Authors:Wojciech Czaja; Benjamin Manning; James M. Murphy; Kevin Stubbs Pages: 1 - 21 Abstract: Publication date: July 2018 Source:Applied and Computational Harmonic Analysis, Volume 45, Issue 1 Author(s): Wojciech Czaja, Benjamin Manning, James M. Murphy, Kevin Stubbs We develop a theory of discrete directional Gabor frames for functions defined on the d-dimensional Euclidean space. Our construction incorporates the concept of ridge functions into the theory of isotropic Gabor systems, in order to develop an anisotropic Gabor system with strong directional sensitivity. We present sufficient conditions on a window function g and a sampling set Λ ω for the corresponding directional Gabor system { g m , t , u } ( m , t , u ) ∈ Λ ω to form a discrete frame. Explicit estimates on the frame bounds are developed. A numerical implementation of our scheme is also presented, and is shown to perform competitively in compression and denoising schemes against state-of-the-art multiscale and anisotropic methods, particularly for images with significant texture components.

Authors:Karin Schnass Pages: 22 - 58 Abstract: Publication date: July 2018 Source:Applied and Computational Harmonic Analysis, Volume 45, Issue 1 Author(s): Karin Schnass In this work we show that iterative thresholding and K means (ITKM) algorithms can recover a generating dictionary with K atoms from noisy S sparse signals up to an error ε ˜ as long as the initialisation is within a convergence radius, that is up to a log K factor inversely proportional to the dynamic range of the signals, and the sample size is proportional to K log K ε ˜ − 2 . The results are valid for arbitrary target errors if the sparsity level is of the order of the square root of the signal dimension d and for target errors down to K − ℓ if S scales as S ≤ d / ( ℓ log K ) .

Authors:Lotfi Hermi; Naoki Saito Pages: 59 - 83 Abstract: Publication date: July 2018 Source:Applied and Computational Harmonic Analysis, Volume 45, Issue 1 Author(s): Lotfi Hermi, Naoki Saito In this article we prove the existence, uniqueness, and simplicity of a negative eigenvalue for a class of integral operators whose kernel is of the form x − y ρ , 0 < ρ ≤ 1 , x , y ∈ [ − a , a ] . We also provide two different ways of producing recursive formulas for the Rayleigh functions (i.e., recursion formulas for power sums) of the eigenvalues of this integral operator when ρ = 1 , providing means of approximating this negative eigenvalue. These methods offer recursive procedures for dealing with the eigenvalues of a one-dimensional Laplacian with non-local boundary conditions which commutes with an integral operator having a harmonic kernel. The problem emerged in recent work by one of the authors [48]. We also discuss extensions in higher dimensions and links with distance matrices.

Authors:Tyrus Berry; John Harlim Pages: 84 - 119 Abstract: Publication date: July 2018 Source:Applied and Computational Harmonic Analysis, Volume 45, Issue 1 Author(s): Tyrus Berry, John Harlim Recently, the theory of diffusion maps was extended to a large class of local kernels with exponential decay which were shown to represent various Riemannian geometries on a data set sampled from a manifold embedded in Euclidean space. Moreover, local kernels were used to represent a diffeomorphism H between a data set and a feature of interest using an anisotropic kernel function, defined by a covariance matrix based on the local derivatives D H . In this paper, we generalize the theory of local kernels to represent degenerate mappings where the intrinsic dimension of the data set is higher than the intrinsic dimension of the feature space. First, we present a rigorous method with asymptotic error bounds for estimating D H from the training data set and feature values. We then derive scaling laws for the singular values of the local linear structure of the data, which allows the identification the tangent space and improved estimation of the intrinsic dimension of the manifold and the bandwidth parameter of the diffusion maps algorithm. Using these numerical tools, our approach to feature identification is to iterate the diffusion map with appropriately chosen local kernels that emphasize the features of interest. We interpret the iterated diffusion map (IDM) as a discrete approximation to an intrinsic geometric flow which smoothly changes the geometry of the data space to emphasize the feature of interest. When the data lies on a manifold which is a product of the feature manifold with an irrelevant manifold, we show that the IDM converges to the quotient manifold which is isometric to the feature manifold, thereby eliminating the irrelevant dimensions. We will also demonstrate empirically that if we apply the IDM to features which are not a quotient of the data manifold, the algorithm identifies an intrinsically lower-dimensional set embedding of the data which better represents the features.

Authors:Yongsheng Han; Ji Li; Lesley A. Ward Pages: 120 - 169 Abstract: Publication date: July 2018 Source:Applied and Computational Harmonic Analysis, Volume 45, Issue 1 Author(s): Yongsheng Han, Ji Li, Lesley A. Ward In this paper, we first show that the remarkable orthonormal wavelet expansion for L p constructed recently by Auscher and Hytönen also converges in certain spaces of test functions and distributions. Hence we establish the theory of product Hardy spaces on spaces X ˜ = X 1 × X 2 × ⋅ ⋅ ⋅ × X n , where each factor X i is a space of homogeneous type in the sense of Coifman and Weiss. The main tool we develop is the Littlewood–Paley theory on X ˜ , which in turn is a consequence of a corresponding theory on each factor space. We define the square function for this theory in terms of the wavelet coefficients. The Hardy space theory developed in this paper includes product H p , the dual CMO p of H p with the special case BMO = CMO 1 , and the predual VMO of H 1 . We also use the wavelet expansion to establish the Calderón–Zygmund decomposition for product H p , and deduce an interpolation theorem. We make no additional assumptions on the quasi-metric or the doubling measure for each factor space, and thus we extend to the full generality of product spaces of homogeneous type the aspects of both one-parameter and multiparameter theory involving the Littlewood–Paley theory and function spaces. Moreover, our methods would be expected to be a powerful tool for developing wavelet analysis on spaces of homogeneous type.

Authors:Yann Traonmilin; Rémi Gribonval Pages: 170 - 205 Abstract: Publication date: July 2018 Source:Applied and Computational Harmonic Analysis, Volume 45, Issue 1 Author(s): Yann Traonmilin, Rémi Gribonval Many inverse problems in signal processing deal with the robust estimation of unknown data from underdetermined linear observations. Low-dimensional models, when combined with appropriate regularizers, have been shown to be efficient at performing this task. Sparse models with the 1-norm or low rank models with the nuclear norm are examples of such successful combinations. Stable recovery guarantees in these settings have been established using a common tool adapted to each case: the notion of restricted isometry property (RIP). In this paper, we establish generic RIP-based guarantees for the stable recovery of cones (positively homogeneous model sets) with arbitrary regularizers. These guarantees are illustrated on selected examples. For block structured sparsity in the infinite-dimensional setting, we use the guarantees for a family of regularizers which efficiency in terms of RIP constant can be controlled, leading to stronger and sharper guarantees than the state of the art.

Authors:Chun-Kit Lai; Shidong Li; Daniel Mondo Pages: 206 - 215 Abstract: Publication date: July 2018 Source:Applied and Computational Harmonic Analysis, Volume 45, Issue 1 Author(s): Chun-Kit Lai, Shidong Li, Daniel Mondo Solving compressed sensing problems relies on the properties of sparse signals. It is commonly assumed that the sparsity s needs to be less than one half of the spark of the sensing matrix A, and then the unique sparsest solution exists, and is recoverable by ℓ 1 -minimization or related procedures. We discover, however, a measure theoretical uniqueness exists for nearly spark-level sparsity from compressed measurements A x = b . Specifically, suppose A is of full spark with m rows, and suppose m 2 < s < m . Then the solution to A x = b is unique for x with ‖ x ‖ 0 ≤ s up to a set of measure 0 in every s-sparse plane. This phenomenon is observed and confirmed by an ℓ 1 -tail minimization procedure, which recovers sparse signals uniquely with s > m 2 in thousands and thousands of random tests. We further show instead that the mere ℓ 1 -minimization would actually fail if s > m 2 even from the same measure theoretical point of view.

Authors:Joel Laity; Barak Shani Pages: 216 - 232 Abstract: Publication date: July 2018 Source:Applied and Computational Harmonic Analysis, Volume 45, Issue 1 Author(s): Joel Laity, Barak Shani A function f : Z n → C can be represented as a linear combination f ( x ) = ∑ α ∈ Z n f ˆ ( α ) χ α , n ( x ) where f ˆ is the (discrete) Fourier transform of f. Clearly, the basis { χ α , n ( x ) : = exp ( 2 π i α x / n ) } depends on the value n. We show that if f has “large” Fourier coefficients, then the function f ˜ : Z m → C , given by f ˜ ( x ) = { f ( x ) when 0 ≤ x < min ( n , m ) , 0 otherwise , also has “large” coefficients. Moreover, they are all contained in a “small” interval around ⌊ m n α ⌉ for each α ∈ Z n such that f ˆ ( α ) is large. One can use this result to recover the large Fourier coefficients of a function f by redefining it on a convenient domain. One can also use this result to reprove a result by Morillo and Ràfols: single-bit functions, defined over any domain, are Fourier concentrated.

Authors:Mark Bodner; Jiří Patera; Marzena Szajewska Pages: 233 - 238 Abstract: Publication date: July 2018 Source:Applied and Computational Harmonic Analysis, Volume 45, Issue 1 Author(s): Mark Bodner, Jiří Patera, Marzena 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 applies to lattice of any density that carries data.

Authors:Roy R. Lederman; Ronen Talmon Pages: 509 - 536 Abstract: Publication date: May 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 3 Author(s): Roy R. Lederman, Ronen Talmon One of the challenges in data analysis is to distinguish between different sources of variability manifested in data. In this paper, we consider the case of multiple sensors measuring the same physical phenomenon, such that the properties of the physical phenomenon are manifested as a hidden common source of variability (which we would like to extract), while each sensor has its own sensor-specific effects (hidden variables which we would like to suppress); the relations between the measurements and the hidden variables are unknown. We present a data-driven method based on alternating products of diffusion operators and show that it extracts the common source of variability. Moreover, we show that it extracts the common source of variability in a multi-sensor experiment as if it were a standard manifold learning algorithm used to analyze a simple single-sensor experiment, in which the common source of variability is the only source of variability.

Authors:Uri Shaham; Alexander Cloninger; Ronald R. Coifman Pages: 537 - 557 Abstract: Publication date: May 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 3 Author(s): Uri Shaham, Alexander Cloninger, Ronald R. Coifman We discuss approximation of functions using deep neural nets. Given a function f on a d-dimensional manifold Γ ⊂ R m , we construct a sparsely-connected depth-4 neural network and bound its error in approximating f. The size of the network depends on dimension and curvature of the manifold Γ, the complexity of f, in terms of its wavelet description, and only weakly on the ambient dimension m. Essentially, our network computes wavelet functions, which are computed from Rectified Linear Units (ReLU).

Authors:Alessandro Cardinali; Guy P. Nason Pages: 558 - 583 Abstract: Publication date: May 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 3 Author(s): Alessandro Cardinali, Guy P. Nason Methods designed for second-order stationary time series can be misleading when applied to nonstationary series, often resulting in inaccurate models and poor forecasts. Hence, testing time series stationarity is important especially with the advent of the ‘data revolution’ and the recent explosion in the number of nonstationary time series analysis tools. Most existing stationarity tests rely on a single basis. We propose new tests that use nondecimated basis libraries which permit discovery of a wider range of nonstationary behaviours, with greater power whilst preserving acceptable statistical size. Our tests work with a wide range of time series including those whose marginal distributions possess heavy tails. We provide freeware R software that implements our tests and a range of graphical tools to identify the location and duration of nonstationarities. Theoretical and simulated power calculations show the superiority of our wavelet packet approach in a number of important situations and, hence, we suggest that the new tests are useful additions to the analyst's toolbox.

Authors:William Leeb Pages: 584 - 610 Abstract: Publication date: May 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 3 Author(s): William Leeb This paper develops a theory of harmonic analysis on spaces with tree metrics, extending previous work in this direction by Gavish, Nadler and Coifman (2010) [30] and Gavish and Coifman (2011, 2012) [28,29]. We show how a natural system of martingales and martingale differences induced by a partition tree leads to simple and effective characterizations of the Lipschitz norm and its dual for functions on a single tree metric space. The restrictions we place on the tree metrics are far more general than those considered in previous work. As the dual norm is equal to the Earth Mover's Distance (EMD) between two probability distributions, we recover a simple formula for EMD with respect to tree distances presented by Charikar (2002) [36]. We also consider the situation where an arbitrary metric is approximated by the average of a family of dominating tree metrics. We show that the Lipschitz norm and its dual for the tree metrics can be combined to yield an approximation to the corresponding norms for the underlying metric. The main contributions of this paper, however, are the generalizations of the aforementioned results to the setting of the product of two or more tree metric spaces. For functions on a product space, the notion of regularity we consider is not the Lipschitz condition, but rather the mixed Lipschitz condition that controls the size of a function's mixed difference quotient. This condition is extremely natural for datasets that can be described as a product of metric spaces, such as word-document databases. We develop effective formulas for norms equivalent to the mixed Lipschitz norm and its dual, and extend our results on combining pairs of trees.

Authors:H.N. Mhaskar Pages: 611 - 644 Abstract: Publication date: May 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 3 Author(s): H.N. Mhaskar We present a general framework for studying harmonic analysis of functions in the settings of various emerging problems in the theory of diffusion geometry. The starting point of the now classical diffusion geometry approach is the construction of a kernel whose discretization leads to an undirected graph structure on an unstructured data set. We study the question of constructing such kernels for directed graph structures, and argue that our construction is essentially the only way to do so using discretizations of kernels. We then use our previous theory to develop harmonic analysis based on the singular value decomposition of the resulting non-self-adjoint operators associated with the directed graph. Next, we consider the question of how functions defined on one space evolve to another space in the paradigm of changing data sets recently introduced by Coifman and Hirn. While the approach of Coifman and Hirn requires that the points on one space should be in a known one-to-one correspondence with the points on the other, our approach allows the identification of only a subset of landmark points. We introduce a new definition of distance between points on two spaces, construct localized kernels based on the two spaces and certain interaction parameters, and study the evolution of smoothness of a function on one space to its lifting to the other space via the landmarks. We develop novel mathematical tools that enable us to study these seemingly different problems in a unified manner.

Authors:Jun Lai; Shidong Jiang Pages: 645 - 664 Abstract: Publication date: May 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 3 Author(s): Jun Lai, Shidong Jiang We present a second kind integral equation (SKIE) formulation for calculating the electromagnetic modes of optical waveguides, where the unknowns are only on material interfaces. The resulting numerical algorithm can handle optical waveguides with a large number of inclusions of arbitrary irregular cross section. It is capable of finding the bound, leaky, and complex modes for optical fibers and waveguides including photonic crystal fibers (PCF), dielectric fibers and waveguides. Most importantly, the formulation is well conditioned even in the case of nonsmooth geometries. Our method is highly accurate and thus can be used to calculate the propagation loss of the electromagnetic modes accurately, which provides the photonics industry a reliable tool for the design of more compact and efficient photonic devices. We illustrate and validate the performance of our method through extensive numerical studies and by comparison with semi-analytical results and previously published results.

Authors:Pengwen Chen; Albert Fannjiang Pages: 665 - 699 Abstract: Publication date: May 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 3 Author(s): Pengwen Chen, Albert Fannjiang The Fourier-domain Douglas–Rachford (FDR) algorithm is analyzed for phase retrieval with a single random mask. Since the uniqueness of phase retrieval solution requires more than a single oversampled coded diffraction pattern, the extra information is imposed in either of the following forms: 1) the sector condition on the object; 2) another oversampled diffraction pattern, coded or uncoded. For both settings, the uniqueness of projected fixed point is proved and for setting 2) the local, geometric convergence is derived with a rate given by a spectral gap condition. Numerical experiments demonstrate global, power-law convergence of FDR from arbitrary initialization for both settings as well as for 3 or more coded diffraction patterns without oversampling. In practice, the geometric convergence can be recovered from the power-law regime by a simple projection trick, resulting in highly accurate reconstruction from generic initialization.

Authors:Joel A. Tropp Pages: 700 - 736 Abstract: Publication date: May 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 3 Author(s): Joel A. Tropp Matrix concentration inequalities give bounds for the spectral-norm deviation of a random matrix from its expected value. These results have a weak dimensional dependence that is sometimes, but not always, necessary. This paper identifies one of the sources of the dimensional term and exploits this insight to develop sharper matrix concentration inequalities. In particular, this analysis delivers two refinements of the matrix Khintchine inequality that use information beyond the matrix variance to improve the dimensional dependence.

Authors:Yingzhou Li; Haizhao Yang; Lexing Ying Pages: 737 - 758 Abstract: Publication date: May 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 3 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:Carmeline J. Dsilva; Ronen Talmon; Ronald R. Coifman; Ioannis G. Kevrekidis Pages: 759 - 773 Abstract: Publication date: May 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 3 Author(s): Carmeline J. Dsilva, Ronen Talmon, Ronald R. Coifman, Ioannis G. Kevrekidis Nonlinear manifold learning algorithms, such as diffusion maps, have been fruitfully applied in recent years to the analysis of large and complex data sets. However, such algorithms still encounter challenges when faced with real data. One such challenge is the existence of “repeated eigendirections,” which obscures the detection of the true dimensionality of the underlying manifold and arises when several embedding coordinates parametrize the same direction in the intrinsic geometry of the data set. We propose an algorithm, based on local linear regression, to automatically detect coordinates corresponding to repeated eigendirections. We construct a more parsimonious embedding using only the eigenvectors corresponding to unique eigendirections, and we show that this reduced diffusion maps embedding induces a metric which is equivalent to the standard diffusion distance. We first demonstrate the utility and flexibility of our approach on synthetic data sets. We then apply our algorithm to data collected from a stochastic model of cellular chemotaxis, where our approach for factoring out repeated eigendirections allows us to detect changes in dynamical behavior and the underlying intrinsic system dimensionality directly from data.

Authors:Alexander Cloninger; Ronald R. Coifman; Nicholas Downing; Harlan M. Krumholz Pages: 774 - 785 Abstract: Publication date: May 2018 Source:Applied and Computational Harmonic Analysis, Volume 44, Issue 3 Author(s): Alexander Cloninger, Ronald R. Coifman, Nicholas Downing, Harlan M. Krumholz In this paper, we build an organization of high-dimensional datasets that cannot be cleanly embedded into a low-dimensional representation due to missing entries and a subset of the features being irrelevant to modeling functions of interest. Our algorithm begins by defining coarse neighborhoods of the points and defining an expected empirical function value on these neighborhoods. We then generate new non-linear features with deep net representations tuned to model the approximate function, and re-organize the geometry of the points with respect to the new representation. Finally, the points are locally z-scored to create an intrinsic geometric organization which is independent of the parameters of the deep net, a geometry designed to assure smoothness with respect to the empirical function. We examine this approach on data from the Center for Medicare and Medicaid Services Hospital Quality Initiative, and generate an intrinsic low-dimensional organization of the hospitals that is smooth with respect to an expert driven function of quality.

Authors:I. Iglewska-Nowak Pages: 201 - 229 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.

Authors:Elena Cordero; Maurice de Gosson; Fabio Nicola Pages: 230 - 245 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.

Authors:Gil Shabat; Yaniv Shmueli; Yariv Aizenbud; Amir Averbuch Pages: 246 - 272 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.

Authors:Sheng-Chun Yang; Hu-Jun Qian; Zhong-Yuan Lu Pages: 273 - 293 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.

Authors:Douglas P. Hardin; Michael C. Northington; Alexander M. Powell Pages: 294 - 311 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 .

Authors:James Bremer Pages: 312 - 349 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.

Authors:D.W. Pravica; N. Randriampiry; M.J. Spurr Pages: 350 - 413 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 ) .

Authors:Kenji Nakahira; Atsushi Miyamoto Pages: 414 - 445 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.

Authors:Gilles Puy; Nicolas Tremblay; Rémi Gribonval; Pierre Vandergheynst Pages: 446 - 475 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.

Authors:Matthew Fickus; John Jasper; Dustin G. Mixon; Jesse D. Peterson; Cody E. Watson Pages: 476 - 496 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.

Authors:Youming Liu; Xiaochen Zeng Abstract: Publication date: Available online 22 May 2018 Source:Applied and Computational Harmonic Analysis Author(s): Youming Liu, Xiaochen Zeng This current paper shows the asymptotic normality for wavelet deconvolution density estimators, when a density function belongs to some L p ( R ) ( p > 2 ) and the noises are moderately ill-posed with the index β. The estimators include both the linear and non-linear wavelet ones. It turns out that the situation for 0 < β ≤ 1 is more complicated than that for β > 1 .

Authors:Yunwen Lei; Ding-Xuan Zhou Abstract: Publication date: Available online 22 May 2018 Source:Applied and Computational Harmonic Analysis Author(s): Yunwen Lei, Ding-Xuan Zhou In this paper we consider online mirror descent (OMD), a class of scalable online learning algorithms exploiting data geometric structures through mirror maps. Necessary and sufficient conditions are presented in terms of the step size sequence { η t } t for the convergence of OMD with respect to the expected Bregman distance induced by the mirror map. The condition is lim t → ∞ η t = 0 , ∑ t = 1 ∞ η t = ∞ in the case of positive variances. It is reduced to ∑ t = 1 ∞ η t = ∞ in the case of zero variance for which linear convergence may be achieved by taking a constant step size sequence. A sufficient condition on the almost sure convergence is also given. We establish tight error bounds under mild conditions on the mirror map, the loss function, and the regularizer. Our results are achieved by some novel analysis on the one-step progress of OMD using smoothness and strong convexity of the mirror map and the loss function.

Authors:Hermine Biermé; Céline Lacaux Abstract: Publication date: Available online 22 May 2018 Source:Applied and Computational Harmonic Analysis Author(s): Hermine Biermé, Céline Lacaux Operator scaling Gaussian random fields, as anisotropic generalizations of self-similar fields, know an increasing interest for theoretical studies in the literature. However, up to now, they were only defined through stochastic integrals, without explicit covariance functions. In this paper we exhibit explicit covariance functions, as anisotropic generalizations of fractional Brownian fields ones, and define corresponding Operator scaling Gaussian random fields. This allows us to propose a fast and exact method of simulation in dimension 2 based on the circulant embedding matrix method, following ideas of Stein [34] for fractional Brownian surfaces syntheses. This is a first piece of work to popularize these models in anisotropic spatial data modeling.

Authors:Ronny Bergmann; Dennis Merkert Abstract: Publication date: Available online 17 May 2018 Source:Applied and Computational Harmonic Analysis Author(s): Ronny Bergmann, Dennis Merkert In this paper we derive a discretisation of the equation of quasi-static elasticity in homogenization in form of a variational formulation and the so-called Lippmann-Schwinger equation, in anisotropic spaces of translates of periodic functions. We unify and extend the truncated Fourier series approach, the constant finite element ansatz and the anisotropic lattice derivation. The resulting formulation of the Lippmann-Schwinger equation in anisotropic translation invariant spaces unifies and analyses for the first time both the Fourier methods and finite element approaches in a common mathematical framework. We further define and characterize the resulting periodised Green operator. This operator coincides in case of a Dirichlet kernel corresponding to a diagonal matrix with the operator derived for the Galerkin projection stemming from the truncated Fourier series approach and to the anisotropic lattice derivation for all other Dirichlet kernels. Additionally, we proof the boundedness of the periodised Green operator. The operator further constitutes a projection if and only if the space of translates is generated by a Dirichlet kernel. Numerical examples for both de la Vallée Poussin means and Box splines illustrate the flexibility of this framework.

Authors:Ralf Banisch; Zofia Trstanova; Andreas Bittracher; Stefan Klus; Péter Koltai Abstract: Publication date: Available online 9 May 2018 Source:Applied and Computational Harmonic Analysis Author(s): Ralf Banisch, Zofia Trstanova, Andreas Bittracher, Stefan Klus, Péter Koltai We present two generalizations of the popular diffusion maps algorithm. The first generalization replaces the drift term in diffusion maps, which is the gradient of the sampling density, with the gradient of an arbitrary density of interest which is known up to a normalization constant. The second generalization allows for a diffusion map type approximation of the forward and backward generators of general Itô diffusions with given drift and diffusion coefficients. We use the local kernels introduced by Berry and Sauer, but allow for arbitrary sampling densities. We provide numerical illustrations to demonstrate that this opens up many new applications for diffusion maps as a tool to organize point cloud data, including biased or corrupted samples, dimension reduction for dynamical systems, detection of almost invariant regions in flow fields, and importance sampling.

Authors:John J. Benedetto; Weilin Li Abstract: Publication date: Available online 4 May 2018 Source:Applied and Computational Harmonic Analysis Author(s): John J. Benedetto, Weilin Li We investigate the super-resolution capabilities of total variation minimization. Namely, given a finite set Λ ⊆ Z d and spectral data F = μ ˆ Λ , where μ is an unknown bounded Radon measure on the torus T d , the problem is to find the measures with smallest norm whose Fourier transforms agree with F on Λ. Our main theorem shows that solutions to the problem depend crucially on a set Γ ⊆ Λ , defined in terms of F and Λ. For example, when # Γ = 0 , the solutions are singular measures supported in the zero set of an analytic function, and when # Γ ≥ 2 , the solutions are singular measures supported in the intersection of ( # Γ 2 ) hyperplanes. By theory and example, we show that the case # Γ = 1 is different from other cases, and is deeply connected with the existence of positive solutions. This theorem has implications to the possibility and impossibility of uniquely recovering μ from F on Λ. We illustrate how to apply our theory to both directions, by computing pertinent analytical examples. These examples are of interest in both super-resolution and deterministic compressed sensing. Our concept of an admissibility range fundamentally connects Beurling's theory of minimal extrapolation [7,8] with Candès and Fernandez-Granda's work on super-resolution [12]. This connection is exploited to address situations where current algorithms fail to compute a numerical solution to the total variation minimization problem.

Authors:Mingjuan Chen; Baoxiang Wang; Shuxia Wang; M.W. Wong Abstract: Publication date: Available online 24 April 2018 Source:Applied and Computational Harmonic Analysis Author(s): Mingjuan Chen, Baoxiang Wang, Shuxia Wang, M.W. Wong Using the local time-frequency analysis techniques, we obtain an equivalent norm on modulation spaces. Secondly, applying this equivalent norm, we consider the Cauchy problem for the dissipative evolutionary pseudo-differential equation ∂ t u + A ( x , D ) u = F ( ( ∂ x α u ) α ⩽ κ ) , u ( 0 , x ) = u 0 ( x ) , where A ( x , D ) is a dissipative pseudo-differential operator and F ( z ) is a multi-polynomial. We will develop the uniform decomposition techniques in both physical and frequency spaces to study its local well posedness in modulation spaces M p , q s and in Sobolev spaces H s . Moreover, the local solution can be extended to a global one in L 2 and in H s ( s > κ + d / 2 ) for certain nonlinearities.

Authors:Xiuyuan Cheng; Manas Rachh; Stefan Steinerberger Abstract: Publication date: Available online 17 April 2018 Source:Applied and Computational Harmonic Analysis Author(s): Xiuyuan Cheng, Manas Rachh, Stefan Steinerberger We study directed, weighted graphs G = ( V , E ) and consider the (not necessarily symmetric) averaging operator ( L u ) ( i ) = − ∑ j ∼ i p i j ( u ( j ) − u ( i ) ) , where p i j are normalized edge weights. Given a vertex i ∈ V , we define the diffusion distance to a set B ⊂ V as the smallest number of steps d B ( i ) ∈ N required for half of all random walks started in i and moving randomly with respect to the weights p i j to visit B within d B ( i ) steps. Our main result is that the eigenfunctions interact nicely with this notion of distance. In particular, if u satisfies L u = λ u on V and ε > 0 is so large that B = { i ∈ V : − ε ≤ u ( i ) ≤ ε } ≠ ∅ , then, for all i ∈ V , d B ( i ) log ( 1 1 − λ ) ≥ log ( u ( i ) ‖ u ‖ L ∞ ) − log ( 1 2 + ε ) . d B ( i ) is a remarkably good approximation of u in the sense of having very high correlation. The result implies that the classical one-dimensional spectral embedding preserves particular aspects of geometry in the presence of clustered data. We also give a continuous variant of the result which has a connection to the hot spots conjecture.

Authors:Ole Christensen; Marzieh Hasannasab Abstract: Publication date: Available online 17 April 2018 Source:Applied and Computational Harmonic Analysis Author(s): Ole Christensen, Marzieh Hasannasab Motivated by recent progress in dynamical sampling we prove that every frame which is norm-bounded below can be represented as a finite union of sequences { ( T j ) n φ j } n = 0 ∞ , j = 1 , … , J for some bounded operators T j and elements φ j in the underlying Hilbert space. The result is optimal, in the sense that it turns out to be problematic to replace the collection of generators φ 1 , … , φ J by a singleton: indeed, for linearly independent frames we prove that we can represent the frame in terms of just one system { T n φ } n = 0 ∞ , but unfortunately this representation often forces the operator T to be unbounded. Several examples illustrate the connection of the results to typical frames like Gabor frames and wavelet frames, as well as generic constructions in arbitrary separable Hilbert spaces.

Authors:Diego H. Díaz Martínez; Facundo Mémoli; Washington Mio Abstract: Publication date: Available online 30 March 2018 Source:Applied and Computational Harmonic Analysis Author(s): Diego H. Díaz Martínez, Facundo Mémoli, Washington Mio We introduce the notion of multiscale covariance tensor fields (CTF) associated with Euclidean random variables as a gateway to the shape of their distributions. Multiscale CTFs quantify variation of the data about every point in the data landscape at all spatial scales, unlike the usual covariance tensor that only quantifies global variation about the mean. Empirical forms of localized covariance previously have been used in data analysis and visualization, for example, in local principal component analysis, but we develop a framework for the systematic treatment of theoretical questions and mathematical analysis of computational models. We prove strong stability theorems with respect to the Wasserstein distance between probability measures, obtain consistency results for estimators, as well as bounds on the rate of convergence of empirical CTFs. These results show that CTFs are robust to sampling, noise and outliers. We provide numerous illustrations of how CTFs let us extract shape from data and also apply CTFs to manifold clustering, the problem of categorizing data points according to their noisy membership in a collection of possibly intersecting smooth submanifolds of Euclidean space. We prove that the proposed manifold clustering method is stable and carry out several experiments to illustrate the method.

Authors:Shuai Lu; Peter Mathé; Sergei V. Pereverzev Abstract: Publication date: Available online 21 March 2018 Source:Applied and Computational Harmonic Analysis Author(s): Shuai Lu, Peter Mathé, Sergei V. Pereverzev We discuss the problem of parameter choice in learning algorithms generated by a general regularization scheme. Such a scheme covers well-known algorithms as regularized least squares and gradient descent learning. It is known that in contrast to classical deterministic regularization methods, the performance of regularized learning algorithms is influenced not only by the smoothness of a target function, but also by the capacity of a space, where regularization is performed. In the infinite dimensional case the latter one is usually measured in terms of the effective dimension. In the context of supervised learning both the smoothness and effective dimension are intrinsically unknown a priori. Therefore we are interested in a posteriori regularization parameter choice, and we propose a new form of the balancing principle. An advantage of this strategy over the known rules such as cross-validation based adaptation is that it does not require any data splitting and allows the use of all available labeled data in the construction of regularized approximants. We provide the analysis of the proposed rule and demonstrate its advantage in simulations.

Authors:Nicki Holighaus; Christoph Wiesmeyr; Peter Balazs Abstract: Publication date: Available online 20 March 2018 Source:Applied and Computational Harmonic Analysis Author(s): Nicki Holighaus, Christoph Wiesmeyr, Peter Balazs We present a novel family of continuous, linear time-frequency transforms adaptable to a multitude of (nonlinear) frequency scales. Similar to classical time-frequency or time-scale representations, the representation coefficients are obtained as inner products with the elements of a continuously indexed family of time-frequency atoms. These atoms are obtained from a single prototype function, by means of modulation, translation and warping. By warping we refer to the process of nonlinear evaluation according to a bijective, increasing function, the warping function. Besides showing that the resulting integral transforms fulfill certain basic, but essential properties, such as continuity and invertibility, we will show that a large subclass of warping functions gives rise to families of generalized coorbit spaces, i.e. Banach spaces of functions whose representations possess a certain localization. Furthermore, we obtain sufficient conditions for subsampled warped time-frequency systems to form atomic decompositions and Banach frames. To this end, we extend results previously presented by Fornasier and Rauhut to a larger class of function systems via a simple, but crucial modification. The proposed method allows for great flexibility, but by choosing particular warping functions Φ we also recover classical time-frequency representations, e.g. Φ ( t ) = c t provides the short-time Fourier transform and Φ ( t ) = log a ( t ) provides wavelet transforms. This is illustrated by a number of examples provided in the manuscript.

Authors:Elona Agora; Jorge Antezana; Mihail N. Kolountzakis Abstract: Publication date: Available online 1 March 2018 Source:Applied and Computational Harmonic Analysis Author(s): Elona Agora, Jorge Antezana, Mihail N. Kolountzakis We study the existence of Gabor orthonormal bases with window the characteristic function of the set Ω = [ 0 , α ] ∪ [ β + α , β + 1 ] of measure 1, with α , β > 0 . By the symmetries of the problem, we can restrict our attention to the case α ≤ 1 / 2 . We prove that either if α < 1 / 2 or ( α = 1 / 2 and β ≥ 1 / 2 ) there exist such Gabor orthonormal bases, with window the characteristic function of the set Ω, if and only if Ω tiles the line. Furthermore, in both cases, we completely describe the structure of the set of time–frequency shifts associated to these bases.

Authors:Stéphane Jaffard; Stéphane Seuret; Herwig Wendt; Roberto Leonarduzzi; Stéphane Roux; Patrice Abry Abstract: Publication date: Available online 24 February 2018 Source:Applied and Computational Harmonic Analysis Author(s): Stéphane Jaffard, Stéphane Seuret, Herwig Wendt, Roberto Leonarduzzi, Stéphane Roux, Patrice Abry We show how a joint multifractal analysis of a collection of signals unravels correlations between the locations of their pointwise singularities. The multivariate multifractal formalism, reformulated in the general setting supplied by multiresolution quantities, provides a framework which allows to estimate joint multifractal spectra. General results on joint multifractal spectra are derived, and illustrated by the theoretical derivation and practical estimation of the joint multifractal spectra of simple mathematical models, including correlated binomial cascades.

Authors:Charles K. Chui; H.N. Mhaskar Abstract: Publication date: Available online 21 February 2018 Source:Applied and Computational Harmonic Analysis Author(s): Charles K. Chui, H.N. Mhaskar In this paper, motivated by diffraction of traveling light waves, a simple mathematical model is proposed, both for the multivariate super-resolution problem and the problem of blind-source separation of real-valued exponential sums. This model facilitates the development of a unified theory and a unified solution of both problems in this paper. Our consideration of the super-resolution problem is aimed at applications to fluorescence microscopy and observational astronomy, and the motivation for our consideration of the second problem is the current need of extracting multivariate exponential features in magnetic resonance spectroscopy (MRS) for the neurologist and radiologist as well as for providing a mathematical tool for isotope separation in Nuclear Chemistry. The unified method introduced in this paper can be easily realized by processing only finitely many data, sampled at locations that are not necessarily prescribed in advance, with computational scheme consisting only of matrix-vector multiplication, peak finding, and clustering.

Authors:Diego H. Díaz Martínez; Christine H. Lee; Peter T. Kim; Washington Mio Abstract: Publication date: Available online 13 February 2018 Source:Applied and Computational Harmonic Analysis Author(s): Diego H. Díaz Martínez, Christine H. Lee, Peter T. Kim, Washington Mio The state of many complex systems, such as ecosystems formed by multiple microbial taxa that interact in intricate ways, is often summarized as a probability distribution on the nodes of a weighted network. This paper develops methods for modeling the organization of such data, as well as their Euclidean counterparts, across spatial scales. Using the notion of diffusion distance, we introduce diffusion Fréchet functions and diffusion Fréchet vectors associated with probability distributions on Euclidean space and the vertex set of a weighted network, respectively. We prove that these functional statistics are stable with respect to the Wasserstein distance between probability measures, thus yielding robust descriptors of their shapes. We provide several examples that illustrate the geometric characteristics of a distribution that are captured by multi-scale Fréchet functions and vectors.

Authors:Zhiqiang 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].