Abstract: Publication date: March 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 2 Author(s): I. Iglewska-Nowak Directional Poisson wavelets, being directional derivatives of Poisson kernel, are introduced on n-dimensional spheres. It is shown that, slightly modified and together with another wavelet family, they are an admissible wavelet pair according to the definition derived from the theory of approximate identities. We investigate some of the properties of directional Poisson wavelets, such as recursive formulae for their Fourier coefficients or explicit representations as functions of spherical variables (for some of the wavelets). We derive also an explicit formula for their Euclidean limits.

Abstract: Publication date: March 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 2 Author(s): Elena Cordero, Maurice de Gosson, Fabio Nicola One of the most popular time–frequency representations is certainly the Wigner distribution. To reduce the interferences coming from its quadratic nature, several related distributions have been proposed, among which is the so-called Born–Jordan distribution. It is well known that in the Born–Jordan distribution the ghost frequencies are in fact damped quite well, and the noise is in general reduced. However, the horizontal and vertical directions escape from this general smoothing effect, so that the interferences arranged along these directions are in general kept. Whereas these features are graphically evident on examples and heuristically well understood in the engineering community, there is no at present mathematical explanation of these phenomena, valid for general signals in L 2 and, more in general, in the space S ′ of temperate distributions. In the present note we provide such a rigorous study using the notion of wave-front set of a distribution. We use techniques from Time–frequency Analysis, such as the modulation and Wiener amalgam spaces, and also results of microlocal regularity of linear partial differential operators.

Abstract: Publication date: March 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 2 Author(s): Gil Shabat, Yaniv Shmueli, Yariv Aizenbud, Amir Averbuch Randomized algorithms play a central role in low rank approximations of large matrices. In this paper, the scheme of the randomized SVD is extended to a randomized LU algorithm. Several error bounds are introduced, that are based on recent results from random matrix theory related to subgaussian matrices. The bounds also improve the existing bounds of already known randomized SVD algorithm. The algorithm is fully parallelized and thus can utilize efficiently GPUs without any CPU–GPU data transfer. Numerical examples, which illustrate the performance of the algorithm and compare it to other decomposition methods, are presented.

Abstract: Publication date: March 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 2 Author(s): Sheng-Chun Yang, Hu-Jun Qian, Zhong-Yuan Lu An efficient calculation of NFFT (nonequispaced fast Fourier transforms) is always a challenging task in a variety of application areas, from medical imaging to radio astronomy to chemical simulation. In this article, a new theoretical derivation is proposed for NFFT based on gridding algorithm and new strategies are proposed for the implementation of both forward NFFT and its inverse on both CPU and GPU. The GPU-based version, namely CUNFFT, adopts CUDA (Compute Unified Device Architecture) technology, which supports a fine-grained parallel computing scheme. The approximation errors introduced in the algorithm are discussed with respect to different window functions. Finally, benchmark calculations are executed to illustrate the accuracy and performance of NFFT and CUNFFT. The results show that CUNFFT is not only with high accuracy, but also substantially faster than conventional NFFT on CPU.

Abstract: Publication date: March 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 2 Author(s): Douglas P. Hardin, Michael C. Northington, Alexander M. Powell A sharp version of the Balian–Low theorem is proven for the generators of finitely generated shift-invariant spaces. If generators { f k } k = 1 K ⊂ L 2 ( R d ) are translated along a lattice to form a frame or Riesz basis for a shift-invariant space V, and if V has extra invariance by a suitable finer lattice, then one of the generators f k must satisfy ∫ R d x f k ( x ) 2 d x = ∞ , namely, f k ˆ ∉ H 1 / 2 ( R d ) . Similar results are proven for frames of translates that are not Riesz bases without the assumption of extra lattice invariance. The best previously existing results in the literature give a notably weaker conclusion using the Sobolev space H d / 2 + ϵ ( R d ) ; our results provide an absolutely sharp improvement with H 1 / 2 ( R d ) . Our results are sharp in the sense that H 1 / 2 ( R d ) cannot be replaced by H s ( R d ) for any s < 1 / 2 .

Abstract: Publication date: March 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 2 Author(s): James Bremer We describe an algorithm for the numerical solution of second order linear ordinary differential equations in the high-frequency regime. It is based on the recent observation that solutions of equations of this type can be accurately represented using nonoscillatory phase functions. Unlike standard solvers for ordinary differential equations, the running time of our algorithm is independent of the frequency of oscillation of the solutions. We illustrate this and other properties of the method with several numerical experiments.

Abstract: Publication date: March 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 2 Author(s): D.W. Pravica, N. Randriampiry, M.J. Spurr For q > 1 , the nth order q-advanced spherical Bessel functions of the first kind, j n ( q ; t ) , are introduced. Smooth perturbations, H q ( ω ) , of the Haar wavelet are derived. The inverse Fourier transforms F − 1 [ j n ( q ; t ) ] ( ω ) are expressed in terms of the Jacobi theta function and are shown to give genesis to the q-advanced Legendre polynomials P ˜ n ( q ; ω ) . The wavelet F − 1 [ sin ( t ) j 0 ( q ; t ) ] ( ω ) is studied and shown to generate H q ( ω ) . For each n ≥ 1 , F − 1 [ j n ( q ; t ) ] ( ω ) is shown to be a Schwartz wavelet with vanishing jth moments for 0 ≤ j ≤ n − 1 and non-vanishing nth moment. Wavelet frame properties are developed. The family { 2 j / 2 H q ( 2 j ω − k ) j , k ∈ Z } is seen to be a nearly orthonormal frame for L 2 ( R ) and a perturbation of the Haar basis. The corresponding multiplicatively advanced differential equations (MADEs) satisfied by these new functions are presented. As the parameter q → 1 + , convergence of the q-advanced functions to their classical counterparts is shown. A q-Wallis formula is given. Symmetry of the Jacobi theta function is shown to preclude Gibb's type phenomena. A Schwartz function with lower moments vanishing is shown to be a mother wavelet for a frame generating L 2 ( R ) .

Abstract: Publication date: March 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 2 Author(s): Kenji Nakahira, Atsushi Miyamoto Wavelets on graphs have been studied for the past few years, and in particular, several approaches have been proposed to design wavelet transforms on hierarchical graphs. Although such methods are computationally efficient and easy to implement, their frames are highly restricted. In this paper, we propose a general framework for the design of wavelet transforms on hierarchical graphs. Our design is guaranteed to be a Parseval tight frame, which preserves the l 2 norm of any input signals. To demonstrate the potential usefulness of our approach, we perform several experiments, in which we learn a wavelet frame based on our framework, and show, in inpainting experiments, that it performs better than a Haar-like hierarchical wavelet transform and a learned treelet. We also show with category theory that the algebraic properties of the proposed transform have a strong relationship with those of the hierarchical graph that represents the structure of the given data.

Abstract: Publication date: March 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 2 Author(s): Gilles Puy, Nicolas Tremblay, Rémi Gribonval, Pierre Vandergheynst We study the problem of sampling k-bandlimited signals on graphs. We propose two sampling strategies that consist in selecting a small subset of nodes at random. The first strategy is non-adaptive, i.e., independent of the graph structure, and its performance depends on a parameter called the graph coherence. On the contrary, the second strategy is adaptive but yields optimal results. Indeed, no more than O ( k log ( k ) ) measurements are sufficient to ensure an accurate and stable recovery of all k-bandlimited signals. This second strategy is based on a careful choice of the sampling distribution, which can be estimated quickly. Then, we propose a computationally efficient decoder to reconstruct k-bandlimited signals from their samples. We prove that it yields accurate reconstructions and that it is also stable to noise. Finally, we conduct several experiments to test these techniques.

Abstract: Publication date: March 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 2 Author(s): Matthew Fickus, John Jasper, Dustin G. Mixon, Jesse D. Peterson, Cody E. Watson An equiangular tight frame (ETF) is a set of unit vectors whose coherence achieves the Welch bound, and so is as incoherent as possible. Though they arise in many applications, only a few methods for constructing them are known. Motivated by the connection between real ETFs and graph theory, we introduce the notion of ETFs that are symmetric about their centroid. We then discuss how well-known constructions, such as harmonic ETFs and Steiner ETFs, can have centroidal symmetry. Finally, we establish a new equivalence between centroid-symmetric real ETFs and certain types of strongly regular graphs (SRGs). Together, these results give the first proof of the existence of certain SRGs, as well as the disproofs of the existence of others.

Abstract: Publication date: March 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 2 Author(s): Zhiqiang Xu The paper presents several results that address a fundamental question in low-rank matrix recovery: how many measurements are needed to recover low-rank matrices' We begin by investigating the complex matrices case and show that 4 n r − 4 r 2 generic measurements are both necessary and sufficient for the recovery of rank-r matrices in C n × n . Thus, we confirm a conjecture which is raised by Eldar, Needell and Plan for the complex case. We next consider the real case and prove that the bound 4 n r − 4 r 2 is tight provided n = 2 k + r , k ∈ Z + . Motivated by Vinzant's work [19], we construct 11 matrices in R 4 × 4 by computer random search and prove they define injective measurements on rank-1 matrices in R 4 × 4 . This disproves the conjecture raised by Eldar, Needell and Plan for the real case. Finally, we use the results in this paper to investigate the phase retrieval by projection and show fewer than 2 n − 1 orthogonal projections are possible for the recovery of x ∈ R n from the norm of them, which gives a negative answer for a question raised in [1].

Abstract: Publication date: Available online 2 February 2018 Source:Applied and Computational Harmonic Analysis Author(s): Xiaodong Li, Shuyang Ling, Thomas Strohmer, Ke Wei We study the question of reconstructing two signals f and g from their convolution y = f ⁎ g . This problem, known as blind deconvolution, pervades many areas of science and technology, including astronomy, medical imaging, optics, and wireless communications. A key challenge of this intricate non-convex optimization problem is that it might exhibit many local minima. We present an efficient numerical algorithm that is guaranteed to recover the exact solution, when the number of measurements is (up to log-factors) slightly larger than the information-theoretical minimum, and under reasonable conditions on f and g . The proposed regularized gradient descent algorithm converges at a geometric rate and is provably robust in the presence of noise. To the best of our knowledge, our algorithm is the first blind deconvolution algorithm that is numerically efficient, robust against noise, and comes with rigorous recovery guarantees under certain subspace conditions. Moreover, numerical experiments do not only provide empirical verification of our theory, but they also demonstrate that our method yields excellent performance even in situations beyond our theoretical framework.

Abstract: Publication date: Available online 31 January 2018 Source:Applied and Computational Harmonic Analysis Author(s): Ronen Talmon, Hau-Tieng Wu The analysis of data sets arising from multiple sensors has drawn significant research attention over the years. Traditional methods, including kernel-based methods, are typically incapable of capturing nonlinear geometric structures. We introduce a latent common manifold model underlying multiple sensor observations for the purpose of multimodal data fusion. A method based on alternating diffusion is presented and analyzed; we provide theoretical analysis of the method under the latent common manifold model. To exemplify the power of the proposed framework, experimental results in several applications are reported.

Abstract: Publication date: Available online 3 January 2018 Source:Applied and Computational Harmonic Analysis Author(s): Francesca Bartolucci, Filippo De Mari, Ernesto De Vito, Francesca Odone We prove that the unitary affine Radon transform intertwines the quasi-regular representation of a class of semidirect products, built by shearlet dilation groups and translations, and the tensor product of a standard wavelet representation with a wavelet-like representation. This yields a formula for shearlet coefficients that involves only integral transforms applied to the affine Radon transform of the signal, thereby opening new perspectives in the inversion of the Radon transform.

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:Maria Charina; Vladimir Yu. Protasov Abstract: Publication date: Available online 8 December 2017 Source:Applied and Computational Harmonic Analysis Author(s): Maria Charina, Vladimir Yu. Protasov This paper presents a detailed regularity analysis of multivariate refinable functions with general dilation matrices, with emphasis on the anisotropic case. In the univariate setting, the smoothness of refinable functions is well understood by means of the matrix approach. In the multivariate setting, this approach has been extended only to the special case of isotropic refinement with the dilation matrix all of whose eigenvalues are equal in the absolute value. The general anisotropic case has resisted to be fully understood: the matrix approach can determine whether a refinable function belongs to C ( R s ) or L p ( R s ) , 1 ≤ p < ∞ , but its Hölder regularity remained mysteriously unattainable. We show how to compute the Hölder regularity in C ( R s ) or L p ( R s ) , 1 ≤ p < ∞ . In the anisotropic case, our expression for the exact Hölder exponent of a refinable function reflects the impact of the variable moduli of the eigenvalues of the corresponding dilation matrix. In the isotropic case, our results reduce to the well-known facts from the literature. We also analyze the higher regularity of anistropic refinable functions. We illustrate our results with several examples.

Authors:Bin Han; Michelle Michelle Abstract: Publication date: Available online 6 December 2017 Source:Applied and Computational Harmonic Analysis Author(s): Bin Han, Michelle Michelle Riesz wavelets in the Sobolev space H m ( R ) with m ∈ N ∪ { 0 } , whose mth-order derivatives are orthogonal among different levels, are of particular interest and importance in computational mathematics, due to their many desirable properties such as small condition numbers and sparse stiffness matrices. We call such Riesz wavelets in the Sobolev space H m ( R ) as mth-order derivative-orthogonal Riesz wavelets. In this paper we shall comprehensively study and completely characterize all compactly supported mth-order derivative-orthogonal Riesz wavelets in the Sobolev space H m ( R ) . More precisely, from any given compactly supported refinable vector function ϕ = ( ϕ 1 , … , ϕ r ) T in H m ( R ) satisfying the refinement equation ϕ ˆ ( 2 ξ ) = a ˆ ( ξ ) ϕ ˆ ( ξ ) for some r × r matrix a ˆ of 2π-periodic trigonometric polynomials, we prove that there exists a compactly supported mth-order derivative-orthogonal Riesz wavelet in H m ( R ) , which is derived from ϕ through the refinable structure, if and only if the refinable vector function ϕ has stable integer shifts and the filter a has at least order 2m sum rules. This double order of sum rules over the smoothness order m is surprising but is necessary for constructing mth-order derivative-orthogonal Riesz wavelets in H m ( R ) . Then we shall present several examples of such derivative-orthogonal spline Riesz wavelets with short support derived from B-splines and Hermite splines. To illustrate the developed theory and its potential usefulness, we shall apply our constructed such mth-order derivative-orthogonal Riesz wavelets for the numerical solutions of differential equations such as Sturm-Liouville equations and biharmonic equations. Our constructed derivative-orthogonal spline Riesz wavelets on the interval [ 0 , 1 ] have a simple structure with only one boundary wavelet at each endpoint and can easily handle different types of boundary conditions. The resulting coefficient matrices are sparse and have very small condition numbers with some examples even having the optimal condition number one.

Authors:Jérémie Bigot; Paul Escande; Pierre Weiss Abstract: Publication date: Available online 5 December 2017 Source:Applied and Computational Harmonic Analysis Author(s): Jérémie Bigot, Paul Escande, Pierre Weiss We provide a new estimator of integral operators with smooth kernels, obtained from a set of scattered and noisy impulse responses. The proposed approach relies on the formalism of smoothing in reproducing kernel Hilbert spaces and on the choice of an appropriate regularization term that takes the smoothness of the operator into account. It is numerically tractable in very large dimensions. We study the estimator's robustness to noise and analyze its approximation properties with respect to the size and the geometry of the dataset. In addition, we show minimax optimality of the proposed estimator.

Authors:Lutz Abstract: Publication date: Available online 2 December 2017 Source:Applied and Computational Harmonic Analysis Author(s): Lutz Kämmerer The paper discusses the construction of high dimensional spatial discretizations for arbitrary multivariate trigonometric polynomials, where the frequency support of the trigonometric polynomial is known. We suggest a construction based on the union of several rank-1 lattices as sampling scheme. We call such schemes multiple rank-1 lattices. This approach automatically makes available a fast discrete Fourier transform (FFT) on the data. The key objective of the construction of spatial discretizations is the unique reconstruction of the trigonometric polynomial using the sampling values at the sampling nodes. We develop different construction methods for multiple rank-1 lattices that allow for this unique reconstruction. The symbol M denotes the total number of sampling nodes within the multiple rank-1 lattice. In addition, we assume that the multivariate trigonometric polynomial is a linear combination of T trigonometric monomials. The ratio of the number M of sampling points that are sufficient for the unique reconstruction to the number T of distinct monomials is called oversampling factor in this context. The presented construction methods for multiple rank-1 lattices allow for estimates of this number M. Roughly speaking, the oversampling factor M / T is independent of the spatial dimension and, with high probability, only logarithmic in T, which is much better than the oversampling factor that is expected for a sampling method that uses one single rank-1 lattice. The newly developed approaches for the construction of spatial discretizations are probabilistic methods. The arithmetic complexity of these algorithms depend only linearly on the spatial dimension and, with high probability, only linearly on T up to some logarithmic factors. Furthermore, we analyze the computational complexities of the resulting FFT algorithms, that exploits the structure of the suggested multiple rank-1 lattice spatial discretizations, in detail and obtain upper bounds in O ( M log M ) , where the constants depend only linearly on the spatial dimension. With high probability, we construct spatial discretizations where M / T ≤ C log T holds, which implies that the complexity of the corresponding FFT converts to O ( T log 2 T ) .

Authors:Matthew Fickus; John Jasper; Dustin G. Mixon; Jesse D. Peterson; Cody E. Watson Abstract: Publication date: Available online 2 December 2017 Source:Applied and Computational Harmonic Analysis Author(s): Matthew Fickus, John Jasper, Dustin G. Mixon, Jesse D. Peterson, Cody E. Watson An equiangular tight frame (ETF) is a type of optimal packing of lines in a finite-dimensional Hilbert space. ETFs arise in various applications, such as waveform design for wireless communication, compressed sensing, quantum information theory and algebraic coding theory. In a recent paper, signature matrices of ETFs were constructed from abelian distance regular covers of complete graphs. We extend this work, constructing ETF synthesis operators from abelian generalized quadrangles, and vice versa. This produces a new infinite family of complex ETFs as well as a new proof of the existence of certain generalized quadrangles. This work involves designing matrices whose entries are polynomials over a finite abelian group. As such, it is related to the concept of a polyphase matrix of a finite filter bank.

Authors:Zheng-Chu Guo; Lei Shi Abstract: Publication date: Available online 2 December 2017 Source:Applied and Computational Harmonic Analysis Author(s): Zheng-Chu Guo, Lei Shi In this paper, we consider the coefficient-based regularized kernel regression. In form, the algorithm minimizes a least-square loss functional adding a coefficient-based ℓ 2 − penalty term over a linear span of features generated by a kernel function. We study the asymptotic behavior of the algorithm under the framework of learning theory. Compared with the classical kernel ridge regression (KRR), the algorithm under consideration does not require the kernel to be positive semi-definite and hence provides a simple paradigm for designing indefinite kernel methods. Another important feature of this algorithm is that, it can improve the saturation effect suffered by KRR. In fact, this algorithm can be viewed as a variant of KRR based on high-order kernels, which provides an effective way to improve the saturation effect. In this paper, we establish a nice convergence analysis by means of a novel integral operator approach. Then we obtain the optimal rates for the algorithm in a mini-max sense. We thus demonstrate that the performance of the concerned algorithm is comparable with that of KRR, and even outperforms it when the regression function has higher regularities.

Authors:Alex Iosevich; Chun-Kit Lai; Azita Mayeli Abstract: Publication date: Available online 24 November 2017 Source:Applied and Computational Harmonic Analysis Author(s): Alex Iosevich, Chun-Kit Lai, Azita Mayeli Let q ≥ 2 be an integer, and F q d , d ≥ 1 , be the vector space over the cyclic space F q . The purpose of this paper is two-fold. First, we obtain sufficient conditions on E ⊂ F q d such that the inverse Fourier transform of 1 E generates a tight wavelet frame in L 2 ( F q d ) . We call these sets (tight) wavelet frame sets. The conditions are given in terms of multiplicative and translational tilings, which is analogous with Theorem 1.1 ([20]) by Wang in the setting of finite fields. In the second part of the paper, we exhibit a constructive method for obtaining tight wavelet frame sets in F q d , d ≥ 2 , q an odd prime and q ≡ 3 (mod 4).

Authors:Tabea Méndez; Andreas Müller Abstract: Publication date: Available online 21 November 2017 Source:Applied and Computational Harmonic Analysis Author(s): Tabea Méndez, Andreas Müller The image registration problem on a group G asks, given two functions f , g : G → R that are related by a translation f ( x ) = g ( s − 1 ⋅ x ) by an element s ∈ G , to find s. For abelian groups, the Fourier transform provides an elegant and fast solution to this problem. For nonabelian groups, the problem is much more involved. This paper shows how this applied problem can shed light on the constructions of noncommutative harmonic analysis, in particular the theory of Gelfand pairs. The abstract theory then suggests a novel two-step approach to solving such problems. The Gelfand pair ( SO ( 3 ) , SO ( 2 ) ) then provides us with an intuitive solution of the registration problem for images on S 2 .

Authors:Yun-Zhang Li; Jian-Ping Zhang Abstract: Publication date: Available online 14 November 2017 Source:Applied and Computational Harmonic Analysis Author(s): Yun-Zhang Li, Jian-Ping Zhang Mixed Oblique Extension Principles (MOEP) provide an important method to construct affine dual frames from refinable functions. This paper addresses MOEP under the setting of reducing subspaces of L 2 ( R d ) . We obtain an MOEP for (non)homogeneous affine dual frames and (non)homogeneous affine Parseval frames.

Authors:Nicholas F. Marshall; Matthew J. Hirn Abstract: Publication date: Available online 7 November 2017 Source:Applied and Computational Harmonic Analysis Author(s): Nicholas F. Marshall, Matthew J. Hirn We consider a collection of n points in R d measured at m times, which are encoded in an n × d × m data tensor. Our objective is to define a single embedding of the n points into Euclidean space which summarizes the geometry as described by the data tensor. In the case of a fixed data set, diffusion maps and related graph Laplacian methods define such an embedding via the eigenfunctions of a diffusion operator constructed on the data. Given a sequence of m measurements of n points, we introduce the notion of time coupled diffusion maps which have natural geometric and probabilistic interpretations. To frame our method in the context of manifold learning, we model evolving data as samples from an underlying manifold with a time-dependent metric, and we describe a connection of our method to the heat equation on such a manifold.

Authors:Richard Mikaël Slevinsky Abstract: Publication date: Available online 7 November 2017 Source:Applied and Computational Harmonic Analysis Author(s): Richard Mikaël Slevinsky A rapid transformation is derived between spherical harmonic expansions and their analogues in a bivariate Fourier series. The change of basis is described in two steps: firstly, expansions in normalized associated Legendre functions of all orders are converted to those of order zero and one; then, these intermediate expressions are re-expanded in trigonometric form. The first step proceeds with a butterfly factorization of the well-conditioned matrices of connection coefficients. The second step proceeds with fast orthogonal polynomial transforms via hierarchically off-diagonal low-rank matrix decompositions. Total pre-computation requires at best O ( n 3 log n ) flops; and, asymptotically optimal execution time of O ( n 2 log 2 n ) is rigorously proved via connection to Fourier integral operators.

Abstract: Publication date: Available online 7 November 2017 Source:Applied and Computational Harmonic Analysis Author(s): I. Iglewska–Nowak We construct spherical wavelets based on approximate identities that are directional, i.e. not rotation-invariant, and have an adaptive angular selectivity. The problem of how to find a proper representation of distinct kinds of details of real images, ranging from highly directional to fully isotropic ones, was quite intensively studied for the case of signals over the Euclidean space. However, the present paper is the first attempt to deal with this task in the case of spherical signals. A multiselectivity scheme, similar to that proposed for R 2 -functions, is presented.

Authors:M.S. Kotzagiannidis; P.L. Dragotti Abstract: Publication date: Available online 18 October 2017 Source:Applied and Computational Harmonic Analysis Author(s): M.S. Kotzagiannidis, P.L. Dragotti With the objective of employing graphs toward a more generalized theory of signal processing, we present a novel sampling framework for (wavelet-)sparse signals defined on circulant graphs which extends basic properties of Finite Rate of Innovation (FRI) theory to the graph domain, and can be applied to arbitrary graphs via suitable approximation schemes. At its core, the introduced Graph-FRI-framework states that any K-sparse signal on the vertices of a circulant graph can be perfectly reconstructed from its dimensionality-reduced representation in the graph spectral domain, the Graph Fourier Transform (GFT), of minimum size 2K. By leveraging the recently developed theory of e-splines and e-spline wavelets on graphs, one can decompose this graph spectral transformation into the multiresolution low-pass filtering operation with a graph e-spline filter, with subsequent transformation to the spectral graph domain; this allows to infer a distinct sampling pattern, and, ultimately, the structure of an associated coarsened graph, which preserves essential properties of the original, including circularity and, where applicable, the graph generating set.

Authors:Rui Zhang; Song Li Abstract: Publication date: Available online 18 October 2017 Source:Applied and Computational Harmonic Analysis Author(s): Rui Zhang, Song Li In this paper, we present a unified analysis of RIP bounds for sparse signals recovery by using ℓ p minimization with 0 < p ≤ 1 and provide optimal RIP bounds which can guarantee sparse signals recovery via ℓ p minimization for each p ∈ ( 0 , 1 ] in both noiseless and noisy settings. It is shown that if the measurement matrix Φ satisfies the RIP condition δ 2 k < δ ( p ) , where δ ( p ) will be specified in the context, then all k-sparse signals x can be recovered stably via the constrained ℓ p minimization based on b = Φ x + z , where z is one type of noise. Furthermore, we show that for any ϵ > 0 , δ 2 k < δ ( p ) + ϵ is not sufficient to guarantee the exact recovery of all k-sparse signals. We also apply the results to the cases of low rank matrix recovery and the reconstruction of sparse vectors in terms of redundant dictionary. In particular, when p = 1 , the corresponding constant δ ( 1 ) = 2 2 , this sharp bound was shown by T. Cai and A. Zhang in their work. Thus, we give a complete characterization for optimal RIP bounds δ 2 k via ℓ p minimization for k-sparse signals recovery with 0 < p ≤ 1 . Our approaches are based on some ideas of sparse representation of a given polytope, which was firstly used by T. Cai and A. Zhang.

Authors:Charles K. Chui; H.N. Mhaskar Abstract: Publication date: Available online 16 October 2017 Source:Applied and Computational Harmonic Analysis Author(s): Charles K. Chui, H.N. Mhaskar Motivated by the interest of observing the growth of cancer cells among normal living cells and exploring how galaxies and stars are truly formed, the objective of this paper is to introduce a rigorous and effective method for counting point-masses, determining their spatial locations, and computing their attributes. Based on computation of Hermite moments that are Fourier-invariant, our approach facilitates the processing of both spatial and Fourier data in any dimension.

Authors:M.S. Kotzagiannidis; P.L. Dragotti Abstract: Publication date: Available online 14 October 2017 Source:Applied and Computational Harmonic Analysis Author(s): M.S. Kotzagiannidis, P.L. Dragotti We present novel families of wavelets and associated filterbanks for the analysis and representation of functions defined on circulant graphs. In this work, we leverage the inherent vanishing moment property of the circulant graph Laplacian operator, and by extension, the e-graph Laplacian, which is established as a parameterization of the former with respect to the degree per node, for the design of vertex-localized and critically-sampled higher-order graph (e-)spline wavelet filterbanks, which can reproduce and annihilate classes of (exponential) polynomial signals on circulant graphs. In addition, we discuss similarities and analogies of the detected properties and resulting constructions with splines and spline wavelets in the Euclidean domain. Ultimately, we consider generalizations to arbitrary graphs in the form of graph approximations, with focus on graph product decompositions. In particular, we proceed to show how the use of graph products facilitates a multi-dimensional extension of the proposed constructions and properties.

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.