. McEwen; Claudio Durastanti; Yves Wiaux Pages: 59 - 88 Abstract: Publication date: January 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 1 Author(s): Jason D. McEwen, Claudio Durastanti, Yves Wiaux Scale-discretised wavelets yield a directional wavelet framework on the sphere where a signal can be probed not only in scale and position but also in orientation. Furthermore, a signal can be synthesised from its wavelet coefficients exactly, in theory and practice (to machine precision). Scale-discretised wavelets are closely related to spherical needlets (both were developed independently at about the same time) but relax the axisymmetric property of needlets so that directional signal content can be probed. Needlets have been shown to satisfy important quasi-exponential localisation and asymptotic uncorrelation properties. We show that these properties also hold for directional scale-discretised wavelets on the sphere and derive similar localisation and uncorrelation bounds in both the scalar and spin settings. Scale-discretised wavelets can thus be considered as directional needlets.

Authors:Matthieu Kowalski; Adrien Meynard; Hau-tieng Wu Pages: 89 - 122 Abstract: Publication date: January 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 1 Author(s): Matthieu Kowalski, Adrien Meynard, Hau-tieng Wu Motivated by the limitation of analyzing oscillatory signals composed of multiple components with fast-varying instantaneous frequency, we approach the time-frequency analysis problem by optimization. Based on the proposed adaptive harmonic model, the time-frequency representation of a signal is obtained by directly minimizing a functional, which involves few properties an “ideal time-frequency representation” should satisfy, for example, the signal reconstruction and concentrative time-frequency representation. FISTA (Fast Iterative Shrinkage-Thresholding Algorithm) is applied to achieve an efficient numerical approximation of the functional. We coin the algorithm as Time-frequency bY COnvex OptimizatioN (Tycoon). The numerical results confirm the potential of the Tycoon algorithm.

Authors:Rayan Saab; Rongrong Wang; Özgür Yılmaz Pages: 123 - 143 Abstract: Publication date: January 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 1 Author(s): Rayan Saab, Rongrong Wang, Özgür Yılmaz In this paper we study the quantization stage that is implicit in any compressed sensing signal acquisition paradigm. We propose using Sigma–Delta (ΣΔ) quantization and a subsequent reconstruction scheme based on convex optimization. We prove that the reconstruction error due to quantization decays polynomially in the number of measurements. Our results apply to arbitrary signals, including compressible ones, and account for measurement noise. Additionally, they hold for sub-Gaussian (including Gaussian and Bernoulli) random compressed sensing measurements, as well as for both high bit-depth and coarse quantizers, and they extend to 1-bit quantization. In the noise-free case, when the signal is strictly sparse we prove that by optimizing the order of the quantization scheme one can obtain root-exponential decay in the reconstruction error due to quantization.

Authors:Hong Chen; Yulong Wang Pages: 144 - 164 Abstract: Publication date: January 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 1 Author(s): Hong Chen, Yulong Wang The correntropy-induced loss (C-loss) has been employed in learning algorithms to improve their robustness to non-Gaussian noise and outliers recently. Despite its success on robust learning, only little work has been done to study the generalization performance of regularized regression with the C-loss. To enrich this theme, this paper investigates a kernel-based regression algorithm with the C-loss and ℓ 1 -regularizer in data dependent hypothesis spaces. The asymptotic learning rate is established for the proposed algorithm in terms of novel error decomposition and capacity-based analysis technique. The sparsity characterization of the derived predictor is studied theoretically. Empirical evaluations demonstrate its advantages over the related approaches.

Authors:Michaël Fanuel; Carlos M. Alaíz; Ángela Fernández; Johan A.K. Suykens Pages: 189 - 199 Abstract: Publication date: January 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 1 Author(s): Michaël Fanuel, Carlos M. Alaíz, Ángela Fernández, Johan A.K. Suykens We propose a framework for the visualization of directed networks relying on the eigenfunctions of the magnetic Laplacian, called here Magnetic Eigenmaps. The magnetic Laplacian is a complex deformation of the well-known combinatorial Laplacian. Features such as density of links and directionality patterns are revealed by plotting the phases of the first magnetic eigenvectors. An interpretation of the magnetic eigenvectors is given in connection with the angular synchronization problem. Illustrations of our method are given for both artificial and real networks.

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.

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.

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.

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.

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.

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 .

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 ) ) .

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.

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.

Authors:Galatia Cleanthous; Athanasios Georgiadis Morten Nielsen Abstract: Publication date: Available online 9 October 2017 Source:Applied and Computational Harmonic Analysis Author(s): Galatia Cleanthous, Athanasios G. Georgiadis, Morten Nielsen Anisotropic homogeneous mixed-norm Besov and Triebel-Lizorkin spaces are introduced and their properties are explored. A discrete adapted φ-transform decomposition is obtained. An associated class of almost diagonal operators is introduced and a boundedness result for such operators is obtained. Molecular decompositions for all the considered spaces are derived with the help of the algebra of almost diagonal operators. As an application, we obtain boundedness results on the considered spaces for Fourier multipliers and for pseudodifferential operators with suitable adapted homogeneous symbols using the molecular decomposition theory.

Authors:Robert Dahl Jacobsen; Jesper Møller; Morten Nielsen; Morten Grud Rasmussen Abstract: Publication date: Available online 2 October 2017 Source:Applied and Computational Harmonic Analysis Author(s): Robert Dahl Jacobsen, Jesper Møller, Morten Nielsen, Morten Grud Rasmussen Generalized sampling is a mathematical technique for obtaining approximations of signals with respect to different representations in a numerically stable manner. This can for example be relevant in processing MRI images, where hardware often enforces initial frequency measurements, but where a wavelet basis may be better suited for representing the image. Recently the theory of generalized sampling was extended to work with arbitrary patterns in R d . In this article we investigate how the choice of the probability distribution generating random sampling schemes in R 2 affects the numerical stability of generalized sampling.

Authors:M. Fanuel; J.A.K. Suykens Abstract: Publication date: Available online 20 September 2017 Source:Applied and Computational Harmonic Analysis Author(s): M. Fanuel, J.A.K. Suykens Deformations of the combinatorial Laplacian are proposed, which generalize several existing Laplacians. As particular cases of this construction, the dilation Laplacians are shown to be useful tools for ranking in directed networks of pairwise comparisons. In the case of a connected graph, the entries of the eigenvector of the dilation Laplacians with the smallest eigenvalue have all the same sign, and provide directly a ranking score of its nodes. The ranking method, phrased in terms of a group synchronization problem, is applied to artificial and real data, and its performance is compared with other ranking strategies. A main feature of this approach is the presence of a deformation parameter enabling the emphasis of the top-k objects in the ranking. Furthermore, inspired by these results, a family of random walks interpolating between the undirected random walk and the Pagerank random walk is also proposed.

Authors:Dimitrios Giannakis Abstract: Publication date: Available online 15 September 2017 Source:Applied and Computational Harmonic Analysis Author(s): Dimitrios Giannakis We develop a framework for dimension reduction, mode decomposition, and nonparametric forecasting of data generated by ergodic dynamical systems. This framework is based on a representation of the Koopman and Perron–Frobenius groups of unitary operators in a smooth orthonormal basis of the L 2 space of the dynamical system, acquired from time-ordered data through the diffusion maps algorithm. Using this representation, we compute Koopman eigenfunctions through a regularized advection-diffusion operator, and employ these eigenfunctions in dimension reduction maps with projectible dynamics and high smoothness for the given observation modality. In systems with pure point spectra, we construct a decomposition of the generator of the Koopman group into mutually commuting vector fields that transform naturally under changes of observation modality, which we reconstruct in data space through a representation of the pushforward map in the Koopman eigenfunction basis. We also establish a correspondence between Koopman operators and Laplace-Beltrami operators constructed from data in Takens delay-coordinate space, and use this correspondence to provide an interpretation of diffusion-mapped delay coordinates for this class of systems. Moreover, we take advantage of a special property of the Koopman eigenfunction basis, namely that the basis elements evolve as simple harmonic oscillators, to build nonparametric forecast models for probability densities and observables. In systems with more complex spectral behavior, including mixing systems, we develop a method inspired from time change in dynamical systems to transform the generator to a new operator with potentially improved spectral properties, and use that operator for vector field decomposition and nonparametric forecasting.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.