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Abstract: Abstract The theory of sampling operators is one of the major areas of analysis in which Paul Leo Butzer and his colleagues/students have achieved long-term and excellent theoretical results. We limit ourselves here to the Mellin sampling operators introduced by Paul Butzer some 25 years ago, but our treatment comes from the point of view of the cosine operator function. PubDate: 2024-01-23

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Abstract: Abstract Gabor phase retrieval is the problem of reconstructing a signal from only the magnitudes of its Gabor transform. Previous findings suggest a possible link between unique solvability of the discrete problem (recovery from measurements on a lattice) and stability of the continuous problem (recovery from measurements on an open subset of \(\mathbb {R}^2\) ). In this paper, we close this gap by proving that such a link cannot be made. More precisely, we establish the existence of functions which break uniqueness from samples without affecting stability of the continuous problem. Furthermore, we prove the novel result that counterexamples to unique recovery from samples are dense in \(L^2(\mathbb {R})\) . Finally, we develop an intuitive argument on the connection between directions of instability in phase retrieval and certain Laplacian eigenfunctions associated to small eigenvalues. PubDate: 2024-01-17

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Abstract: Abstract In 2023, it is 100 years after the introduction of Walsh functions and 50 years after the death of their introducer Joseph Leonard Walsh. This paper is an homage to J. L. Walsh and presents some insights into the development of Walsh and dyadic analysis with applications of Walsh functions in both mathematics and various areas of computing and engineering. The presentation is based on reminiscences of the authors acting in these areas. PubDate: 2024-01-06

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Abstract: Abstract We introduce a new method based on nonnegative matrix factorization, Neural NMF, for detecting latent hierarchical structure in data. Datasets with hierarchical structure arise in a wide variety of fields, such as document classification, image processing, and bioinformatics. Neural NMF recursively applies NMF in layers to discover overarching topics encompassing the lower-level features. We derive a backpropagation optimization scheme that allows us to frame hierarchical NMF as a neural network. We test Neural NMF on a synthetic hierarchical dataset, the 20 Newsgroups dataset, and the MyLymeData symptoms dataset. Numerical results demonstrate that Neural NMF outperforms other hierarchical NMF methods on these data sets and offers better learned hierarchical structure and interpretability of topics. PubDate: 2023-12-19

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Abstract: Abstract We present an approach to Fuglede’s conjecture in \(\mathbb {Z}_p^3\) using linear programming bounds, obtaining the following partial result: if \(A\subseteq \mathbb {Z}_p^3\) with \(p^2-p\sqrt{p}+\sqrt{p}< A <p^2\) , then A is not spectral. PubDate: 2023-12-11

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Abstract: Abstract Our previous multiscale graph basis dictionaries/graph signal transforms—Generalized Haar-Walsh Transform (GHWT); Hierarchical Graph Laplacian Eigen Transform (HGLET); Natural Graph Wavelet Packets (NGWPs); and their relatives—were developed for analyzing data recorded on vertices of a given graph. In this article, we propose their generalization for analyzing data recorded on edges, faces (i.e., triangles), or more generally \(\kappa \) -dimensional simplices of a simplicial complex (e.g., a triangle mesh of a manifold). The key idea is to use the Hodge Laplacians and their variants for hierarchical partitioning of a set of \(\kappa \) -dimensional simplices in a given simplicial complex, and then build localized basis functions on these partitioned subsets. We demonstrate their usefulness for data representation on both illustrative synthetic examples and real-world simplicial complexes generated from a co-authorship/citation dataset and an ocean current/flow dataset. PubDate: 2023-12-08

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Abstract: Abstract We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary p-groups \((\mathbb {Z}_p)^d\) , we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets. We characterize such 4-tuples for \(d=1, 2\) , and prove some partial results for \(d=3\) . PubDate: 2023-12-04

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Abstract: Abstract In the (special) smoothing spline problem one considers a variational problem with a quadratic data fidelity penalty and Laplacian regularization. Higher order regularity can be obtained via replacing the Laplacian regulariser with a poly-Laplacian regulariser. The methodology is readily adapted to graphs and here we consider graph poly-Laplacian regularization in a fully supervised, non-parametric, noise corrupted, regression problem. In particular, given a dataset \(\{x_i\}_{i=1}^n\) and a set of noisy labels \(\{y_i\}_{i=1}^n\subset \mathbb {R}\) we let \(u_n{:}\{x_i\}_{i=1}^n\rightarrow \mathbb {R}\) be the minimizer of an energy which consists of a data fidelity term and an appropriately scaled graph poly-Laplacian term. When \(y_i = g(x_i)+\xi _i\) , for iid noise \(\xi _i\) , and using the geometric random graph, we identify (with high probability) the rate of convergence of \(u_n\) to g in the large data limit \(n\rightarrow \infty \) . Furthermore, our rate is close to the known rate of convergence in the usual smoothing spline model. PubDate: 2023-11-27 DOI: 10.1007/s43670-023-00075-5

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Abstract: Abstract The purpose of the present paper is to place a number of geometric (and hands-on) configurations relating to spectrum and geometry inside a general framework for the Fuglede conjecture. Note that in its general form, the Fuglede conjecture concerns general Borel sets \(\Omega \) in a fixed number of dimensions d such that \(\Omega \) has finite positive Lebesgue measure. The conjecture proposes a correspondence between two properties for \(\Omega \) , one takes the form of spectrum, while the other refers to a translation-tiling property. We focus here on the case of dimension one, and the connections between the Fuglede conjecture and properties of the self-adjoint extensions of the momentum operator \(\frac{1}{2\pi i}\frac{d}{dx}\) , realized in \(L^2\) of a union of intervals. PubDate: 2023-11-10 DOI: 10.1007/s43670-023-00072-8

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Abstract: Abstract The purpose of this paper is to place the classical formulation of the Fuglede conjecture (i.e., spectrum in \(L^{2}(\Omega )\) vs tiling properties for \(\Omega \) ) in a wider context of general questions in inverse problems taking the form of links between spectrum and geometry. By “spectrum” we refer here to existence of an orthogonal multivariable Fourier expansions for \(L^{2}(\Omega )\) . We accomplish this with reference to the early history of the problem, i.e., the context of classical theorems (von Neumann, Stone, and Segal) dealing with spectral resolutions, orthogonal expansions, and the theory of commuting selfadjoint extensions for partial differential operators which are defined from systems of local coordinates in \(\Omega \) . PubDate: 2023-11-03 DOI: 10.1007/s43670-023-00074-6

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Abstract: Abstract Label Shift has been widely believed to be harmful to the generalization performance of machine learning models. Researchers have proposed many approaches to mitigate the impact of the label shift, e.g., balancing the training data. However, these methods often consider the underparametrized regime, where the sample size is much larger than the data dimension. The research under the overparametrized regime is very limited. To bridge this gap, we propose a new asymptotic analysis of the Fisher Linear Discriminant classifier for binary classification with label shift. Specifically, we prove that there exists a phase transition phenomenon: Under certain overparametrized regime, the classifier trained using imbalanced data outperforms the counterpart with reduced balanced data. Moreover, we investigate the impact of regularization to the label shift: The aforementioned phase transition vanishes as the regularization becomes strong. PubDate: 2023-10-25 DOI: 10.1007/s43670-023-00071-9

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Abstract: Abstract A set \(\Omega \subset \mathbb {R}^d\) is said to be spectral if the space \(L^2(\Omega )\) admits an orthogonal basis of exponential functions. Fuglede (1974) conjectured that \(\Omega \) is spectral if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it was recently proved that the Fuglede conjecture does hold for the class of convex bodies in \(\mathbb {R}^d\) . The proof was based on a new geometric necessary condition for spectrality, called “weak tiling”. In this paper we study further properties of the weak tiling notion, and present applications to convex bodies, non-convex polytopes, product domains and Cantor sets of positive measure. PubDate: 2023-10-13 DOI: 10.1007/s43670-023-00070-w

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Abstract: Abstract We introduce an expansion scheme in reproducing kernel Hilbert spaces, which as a special case covers the celebrated Blaschke unwinding series expansion for analytic functions. The expansion scheme is further generalized to cover Hardy spaces \(H^p\) , \(1<p<\infty \) , viewed as Banach spaces of analytic functions with bounded evaluation functionals. In this setting a dichotomy is more transparent: depending on the multipliers used, the expansion of \(f \in H^p\) converges either to f in \(H^p\) -norm or to its projection onto a model space generated by the corresponding multipliers. Some explicit instances of the general expansion scheme, which are not covered by the previously known methods, are also discussed. PubDate: 2023-09-29 DOI: 10.1007/s43670-023-00069-3

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Abstract: Abstract Let G be a closed subgroup of \({\mathbb {R}}^d\) and let \(\nu \) be a Borel probability measure admitting a Riesz basis of exponentials with frequency sets in the dual group \(G^{\perp }\) . We form a multi-tiling measure \(\mu = \mu _1+\cdots +\mu _N\) where \(\mu _i\) is translationally equivalent to \(\nu \) and different \(\mu _i\) and \(\mu _j\) have essentially disjoint support. We obtain some necessary and sufficient conditions for \(\mu \) to admit a Riesz basis of exponentials . As an application, the square boundary, after a rotation, is a union of two fundamental domains of \(G = {\mathbb {Z}}\times {\mathbb {R}}\) and can be regarded as a multi-tiling measure. We show that, unfortunately, the square boundary does not admit a Riesz basis of exponentials of the form as a union of translate of discrete subgroups \({\mathbb {Z}}\times \{0\}\) . This rules out a natural candidate of potential Riesz basis for the square boundary. PubDate: 2023-09-08 DOI: 10.1007/s43670-023-00068-4

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Abstract: Abstract Learning nonlinear functions from time-varying measurements is always difficult due to the high correlation among observations. This task is more challenging when the target function is high dimensional. In this work, we propose a new method to learn high dimensional functions which depends only on a few unknown coordinates from a set of time-varying measurements. More precisely, we approximate the target function by a neural network and enforce an adaptive group Lasso constraint on the suitable weight matrix to represent the low-dimensional property of the unknown function. Using the non-negative property of the Bregman distance, we show that the proposed optimization procedure achieves loss decay. Our empirical studies show that the proposed method outperforms recent state-of-the-art methods including the sparse dictionary matrix method, and neural networks with or without group Lasso penalty. PubDate: 2023-08-29 DOI: 10.1007/s43670-023-00067-5

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Abstract: Abstract We propose a random feature model for approximating high-dimensional sparse additive functions called the hard-ridge random feature expansion method (HARFE). This method utilizes a hard-thresholding pursuit-based algorithm applied to the sparse ridge regression (SRR) problem to approximate the coefficients with respect to the random feature matrix. The SRR formulation balances between obtaining sparse models that use fewer terms in their representation and ridge-based smoothing that tend to be robust to noise and outliers. In addition, we use a random sparse connectivity pattern in the random feature matrix to match the additive function assumption. We prove that the HARFE method is guaranteed to converge with a given error bound depending on the noise and the parameters of the sparse ridge regression model. In addition, we provide a risk bound on the learned model. Based on numerical results on synthetic data as well as on real datasets, the HARFE approach obtains lower (or comparable) error than other state-of-the-art algorithms. PubDate: 2023-08-09 DOI: 10.1007/s43670-023-00063-9

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Abstract: Abstract Many empirical studies suggest that samples of continuous-time signals taken at locations randomly deviated from an equispaced grid (i.e., off-the-grid) can benefit signal acquisition, e.g., undersampling and anti-aliasing. However, explicit statements of such advantages and their respective conditions are scarce in the literature. This paper provides some insight on this topic when the sampling positions are known, with grid deviations generated i.i.d. from a variety distributions. By solving a square-root LASSO decoder with an interpolation kernel we demonstrate the capabilities of nonuniform samples for compressive sampling, an effective paradigm for undersampling and anti-aliasing. For functions in the Wiener algebra that admit a discrete s-sparse representation in some transform domain, we show that \({\mathcal {O}}(s{{\,\mathrm{poly\,\hspace{-2pt}log}\,}}N)\) random off-the-grid samples are sufficient to recover an accurate \(\frac{N}{2}\) -bandlimited approximation of the signal. For sparse signals (i.e., \(s \ll N\) ), this sampling complexity is a great reduction in comparison to equispaced sampling where \({\mathcal {O}}(N)\) measurements are needed for the same quality of reconstruction (Nyquist–Shannon sampling theorem). We further consider noise attenuation via oversampling (relative to a desired bandwidth), a standard technique with limited theoretical understanding when the sampling positions are non-equispaced. By solving a least squares problem, we show that \({\mathcal {O}}(N\log N)\) i.i.d. randomly deviated samples provide an accurate \(\frac{N}{2}\) -bandlimited approximation of the signal with suppression of the noise energy by a factor \(\sim \frac{1}{\sqrt{\log (N)}}.\) PubDate: 2023-08-07 DOI: 10.1007/s43670-023-00065-7

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Abstract: Abstract Neural networks functions are supposed to be able to encode the desired solution of an inverse problem very efficiently. In this paper, we consider the problem of solving linear inverse problems with neural network coders. First we establish some correspondences of this formulation with existing concepts in regularization theory, in particular with state space regularization, operator decomposition and iterative regularization methods. A Gauss–Newton method is suitable for solving encoded linear inverse problems, which is supported by a local convergence result. The convergence studies, however, are not complete, and are based on a conjecture on linear independence of activation functions and its derivatives. Some numerical experiments are presented to support the theoretical findings. PubDate: 2023-08-04 DOI: 10.1007/s43670-023-00066-6

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Abstract: Abstract The Graph Fourier transform (GFT) is a fundamental tool in graph signal processing. In this paper, based on singular value decomposition of the Laplacian, we introduce a novel definition of GFT on directed graphs, and use the singular values of the Laplacian to carry the notion of graph frequencies. We show that the proposed GFT has its frequencies and frequency components evaluated by solving some constrained minimization problems with low computational cost, and it could represent graph signals with different modes of variation efficiently. Moreover, the proposed GFT is consistent with the conventional GFT in the undirected graph setting, and on directed circulant graphs, it is the classical discrete Fourier transform, up to some rotation, permutation and phase adjustment. PubDate: 2023-08-02 DOI: 10.1007/s43670-023-00062-w

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Abstract: Abstract The double Fourier sphere (DFS) method uses a clever trick to transform a function defined on the unit sphere to the torus and subsequently approximate it by a Fourier series, which can be evaluated efficiently via fast Fourier transforms. Similar approaches have emerged for approximation problems on the disk, the ball, and the cylinder. In this paper, we introduce a generalized DFS method applicable to various manifolds, including all the above-mentioned cases and many more, such as the rotation group. This approach consists in transforming a function defined on a manifold to the torus of the same dimension. We show that the Fourier series of the transformed function can be transferred back to the manifold, where it converges uniformly to the original function. In particular, we obtain analytic convergence rates in case of Hölder-continuous functions on the manifold. PubDate: 2023-07-25 DOI: 10.1007/s43670-023-00064-8