Abstract: Publication date: July 2018Source: Applied and Computational Harmonic Analysis, Volume 45, Issue 1Author(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.

Abstract: Publication date: July 2018Source: Applied and Computational Harmonic Analysis, Volume 45, Issue 1Author(s): Joel Laity, Barak Shani A function f:Zn→C can be represented as a linear combination f(x)=∑α∈Znfˆ(α)χα,n(x) where fˆ is the (discrete) Fourier transform of f. Clearly, the basis {χα,n(x):=exp(2πiαx/n)} depends on the value n.We show that if f has “large” Fourier coefficients, then the function f˜:Zm→C, given byf˜(x)={f(x)when 0≤x

Abstract: Publication date: July 2018Source: Applied and Computational Harmonic Analysis, Volume 45, Issue 1Author(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.

Abstract: Publication date: July 2018Source: Applied and Computational Harmonic Analysis, Volume 45, Issue 1Author(s): Yongsheng Han, Ji Li, Lesley A. Ward In this paper, we first show that the remarkable orthonormal wavelet expansion for Lp 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˜=X1×X2×⋅⋅⋅×Xn, where each factor Xi 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 Hp, the dual CMOp of Hp with the special case BMO=CMO1, and the predual VMO of H1. We also use the wavelet expansion to establish the Calderón–Zygmund decomposition for product Hp, 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.

Abstract: Publication date: July 2018Source: Applied and Computational Harmonic Analysis, Volume 45, Issue 1Author(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 DH. 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 DH 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.

Abstract: Publication date: July 2018Source: Applied and Computational Harmonic Analysis, Volume 45, Issue 1Author(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

Abstract: Publication date: July 2018Source: Applied and Computational Harmonic Analysis, Volume 45, Issue 1Author(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 logK factor inversely proportional to the dynamic range of the signals, and the sample size is proportional to KlogKε˜−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/(ℓlogK).

Abstract: Publication date: July 2018Source: Applied and Computational Harmonic Analysis, Volume 45, Issue 1Author(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 Λω for the corresponding directional Gabor system {gm,t,u}(m,t,u)∈Λω 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.

Abstract: Publication date: July 2018Source: Applied and Computational Harmonic Analysis, Volume 45, Issue 1Author(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 ℓ1-minimization or related procedures. We discover, however, a measure theoretical uniqueness exists for nearly spark-level sparsity from compressed measurements Ax=b. Specifically, suppose A is of full spark with m rows, and suppose m2m2 even from the same measure theoretical point of view.

Abstract: Publication date: Available online 19 June 2018Source: Applied and Computational Harmonic AnalysisAuthor(s): M. Maslouhi, S. Loukili In this paper we solve an open problem posed in [10] related to the representation of Positive Operator-Valued Measures by means of tight probabilistic frames in Rd. Also, we investigate how far is the closest probabilistic tight frame from a given probability measure where the distance used is the quadratic Wasserstein metric W2 used for optimal transportation problem for measures. In particular, we study the optimization problem I(μ,K):=infν∈T(K)W22(μ,ν), where T(K) is the set of all probabilistic tight frames whose supports are contained in K=Rd or K=Sd−1. This problem is solved and its optimum is given when the mean vector of μ is zero. In the other cases, we give concise upper and lower bounds for I(μ,K).

Abstract: Publication date: Available online 18 June 2018Source: Applied and Computational Harmonic AnalysisAuthor(s): Mark A. Iwen, Brian Preskitt, Rayan Saab, Aditya Viswanathan We improve a phase retrieval approach that uses correlation-based measurements with compactly supported measurement masks [30]. Our approach admits deterministic measurement constructions together with a robust, fast recovery algorithm that consists of solving a system of linear equations in a lifted space, followed by finding an eigenvector (e.g., via an inverse power iteration). Theoretical reconstruction error guarantees from [30] are improved as a result for the new and more robust reconstruction approach proposed herein. Numerical experiments demonstrate robustness and computational efficiency that compete with other approaches on large problems. Along the way, we show that this approach also trivially extends to phase retrieval problems based on windowed Fourier measurements.

Abstract: Publication date: Available online 15 June 2018Source: Applied and Computational Harmonic AnalysisAuthor(s): A. Aldroubi, I. Krishtal, S. Tang Phaseless reconstruction from space–time samples is a nonlinear problem of recovering a function x in a Hilbert space H from the modulus of linear measurements { 〈x,ϕi〉 , …, 〈ALix,ϕi〉 :i∈I}, where {ϕi;i∈I}⊂H is a set of functionals on H, and A is a bounded operator on H that acts as an evolution operator. In this paper, we provide various sufficient or necessary conditions for solving this problem, which has connections to X-ray crystallography, the scattering transform, and deep learning.

Abstract: Publication date: Available online 7 June 2018Source: Applied and Computational Harmonic AnalysisAuthor(s): Markus Faulhuber, Maurice A. de Gosson, David Rottensteiner Gaussian states are at the heart of quantum mechanics and play an essential role in quantum information processing. In this paper we provide approximation formulas for the expansion of a general Gaussian symbol in terms of elementary Gaussian functions. For this purpose we introduce the notion of a “phase space frame” associated with a Weyl–Heisenberg frame. Our results give explicit formulas for approximating general Gaussian symbols in phase space by phase space shifted standard Gaussians as well as explicit error estimates and the asymptotic behavior of the approximation.

Abstract: Publication date: Available online 22 May 2018Source: Applied and Computational Harmonic AnalysisAuthor(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.

Abstract: Publication date: Available online 22 May 2018Source: Applied and Computational Harmonic AnalysisAuthor(s): Youming Liu, Xiaochen Zeng This current paper shows the asymptotic normality for wavelet deconvolution density estimators, when a density function belongs to some Lp(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 01.

Abstract: Publication date: Available online 22 May 2018Source: Applied and Computational Harmonic AnalysisAuthor(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 {ηt}t for the convergence of OMD with respect to the expected Bregman distance induced by the mirror map. The condition is limt→∞ηt=0,∑t=1∞ηt=∞ in the case of positive variances. It is reduced to ∑t=1∞ηt=∞ 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.

Abstract: Publication date: Available online 17 May 2018Source: Applied and Computational Harmonic AnalysisAuthor(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.

Abstract: Publication date: Available online 9 May 2018Source: Applied and Computational Harmonic AnalysisAuthor(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.

Abstract: Publication date: Available online 4 May 2018Source: Applied and Computational Harmonic AnalysisAuthor(s): John J. Benedetto, Weilin Li We investigate the super-resolution capabilities of total variation minimization. Namely, given a finite set Λ⊆Zd and spectral data F=μˆ Λ, where μ is an unknown bounded Radon measure on the torus Td, 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 (#Γ2) 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.

Abstract: Publication date: May 2018Source: Applied and Computational Harmonic Analysis, Volume 44, Issue 3Author(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.

Abstract: Publication date: May 2018Source: Applied and Computational Harmonic Analysis, Volume 44, Issue 3Author(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.

Abstract: Publication date: May 2018Source: Applied and Computational Harmonic Analysis, Volume 44, Issue 3Author(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.

Abstract: Publication date: May 2018Source: Applied and Computational Harmonic Analysis, Volume 44, Issue 3Author(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.

Abstract: Publication date: May 2018Source: Applied and Computational Harmonic Analysis, Volume 44, Issue 3Author(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.

Abstract: Publication date: May 2018Source: Applied and Computational Harmonic Analysis, Volume 44, Issue 3Author(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.

Abstract: Publication date: May 2018Source: Applied and Computational Harmonic Analysis, Volume 44, Issue 3Author(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.

Abstract: Publication date: May 2018Source: Applied and Computational Harmonic Analysis, Volume 44, Issue 3Author(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⊂Rm, 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).

Abstract: Publication date: May 2018Source: Applied and Computational Harmonic Analysis, Volume 44, Issue 3Author(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.

Abstract: Publication date: May 2018Source: Applied and Computational Harmonic Analysis, Volume 44, Issue 3Author(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.

Abstract: Publication date: May 2018Source: Applied and Computational Harmonic Analysis, Volume 44, Issue 3Author(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(logN) 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.

Abstract: Publication date: Available online 24 April 2018Source: Applied and Computational Harmonic AnalysisAuthor(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∂tu+A(x,D)u=F((∂xαu) α ⩽κ),u(0,x)=u0(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 Mp,qs and in Sobolev spaces Hs. Moreover, the local solution can be extended to a global one in L2 and in Hs (s>κ+d/2) for certain nonlinearities.

Abstract: Publication date: Available online 17 April 2018Source: Applied and Computational Harmonic AnalysisAuthor(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 {(Tj)nφj}n=0∞,j=1,…,J for some bounded operators Tj and elements φj 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 φ1,…,φJ by a singleton: indeed, for linearly independent frames we prove that we can represent the frame in terms of just one system {Tnφ}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.

Abstract: Publication date: Available online 17 April 2018Source: Applied and Computational Harmonic AnalysisAuthor(s): Xiuyuan Cheng, Manas Rachh, Stefan Steinerberger We study directed, weighted graphs G=(V,E) and consider the (not necessarily symmetric) averaging operator(Lu)(i)=−∑j∼ipij(u(j)−u(i)), where pij 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 dB(i)∈N required for half of all random walks started in i and moving randomly with respect to the weights pij to visit B within dB(i) steps. Our main result is that the eigenfunctions interact nicely with this notion of distance. In particular, if u satisfies Lu=λu on V and ε>0 is so large thatB={i∈V:−ε≤u(i)≤ε}≠∅, then, for all i∈V,dB(i)log(1 1−λ )≥log( u(i) ‖u‖L∞)−log(12+ε).dB(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.

Abstract: Publication date: Available online 30 March 2018Source: Applied and Computational Harmonic AnalysisAuthor(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.

Abstract: Publication date: Available online 21 March 2018Source: Applied and Computational Harmonic AnalysisAuthor(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.

Abstract: Publication date: Available online 20 March 2018Source: Applied and Computational Harmonic AnalysisAuthor(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)=ct provides the short-time Fourier transform and Φ(t)=loga(t) provides wavelet transforms. This is illustrated by a number of examples provided in the manuscript.

Abstract: Publication date: Available online 1 March 2018Source: Applied and Computational Harmonic AnalysisAuthor(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

Abstract: Publication date: Available online 24 February 2018Source: Applied and Computational Harmonic AnalysisAuthor(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.

Abstract: Publication date: Available online 21 February 2018Source: Applied and Computational Harmonic AnalysisAuthor(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.

Abstract: Publication date: Available online 13 February 2018Source: Applied and Computational Harmonic AnalysisAuthor(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.

Abstract: Publication date: Available online 8 February 2018Source: Applied and Computational Harmonic AnalysisAuthor(s): Simon Marshall, Shayne Waldron There is a finite number hn,d of tight frames of n distinct vectors for Cd which are the orbit of a vector under a unitary action of the cyclic group Zn. These cyclic harmonic frames (or geometrically uniform tight frames) are used in signal analysis and quantum information theory, and provide many tight frames of particular interest. Here we investigate the conjecture that hn,d grows like nd−1. By using a result of Laurent which describes the set of solutions of algebraic equations in roots of unity, we prove the asymptotic estimatehn,d≈ndφ(n)≥nd−1,n→∞. By using a group theoretic approach, we also give some exact formulas for hn,d, and estimate the number of cyclic harmonic frames up to projective unitary equivalence.

Abstract: Publication date: Available online 8 February 2018Source: Applied and Computational Harmonic AnalysisAuthor(s): Chendi Huang, Xinwei Sun, Jiechao Xiong, Yuan Yao Boosting as gradient descent algorithms is one popular method in machine learning. In this paper a novel Boosting-type algorithm is proposed based on restricted gradient descent with structural sparsity control whose underlying dynamics are governed by differential inclusions. In particular, we present an iterative regularization path with structural sparsity where the parameter is sparse under some linear transforms, based on variable splitting and the Linearized Bregman Iteration. Hence it is called Split LBI. Despite its simplicity, Split LBI outperforms the popular generalized Lasso in both theory and experiments. A theory of path consistency is presented that equipped with a proper early stopping, Split LBI may achieve model selection consistency under a family of Irrepresentable Conditions which can be weaker than the necessary and sufficient condition for generalized Lasso. Furthermore, some ℓ2 error bounds are also given at the minimax optimal rates. The utility and benefit of the algorithm are illustrated by several applications including image denoising, partial order ranking of sport teams, and world university grouping with crowdsourced ranking data.

Abstract: Publication date: Available online 8 February 2018Source: Applied and Computational Harmonic AnalysisAuthor(s): Yu Guang Wang, Xiaosheng Zhuang Tight framelets on a smooth and compact Riemannian manifold M provide a tool of multiresolution analysis for data from geosciences, astrophysics, medical sciences, etc. This work investigates the construction, characterizations, and applications of tight framelets on such a manifold M. Characterizations of the tightness of a sequence of framelet systems for L2(M) in both the continuous and semi-discrete settings are provided. Tight framelets associated with framelet filter banks on M can then be easily designed and fast framelet filter bank transforms on M are shown to be realizable with nearly linear computational complexity. Explicit construction of tight framelets on the sphere S2 as well as numerical examples are given.

Abstract: Publication date: Available online 7 February 2018Source: Applied and Computational Harmonic AnalysisAuthor(s): Wengu Chen, Yaling Li In this paper, the sufficient condition in terms of the RIC and ROC for the stable and robust recovery of signals in both noiseless and noisy settings was established via weighted l1 minimization when there is partial prior information on support of signals. An improved performance guarantee has been derived. We can obtain a less restricted sufficient condition for signal reconstruction and a tighter recovery error bound under some conditions via weighted l1 minimization. When prior support estimate is at least 50% accurate, the sufficient condition is weaker than the analogous condition by standard l1 minimization method, meanwhile the reconstruction error upper bound is provably to be smaller under additional conditions. Furthermore, the sufficient condition is also proved sharp.

Abstract: Publication date: Available online 7 February 2018Source: Applied and Computational Harmonic AnalysisAuthor(s): Jinming Wen, Zhengchun Zhou, Zilong Liu, Ming-Jun Lai, Xiaohu Tang In this paper, we use the block orthogonal matching pursuit (BOMP) algorithm to recover block sparse signals x from measurements y=Ax+v, where v is an ℓ2-bounded noise vector (i.e., ‖v‖2≤ϵ for some constant ϵ). We investigate some sufficient conditions based on the block restricted isometry property (block-RIP) for exact (when v=0) and stable (when v≠0) recovery of block sparse signals x. First, on the one hand, we show that if A satisfies the block-RIP with δK+1

Abstract: Publication date: Available online 6 February 2018Source: Applied and Computational Harmonic AnalysisAuthor(s): Gregory Beylkin, Lucas Monzón, Ignas Satkauskas We introduce a new functional representation of probability density functions (PDFs) of non-negative random variables via a product of a monomial factor and linear combinations of decaying exponentials with complex exponents. This approximate representation of PDFs is obtained for any finite, user-selected accuracy. Using a fast algorithm involving Hankel matrices, we develop a general numerical method for computing the PDF of the sums, products, or quotients of any number of non-negative independent random variables yielding the result in the same type of functional representation. We present several examples to demonstrate the accuracy of the approach.

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

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

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