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

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

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

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

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

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

Authors:Aline Bonami; Abderrazek Karoui Pages: 361 - 377 Abstract: Publication date: May 2017 Source:Applied and Computational Harmonic Analysis, Volume 42, Issue 3 Author(s): Aline Bonami, Abderrazek Karoui Recently, there is a growing interest in the spectral approximation by the Prolate Spheroidal Wave Functions (PSWFs) ψ n , c , c > 0 . This is due to the promising new contributions of these functions in various classical as well as emerging applications from Signal Processing, Geophysics, Numerical Analysis, etc. The PSWFs form a basis with remarkable properties not only for the space of band-limited functions with bandwidth c, but also for the Sobolev space H s ( [ − 1 , 1 ] ) . The quality of the spectral approximation and the choice of the parameter c when approximating a function in H s ( [ − 1 , 1 ] ) by its truncated PSWFs series expansion, are the main issues. By considering a function f ∈ H s ( [ − 1 , 1 ] ) as the restriction to [ − 1 , 1 ] of an almost time-limited and band-limited function, we try to give satisfactory answers to these two issues. Also, we illustrate the different results of this work by some numerical examples.

Authors:A. Aldroubi; C. Cabrelli; U. Molter; S. Tang Pages: 378 - 401 Abstract: Publication date: May 2017 Source:Applied and Computational Harmonic Analysis, Volume 42, Issue 3 Author(s): A. Aldroubi, C. Cabrelli, U. Molter, S. Tang Let Y = { f ( i ) , Af ( i ) , … , A l i f ( i ) : i ∈ Ω } , where A is a bounded operator on ℓ 2 ( I ) . The problem under consideration is to find necessary and sufficient conditions on A , Ω , { l i : i ∈ Ω } in order to recover any f ∈ ℓ 2 ( I ) from the measurements Y. This is the so-called dynamical sampling problem in which we seek to recover a function f by combining coarse samples of f and its futures states A l f . We completely solve this problem in finite dimensional spaces, and for a large class of self adjoint operators in infinite dimensional spaces. In the latter case, although Y can be complete, using the Müntz–Szász Theorem we show it can never be a basis. We can also show that, when Ω is finite, Y is not a frame except for some very special cases. The existence of these special cases is derived from Carleson's Theorem for interpolating sequences in the Hardy space H 2 ( D ) . Finally, using the recently proved Kadison–Singer/Feichtinger theorem we show that the set obtained by normalizing the vectors of Y can never be a frame when Ω is finite.

Authors:Clarice Poon Pages: 402 - 451 Abstract: Publication date: May 2017 Source:Applied and Computational Harmonic Analysis, Volume 42, Issue 3 Author(s): Clarice Poon Many of the applications of compressed sensing have been based on variable density sampling, where certain sections of the sampling coefficients are sampled more densely. Furthermore, it has been observed that these sampling schemes are dependent not only on sparsity but also on the sparsity structure of the underlying signal. This paper extends the result of Adcock, Hansen, Poon and Roman (arXiv:1302.0561, 2013) [2] to the case where the sparsifying system forms a tight frame. By dividing the sampling coefficients into levels, our main result will describe how the amount of subsampling in each level is determined by the local coherences between the sampling and sparsifying operators and the localized level sparsities – the sparsity in each level under the sparsifying operator.

Authors:Jianbin Yang; Dominik Stahl; Zuowei Shen Pages: 480 - 507 Abstract: Publication date: May 2017 Source:Applied and Computational Harmonic Analysis, Volume 42, Issue 3 Author(s): Jianbin Yang, Dominik Stahl, Zuowei Shen In real world applications many signals contain singularities, like edges in images. Recent wavelet frame based approaches were successfully applied to reconstruct scattered data from such functions while preserving these features. In this paper we present a recent approach which determines the approximant from shift invariant subspaces by minimizing an ℓ 1 -regularized least squares problem which makes additional use of the wavelet frame transform in order to preserve sharp edges. We give a detailed analysis of this approach, i.e., how the approximation error behaves dependent on data density and noise level. Moreover, a link to wavelet frame based image restoration models is established and the convergence of these models is analyzed. In the end, we present some numerical examples, for instance how to apply this approach to handle coarse-grained models in molecular dynamics.

Authors:Ben Adcock; Milana Gataric; Anders C. Hansen Pages: 508 - 535 Abstract: Publication date: May 2017 Source:Applied and Computational Harmonic Analysis, Volume 42, Issue 3 Author(s): Ben Adcock, Milana Gataric, Anders C. Hansen In this paper, we consider the problem of recovering a compactly supported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly. We do this by using the concept of weighted Fourier frames. A seminal result of Beurling shows that sampling points give rise to a classical Fourier frame provided they are relatively separated and of sufficient density. However, this result does not allow for arbitrary clustering of sampling points, as is often the case in practice. Whilst keeping the density condition sharp and dimension independent, our first result removes the separation condition and shows that density alone suffices. However, this result does not lead to estimates for the frame bounds. A known result of Gröchenig provides explicit estimates, but only subject to a density condition that deteriorates linearly with dimension. In our second result we improve these bounds by reducing the dimension dependence. In particular, we provide explicit frame bounds which are dimensionless for functions having compact support contained in a sphere. Next, we demonstrate how our two main results give new insight into a reconstruction algorithm—based on the existing generalized sampling framework—that allows for stable and quasi-optimal reconstruction in any particular basis from a finite collection of samples. Finally, we construct sufficiently dense sampling schemes that are often used in practice—jittered, radial and spiral sampling schemes—and provide several examples illustrating the effectiveness of our approach when tested on these schemes.

Authors:Elena A. Lebedeva Pages: 536 - 549 Abstract: Publication date: May 2017 Source:Applied and Computational Harmonic Analysis, Volume 42, Issue 3 Author(s): Elena A. Lebedeva An inequality refining the lower bound for a periodic (Breitenberger) uncertainty constant is proved for a wide class of functions. A connection of uncertainty constants for periodic and non-periodic functions is extended to this class. A particular minimization problem for a non-periodic (Heisenberg) uncertainty constant is studied.

Authors:Paul Hand Pages: 550 - 562 Abstract: Publication date: May 2017 Source:Applied and Computational Harmonic Analysis, Volume 42, Issue 3 Author(s): Paul Hand Consider the task of recovering an unknown n-vector from phaseless linear measurements. This nonconvex problem may be convexified into a semidefinite rank-one matrix recovery problem, known as PhaseLift. Under a linear number of Gaussian measurements, PhaseLift recovers the unknown vector exactly with high probability. Under noisy measurements, the solution to a variant of PhaseLift has error proportional to the ℓ 1 norm of the noise. In the present paper, we study the robustness of this variant of PhaseLift to gross, arbitrary corruptions. We prove that PhaseLift can tolerate noise and a small, fixed fraction of gross errors, even in the highly underdetermined regime where there are only O ( n ) measurements. The lifted phase retrieval problem can be viewed as a rank-one robust Principal Component Analysis (PCA) problem under generic rank-one measurements. From this perspective, the proposed convex program is simpler than the semidefinite version of the sparse-plus-low-rank formulation standard in the robust PCA literature. Specifically, the rank penalization through a trace term is unnecessary, and the resulting optimization program has no parameters that need to be chosen.

Authors:B.A. Siudeja Pages: 167 - 198 Abstract: Publication date: March 2017 Source:Applied and Computational Harmonic Analysis, Volume 42, Issue 2 Author(s): B.A. Siudeja We prove sharp upper bounds for sums of eigenvalues (and other spectral functionals) of Laplace-like operators, including bi-Laplacians and fractional Laplacians. We show that among linear images of a highly symmetric domain, our spectral functionals are maximal on the original domain. We exploit the symmetries of the domain, and the operator, avoiding the necessity of finding good test functions for variational problems. This is especially important for fractional Laplacians, since exact solutions are not even known on intervals, making it hard to find good test functions. To achieve our goals we generalize tight p-fusion frames, to extract the best possible geometric results for domains with isometry groups admitting tight p-frames. Any such group generates a tight p-fusion frame via conjugations of a fixed projection matrix. We show that generalized tight p-frames can also be obtained by conjugations of an arbitrary rectangular matrix, with the frame constant depending on the singular values of the matrix.

Authors:Martin Storath; Laurent Demaret; Peter Massopust Pages: 199 - 223 Abstract: Publication date: March 2017 Source:Applied and Computational Harmonic Analysis, Volume 42, Issue 2 Author(s): Martin Storath, Laurent Demaret, Peter Massopust We propose a signal analysis tool based on the sign (or the phase) of complex wavelet coefficients, which we call a signature. The signature is defined as the fine-scale limit of the signs of a signal's complex wavelet coefficients. We show that the signature equals zero at sufficiently regular points of a signal whereas at salient features, such as jumps or cusps, it is non-zero. At such feature points, the orientation of the signature in the complex plane can be interpreted as an indicator of local symmetry and antisymmetry. We establish that the signature rotates in the complex plane under fractional Hilbert transforms. We show that certain random signals, such as white Gaussian noise and Brownian motions, have a vanishing signature. We derive an appropriate discretization and show the applicability to signal analysis.

Authors:Yiming Ying; Ding-Xuan Zhou Pages: 224 - 244 Abstract: Publication date: March 2017 Source:Applied and Computational Harmonic Analysis, Volume 42, Issue 2 Author(s): Yiming Ying, Ding-Xuan Zhou In this paper, we consider unregularized online learning algorithms in a Reproducing Kernel Hilbert Space (RKHS). Firstly, we derive explicit convergence rates of the unregularized online learning algorithms for classification associated with a general α-activating loss (see Definition 1 below). Our results extend and refine the results in [30] for the least square loss and the recent result [3] for the loss function with a Lipschitz-continuous gradient. Moreover, we establish a very general condition on the step sizes which guarantees the convergence of the last iterate of such algorithms. Secondly, we establish, for the first time, the convergence of the unregularized pairwise learning algorithm with a general loss function and derive explicit rates under the assumption of polynomially decaying step sizes. Concrete examples are used to illustrate our main results. The main techniques are tools from convex analysis, refined inequalities of Gaussian averages [5], and an induction approach.

Authors:Gitta Kutyniok; Philipp Petersen Pages: 245 - 293 Abstract: Publication date: March 2017 Source:Applied and Computational Harmonic Analysis, Volume 42, Issue 2 Author(s): Gitta Kutyniok, Philipp Petersen We analyze the detection and classification of singularities of functions f = χ S , where S ⊂ R d and d = 2 , 3 . It will be shown how the set ∂S can be extracted by a continuous shearlet transform associated with compactly supported shearlets. Furthermore, if ∂S is a d − 1 dimensional piecewise smooth manifold with d = 2 or 3, we will classify smooth and non-smooth components of ∂S. This improves previous results given for shearlet systems with a certain band-limited generator, since the estimates we derive are uniform. Moreover, we will show that our bounds are optimal. Along the way, we also obtain novel results on the characterization of wavefront sets in 3 dimensions by compactly supported shearlets. Finally, geometric properties of ∂S such as curvature are described in terms of the continuous shearlet transform of f.

Authors:Giovanni S. Alberti; Habib Ammari Pages: 319 - 349 Abstract: Publication date: March 2017 Source:Applied and Computational Harmonic Analysis, Volume 42, Issue 2 Author(s): Giovanni S. Alberti, Habib Ammari The main focus of this work is the reconstruction of the signals f and g i , i = 1 , … , N , from the knowledge of their sums h i = f + g i , under the assumption that f and the g i 's can be sparsely represented with respect to two different dictionaries A f and A g . This generalizes the well-known “morphological component analysis” to a multi-measurement setting. The main result of the paper states that f and the g i 's can be uniquely and stably reconstructed by finding sparse representations of h i for every i with respect to the concatenated dictionary [ A f , A g ] , provided that enough incoherent measurements g i are available. The incoherence is measured in terms of their mutual disjoint sparsity. This method finds applications in the reconstruction procedures of several hybrid imaging inverse problems, where internal data are measured. These measurements usually consist of the main unknown multiplied by other unknown quantities, and so the disjoint sparsity approach can be directly applied. As an example, we show how to apply the method to the reconstruction in quantitative photoacoustic tomography, also in the case when the Grüneisen parameter, the optical absorption and the diffusion coefficient are all unknown.

Authors:Dan Edidin Pages: 350 - 359 Abstract: Publication date: March 2017 Source:Applied and Computational Harmonic Analysis, Volume 42, Issue 2 Author(s): Dan Edidin We characterize collections of orthogonal projections for which it is possible to reconstruct a vector from the magnitudes of the corresponding projections. As a result we are able to show that in an M-dimensional real vector space a vector can be reconstructed from the magnitudes of its projections onto a generic collection of N ≥ 2 M − 1 subspaces. We also show that this bound is sharp when N = 2 k + 1 . The results of this paper answer a number of questions raised in [5].

Abstract: Publication date: Available online 18 May 2017 Source:Applied and Computational Harmonic Analysis Author(s): John Aston, Florent Autin, Gerda Claeskens, Jean-Marc Freyermuth, Christophe Pouet We present a novel method for detecting some structural characteristics of multidimensional functions. We consider the multidimensional Gaussian white noise model with an anisotropic estimand. Using the relation between the Sobol decomposition and the geometry of multidimensional wavelet basis we can build test statistics for any of the Sobol functional components. We assess the asymptotical minimax optimality of these test statistics and show that they are optimal in presence of anisotropy with respect to the newly determined minimax rates of separation. An appropriate combination of these test statistics allows to test some general structural characteristics such as the atomic dimension or the presence of some variables. Numerical experiments show the potential of our method for studying spatio-temporal processes.

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.

Authors:Jian Zhai; Bowen Zheng Abstract: Publication date: Available online 15 May 2017 Source:Applied and Computational Harmonic Analysis Author(s): Jian Zhai, Bowen Zheng This paper studies the Strichartz estimate of the solutions for the free fractional Schrödinger equation with spatial variable coefficient, which arises from the integrable model of the inhomogeneous spherically symmetric HFSS. It is different from the classical fractional Schrödinger equation in which the Strichartz estimates have been proved. We establish some harmonic analysis tools such as the equivalence of the Sobolev type spaces W ˙ σ , 2 and H ˙ σ and Hankel type Littlewood-Paley theory. By utilizing these ingredients, we prove the Strichartz estimate.

Authors:Kirill Serkh Abstract: Publication date: Available online 15 May 2017 Source:Applied and Computational Harmonic Analysis Author(s): Kirill Serkh In this paper we investigate the solution of boundary value problems on polygonal domains for elliptic partial differential equations. Previously, we observed that when the boundary value problems are formulated as boundary integral equations of classical potential theory, the solutions are representable by series of elementary functions, to arbitrary order, for all but finitely many values of the angles. Here, we extend this observation to all values of the angles. We show that the solutions near corners are representable, to arbitrary order, by linear combinations of certain non-integer powers and non-integer powers multiplied by logarithms.

Authors:Ben Adcock; Milana Gataric; José Luis Romero Abstract: Publication date: Available online 12 May 2017 Source:Applied and Computational Harmonic Analysis Author(s): Ben Adcock, Milana Gataric, José Luis Romero We study the problem of recovering an unknown compactly-supported multivariate function from samples of its Fourier transform that are acquired nonuniformly, i.e. not necessarily on a uniform Cartesian grid. Reconstruction problems of this kind arise in various imaging applications, where Fourier samples are taken along radial lines or spirals for example. Specifically, we consider finite-dimensional reconstructions, where a limited number of samples is available, and investigate the rate of convergence of such approximate solutions and their numerical stability. We show that the proportion of Fourier samples that allow for stable approximations of a given numerical accuracy is independent of the specific sampling geometry and is therefore universal for different sampling scenarios. This allows us to relate both sufficient and necessary conditions for different sampling setups and to exploit several results that were previously available only for very specific sampling geometries. The results are obtained by developing: (i) a transference argument for different measures of the concentration of the Fourier transform and Fourier samples; (ii) frame bounds valid up to the critical sampling density, which depend explicitly on the sampling set and the spectrum. As an application, we identify sufficient and necessary conditions for stable and accurate reconstruction of algebraic polynomials or wavelet coefficients from nonuniform Fourier data.

Authors:Sylvain Durand Abstract: Publication date: Available online 20 April 2017 Source:Applied and Computational Harmonic Analysis Author(s): Sylvain Durand This article addresses the issue of designing bases for L 2 ( R 2 ) that are generated by translations, rotations and dilations of a single mother wavelet ψ. We show how this construction can be simplified by setting an odd number of directions and by choosing properly the phase of the Fourier transform of ψ. A large part of the article is devoted to the proof of theorems that give sufficient conditions for ψ to generate a Riesz sequence and a Riesz basis for L 2 ( R 2 ) . An example of Riesz sequence whose restriction to each scale is orthonormal is set. Theoretical results are confirmed by numerical experiments where a discrete directional wavelet transform is introduced.

Authors:Joachim Toft Abstract: Publication date: Available online 20 April 2017 Source:Applied and Computational Harmonic Analysis Author(s): Joachim Toft We extend Feichtinger's minimality property on the smallest non-trivial time-frequency shift invariant Banach space, to the quasi-Banach case. Analogous properties are deduced for certain matrix spaces. We use these results to prove that the pseudo-differential operator Op ( a ) is a Schatten-q operator from M ∞ to M p and r-nuclear operator from M ∞ to M r when a ∈ M r for suitable p, q and r in ( 0 , ∞ ] .

Authors:Jian-Feng Cai; Tianming Wang; Ke Wei Abstract: Publication date: Available online 19 April 2017 Source:Applied and Computational Harmonic Analysis Author(s): Jian-Feng Cai, Tianming Wang, Ke Wei A spectrally sparse signal of order r is a mixture of r damped or undamped complex sinusoids. This paper investigates the problem of reconstructing spectrally sparse signals from a random subset of n regular time domain samples, which can be reformulated as a low rank Hankel matrix completion problem. We introduce an iterative hard thresholding (IHT) algorithm and a fast iterative hard thresholding (FIHT) algorithm for efficient reconstruction of spectrally sparse signals via low rank Hankel matrix completion. Theoretical recovery guarantees have been established for FIHT, showing that O ( r 2 log 2 ( n ) ) number of samples are sufficient for exact recovery with high probability. Empirical performance comparisons establish significant computational advantages for IHT and FIHT. In particular, numerical simulations on 3D arrays demonstrate the capability of FIHT on handling large and high-dimensional real data.

Authors:Yingzhou Li; Haizhao Yang; Lexing Ying Abstract: Publication date: Available online 18 April 2017 Source:Applied and Computational Harmonic Analysis Author(s): Yingzhou Li, Haizhao Yang, Lexing Ying This paper introduces the multidimensional butterfly factorization as a data-sparse representation of multidimensional kernel matrices that satisfy the complementary low-rank property. This factorization approximates such a kernel matrix of size N × N with a product of O ( log N ) sparse matrices, each of which contains O ( N ) nonzero entries. We also propose efficient algorithms for constructing this factorization when either (i) a fast algorithm for applying the kernel matrix and its adjoint is available or (ii) every entry of the kernel matrix can be evaluated in O ( 1 ) operations. For the kernel matrices of multidimensional Fourier integral operators, for which the complementary low-rank property is not satisfied due to a singularity at the origin, we extend this factorization by combining it with either a polar coordinate transformation or a multiscale decomposition of the integration domain to overcome the singularity. Numerical results are provided to demonstrate the efficiency of the proposed algorithms.

Authors:Yuanxin Li; Yuejie Chi Abstract: Publication date: Available online 1 April 2017 Source:Applied and Computational Harmonic Analysis Author(s): Yuanxin Li, Yuejie Chi We consider simultaneously identifying the membership and locations of point sources that are convolved with different band-limited point spread functions, from the observation of their superpositions. This problem arises in three-dimensional super-resolution single-molecule imaging, neural spike sorting, multi-user channel identification, among other applications. We propose a novel algorithm, based on convex programming, and establish its near-optimal performance guarantee for exact recovery in the noise-free setting by exploiting the spectral sparsity of the point source models as well as the incoherence between point spread functions. Furthermore, robustness of the recovery algorithm in the presence of bounded noise is also established. Numerical examples are provided to demonstrate the effectiveness of the proposed approach.

Authors:Fredrik Andersson; Marcus Carlsson Abstract: Publication date: Available online 1 April 2017 Source:Applied and Computational Harmonic Analysis Author(s): Fredrik Andersson, Marcus Carlsson We develop fixed-point algorithms for the approximation of structured matrices with rank penalties. In particular we use these fixed-point algorithms for making approximations by sums of exponentials, i.e., frequency estimation. For the basic formulation of the fixed-point algorithm we show that it converges to the solution of a related minimization problem, namely the one obtained by replacing the original objective function with its convex envelope and keeping the structured matrix constraint unchanged. It often happens that this solution agrees with the solution to the original minimization problem, and we provide a simple criterion for when this is true. We also provide more general fixed-point algorithms that can be used to treat the problems of making weighted approximations by sums of exponentials given equally or unequally spaced sampling. We apply the method to the case of missing data, although the above mentioned convergence results do not hold in this case. However, it turns out that the method often gives perfect reconstruction (up to machine precision) in such cases. We also discuss multidimensional extensions, and illustrate how the proposed algorithms can be used to recover sums of exponentials in several variables, but when samples are available only along a curve.

Authors:Holger Boche; Volker Pohl Abstract: Publication date: Available online 21 March 2017 Source:Applied and Computational Harmonic Analysis Author(s): Holger Boche, Volker Pohl This paper studies the problem of calculating the finite Hilbert transform f ˜ = H f of functions f from the set B of continuous functions with a continuous conjugate f ˜ based on discrete samples of f. It is shown that all sampling based linear approximations which satisfy three natural axioms diverge strongly on B in the uniform norm. More precisely, we consider sequences { H N } N ∈ N of linear approximation operators H N : B → B such that the calculation of H N f is based on discrete samples of f and which satisfies two additional natural axioms. We show that for all such sequences { H N } N ∈ N there always exists an f ∈ B such that lim N → ∞ ‖ H N f ‖ ∞ = + ∞ . Moreover, it is shown that on the subset B 1 / 2 of all f ∈ B with finite Dirichlet energy an even larger class of sampling based approximation sequences diverges weakly, i.e. for all such sequences there always exists an f ∈ B 1 / 2 such that lim sup N → ∞ ‖ H N f ‖ ∞ = + ∞ .

Authors:Bin Dong Abstract: Publication date: May 2017 Source:Applied and Computational Harmonic Analysis, Volume 42, Issue 3 Author(s): Bin Dong In this paper, we introduce a new (constructive) characterization of tight wavelet frames on non-flat domains in both continuum setting, i.e. on manifolds, and discrete setting, i.e. on graphs; we discuss how fast tight wavelet frame transforms can be computed and how they can be effectively used to process graph data. We start with defining the quasi-affine systems on a given manifold M . The quasi-affine system is formed by generalized dilations and shifts of a finite collection of wavelet functions Ψ : = { ψ j : 1 ≤ j ≤ r } ⊂ L 2 ( R ) . We further require that ψ j is generated by some refinable function ϕ with mask a j . We present the condition needed for the masks { a j : 0 ≤ j ≤ r } , as well as regularity conditions needed for ϕ and ψ j , so that the associated quasi-affine system generated by Ψ is a tight frame for L 2 ( M ) . The condition needed for the masks is a simple set of algebraic equations which are not only easy to verify for a given set of masks { a j } , but also make the construction of { a j } entirely painless. Then, we discuss how the transition from the continuum (manifolds) to the discrete setting (graphs) can be naturally done. In order for the proposed discrete tight wavelet frame transforms to be useful in applications, we show how the transforms can be computed efficiently and accurately by proposing the fast tight wavelet frame transforms for graph data (WFTG). Finally, we consider two specific applications of the proposed WFTG: graph data denoising and semi-supervised clustering. Utilizing the sparse representation provided by the WFTG, we propose ℓ 1 -norm based optimization models on graphs for denoising and semi-supervised clustering. On one hand, our numerical results show significant advantage of the WFTG over the spectral graph wavelet transform (SGWT) by [1] for both applications. On the other hand, numerical experiments on two real data sets show that the proposed semi-supervised clustering model using the WFTG is overall competitive with the state-of-the-art methods developed in the literature of high-dimensional data classification, and is superior to some of these methods.

Authors:Axel Flinth; Gitta Kutyniok Abstract: Publication date: Available online 10 March 2017 Source:Applied and Computational Harmonic Analysis Author(s): Axel Flinth, Gitta Kutyniok Applications such as wireless communications require efficient sensing techniques of signals with the a priori knowledge of those being lattice-valued. In this paper, we study the impact of this prior information on compressed sensing methodologies, and introduce and analyze PROMP (“PReprojected Orthogonal Matching Pursuit”) as a novel algorithmic approach for sparse recovery of lattice-valued signals. More precisely, we first show that the straightforward approach to project the solution of Basis Pursuit onto a prespecified lattice does not improve the performance of Basis Pursuit in this situation. We then introduce PROMP as a novel sparse recovery algorithm for lattice-valued signals which has very low computational complexity, alongside a detailed mathematical analysis of its performance and stability under noise. Finally, we present numerical experiments which show that PROMP outperforms standard sparse recovery approaches in the lattice-valued signal regime.

Authors:Rodrigo Mendoza-Smith; Jared Tanner Abstract: Publication date: Available online 9 March 2017 Source:Applied and Computational Harmonic Analysis Author(s): Rodrigo Mendoza-Smith, Jared Tanner We introduce two new algorithms, Serial- ℓ 0 and Parallel- ℓ 0 for solving a large underdetermined linear system of equations y = A x ∈ R m when it is known that x ∈ R n has at most k < m nonzero entries and that A is the adjacency matrix of an unbalanced left d-regular expander graph. The matrices in this class are sparse and allow a highly efficient implementation. A number of algorithms have been designed to work exclusively under this setting, composing the branch of combinatorial compressed-sensing (CCS). Serial- ℓ 0 and Parallel- ℓ 0 iteratively minimise ‖ y − A x ˆ ‖ 0 by successfully combining two desirable features of previous CCS algorithms: the coordinate selection strategy of ER [1] for updating x ˆ , and the parallel updating mechanism of SMP [2]. We are able to link these elements and guarantee convergence in O ( d n log k ) operations by introducing a randomized scaling of columns in A, with scaling chosen independent of the measured vector. We also empirically observe that the recovery properties of Serial- ℓ 0 and Parallel- ℓ 0 degrade gracefully as the signal is allowed to have its non-zero values chosen adversarially. Moreover, we observe Serial- ℓ 0 and Parallel- ℓ 0 to be able to solve large scale problems with a larger fraction of nonzeros than other algorithms when the number of measurements is substantially less than the signal length; in particular, they are able to reliably solve for a k-sparse vector x ∈ R n from m expander measurements with n / m = 10 3 and k / m up to four times greater than what is achievable by ℓ 1 -regularization from dense Gaussian measurements. Additionally, due to their low computational complexity, Serial- ℓ 0 and Parallel- ℓ 0 are observed to be able to solve large problems sizes in substantially less time than other algorithms for compressed sensing. In particular, Parallel- ℓ 0 is structured to take advantage of massively parallel architectures.

Authors:M. Bodner; J. Patera; M. Szajewska Abstract: Publication date: Available online 6 March 2017 Source:Applied and Computational Harmonic Analysis Author(s): M. Bodner, J. Patera, M. Szajewska A method for the decomposition of data functions sampled on a finite fragment of triangular lattice is described for the cases of lattices of any density corresponding to the simple Lie group G ( 2 ) . Its main advantage is the fact that the decomposition matrix needs to be calculated only once for arbitrary sets of data sampled on the same set of discrete points. The decomposition matrix refers to lattice of any density that carries data.

Authors:Jackie Abstract: Publication date: March 2017 Source:Applied and Computational Harmonic Analysis, Volume 42, Issue 2 Author(s): Jackie Ma In this paper we study the general reconstruction of a compactly supported function from its Fourier coefficients using compactly supported shearlet systems. We assume that only finitely many Fourier samples of the function are accessible and based on this finite collection of measurements an approximation is sought in a finite dimensional shearlet reconstruction space. We analyze this sampling and reconstruction process by a recently introduced method called generalized sampling. In particular by studying the stable sampling rate of generalized sampling we then show stable recovery of the signal is possible using an almost linear rate. Furthermore, we compare the result to the previously obtained rates for wavelets.

Authors:Se Rim Park; Soheil Kolouri; Shinjini Kundu; Gustavo K. Rohde Abstract: Publication date: Available online 22 February 2017 Source:Applied and Computational Harmonic Analysis Author(s): Se Rim Park, Soheil Kolouri, Shinjini Kundu, Gustavo K. Rohde Discriminating data classes emanating from sensors is an important problem with many applications in science and technology. We describe a new transform for pattern representation that interprets patterns as probability density functions, and has special properties with regards to classification. The transform, which we denote as the Cumulative Distribution Transform (CDT), is invertible, with well defined forward and inverse operations. We show that it can be useful in ‘parsing out’ variations (confounds) that are ‘Lagrangian’ (displacement and intensity variations) by converting these to ‘Eulerian’ (intensity variations) in transform space. This conversion is the basis for our main result that describes when the CDT can allow for linear classification to be possible in transform space. We also describe several properties of the transform and show, with computational experiments that used both real and simulated data, that the CDT can help render a variety of real world problems simpler to solve.

Authors:Zhiqiang Abstract: Publication date: Available online 16 February 2017 Source:Applied and Computational Harmonic Analysis 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].

Authors:Paul L. Butzer; Gerhard Schmeisser; Rudolf L. Stens Abstract: Publication date: Available online 10 February 2017 Source:Applied and Computational Harmonic Analysis Author(s): Paul L. Butzer, Gerhard Schmeisser, Rudolf L. Stens The general Poisson summation formula of harmonic analysis and analytic number theory can be regarded as a quadrature formula with remainder. The purpose of this investigation is to give estimates for this remainder based on the classical modulus of smoothness and on an appropriate metric for describing the distance of a function from a Bernstein space. Moreover, to be more flexible when measuring the smoothness of a function, we consider Riesz derivatives of fractional order. It will be shown that the use of the above metric in connection with fractional order derivatives leads to estimates for the remainder, which are best possible with respect to the order and the constants.

Authors:Fanuel Carlos; Johan A.K. Suykens Abstract: Publication date: Available online 1 February 2017 Source:Applied and Computational Harmonic Analysis 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:Charles K. Chui; H.N. Mhaskar; Xiaosheng Zhuang Abstract: Publication date: Available online 26 January 2017 Source:Applied and Computational Harmonic Analysis Author(s): Charles K. Chui, H.N. Mhaskar, Xiaosheng Zhuang This paper is an extension of the previous work of Chui et al. (2015) [4], not only from numeric data to include non-numeric data as in that paper, but also from undirected graphs to directed graphs (called digraphs, for simplicity). Besides theoretical development, this paper introduces effective mathematical tools in terms of certain data-dependent orthogonal systems for function representation and analysis directly on the digraphs. In addition, this paper also includes algorithmic development and discussion of various experimental results on such data-sets as CORA, Proposition, and Wiki-votes.

Authors:Jerrod Ankenman; William Leeb Abstract: Publication date: Available online 25 January 2017 Source:Applied and Computational Harmonic Analysis Author(s): Jerrod Ankenman, William Leeb This paper concerns two related problems in the analysis of data matrices whose rows and columns are equipped with tree metrics. First is the problem of recovering a matrix that has been corrupted by additive noise. Under the assumption that the clean matrix exhibits a specific regularity condition, known as the mixed Hölder condition, we adapt the well-known Donoho–Johnstone wavelet shrinkage methods from classical nonparametric statistics to obtain estimators that are within a logarithmic factor of the minimax error rate with respect to mean squared error loss. The second part of this paper develops a theory of Besov spaces on products of tree geometries. We show that matrices with small Besov norm can be written as a sum of a mixed Hölder matrix and a matrix with small support. Such decompositions are known as Calderón–Zygmund decompositions and are of general interest in harmonic analysis. The decompositions we establish impose fewer conditions on the function with small support than previous decompositions of this type while maintaining the same guarantees on the mixed Hölder matrix. As such, they are applicable to a greater variety of matrices and should find use in many data organization problems. As part of our analysis, we provide characterizations of the underlying Besov spaces using wavelets and other multiscale difference operators that are analogous to those from the classical Euclidean theory.

Authors:Alexander Cloninger; Stefan Steinerberger Abstract: Publication date: Available online 17 January 2017 Source:Applied and Computational Harmonic Analysis 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:Haizhao Yang Abstract: Publication date: Available online 5 January 2017 Source:Applied and Computational Harmonic Analysis Author(s): Haizhao Yang Synchrosqueezed transforms are non-linear processes for a sharpened time–frequency representation of wave-like components. They are efficient tools for identifying and analyzing wave-like components from their superposition. This paper is concerned with the statistical properties of compactly supported synchrosqueezed transforms for wave-like components embedded in a generalized Gaussian random process in multidimensional spaces. Guided by the theoretical analysis of these properties, new numerical implementations are proposed to reduce the noise fluctuations of these transforms on noisy data. A MATLAB package SynLab together with several heavily noisy examples is provided to support these theoretical claims.

Authors:Robert Beinert; Gerlind Plonka Abstract: Publication date: Available online 4 January 2017 Source:Applied and Computational Harmonic Analysis Author(s): Robert Beinert, Gerlind Plonka Considering the ambiguousness of the discrete-time phase retrieval problem to recover a signal from its Fourier intensities, one can ask the question: what additional information about the unknown signal do we need to select the correct solution within the large solution set? Based on a characterization of the occurring ambiguities, we investigate different a priori conditions in order to reduce the number of ambiguities or even to receive a unique solution. Particularly, if we have access to additional magnitudes of the unknown signal in the time domain, we can show that almost all signals with finite support can be uniquely recovered. Moreover, we prove that an analogous result can be obtained by exploiting additional phase information.

Authors:Mateo Díaz; Mauricio Junca; Felipe Rincón; Mauricio Velasco Abstract: Publication date: Available online 3 January 2017 Source:Applied and Computational Harmonic Analysis Author(s): Mateo Díaz, Mauricio Junca, Felipe Rincón, Mauricio Velasco Compressed sensing is a technique for recovering an unknown sparse signal from a small number of linear measurements. When the measurement matrix is random, the number of measurements required for perfect recovery exhibits a phase transition: there is a threshold on the number of measurements after which the probability of exact recovery quickly goes from very small to very large. In this work we are able to reduce this threshold by incorporating statistical information about the data we wish to recover. Our algorithm works by minimizing a suitably weighted ℓ 1 -norm, where the weights are chosen so that the expected statistical dimension of the corresponding descent cone is minimized. We also provide new discrete-geometry-based Monte Carlo algorithms for computing intrinsic volumes of such descent cones, allowing us to bound the failure probability of our methods.