Applied and Computational Harmonic Analysis最新文献

Atomic norm methods have recently been proposed for spectral super-resolution with flexibility in dealing with missing data and miscellaneous noises. A notorious drawback of these convex optimization methods however is their lower resolution in the high signal-to-noise (SNR) regime as compared to conventional methods such as ESPRIT. In this paper, we devise a simple weighting scheme in existing atomic norm methods and show that in theory the resolution of the resulting convex optimization method can be made arbitrarily high in the absence of noise, achieving the so-called separation-free super-resolution. This is proved by a novel, kernel-free construction of the dual certificate whose existence guarantees exact super-resolution using the proposed method. Numerical results corroborating our analysis are provided.

The recovery of multivariate functions and estimating their integrals from finitely many samples is one of the central tasks in modern approximation theory. Marcinkiewicz–Zygmund inequalities provide answers to both the recovery and the quadrature aspect. In this paper, we put ourselves on the q-dimensional sphere $S^q$, and investigate how well continuous $L_p$-norms of polynomials f of maximum degree n on the sphere $S^q$ can be discretized by positively weighted $L_p$-sum of finitely many samples, and discuss the distortion between the continuous and discrete quantities, the number and distribution of the (deterministic or randomly chosen) sample points ${\xi }_1,\dots ,{\xi }_N$ on $S^q$, the dimension q, and the degree n of the polynomials.

For a pair of sets $T,\Omega \subset R$ the time-frequency localization operator is defined as $S_{T,\Omega }=P_T{F}^{-1}{P}_{\Omega }F{P}_T$, where $F$ is the Fourier transform and $P_T,P_{\Omega }$ are projection operators onto T and Ω, respectively. We show that in the case when both T and Ω are intervals, the eigenvalues of $S_{T,\Omega }$ satisfy ${\lambda }_n(T,\Omega )\ge 1-{\delta }^{\left|T\right|\left|\Omega \right|}$ if $n\le (1-\epsilon )\left|T\right|\left|\Omega \right|$, where $\epsilon >0$ is arbitrary and $\delta =\delta \left(\epsilon \right)<1$, provided that $\left|T\right|\left|\Omega \right|>c_{\epsilon }$. This improves the result of Bonami, Jaming and Karoui, who proved it for $\epsilon \ge 0.42$. The proof is based on the properties of the Bargmann transform.

The high efficiency of a recently proposed method for computing with Gaussian processes relies on expanding a (translationally invariant) covariance kernel into complex exponentials, with frequencies lying on a Cartesian equispaced grid. Here we provide rigorous error bounds for this approximation for two popular kernels—Matérn and squared exponential—in terms of the grid spacing and size. The kernel error bounds are uniform over a hypercube centered at the origin. Our tools include a split into aliasing and truncation errors, and bounds on sums of Gaussians or modified Bessel functions over various lattices. For the Matérn case, motivated by numerical study, we conjecture a stronger Frobenius-norm bound on the covariance matrix error for randomly-distributed data points. Lastly, we prove bounds on, and study numerically, the ill-conditioning of the linear systems arising in such regression problems.

We give an asymptotic expansion of the relative entropy between the heat kernel $q_Z(t,z,w)$ of a compact Riemannian manifold Z and the normalized Riemannian volume for small values of t and for a fixed element $z\in Z$. We prove that coefficients in the expansion can be expressed as universal polynomials in the components of the curvature tensor and its covariant derivatives at z, when they are expressed in terms of normal coordinates. We describe a method to compute the coefficients, and we use the method to compute the first three coefficients. The asymptotic expansion is necessary for an unsupervised machine-learning algorithm called the Diffusion Variational Autoencoder.

We introduce a new concept of variable bandwidth that is based on the frequency truncation of Wilson expansions. For this model we derive sampling theorems, a complete reconstruction of f from its samples, and necessary density conditions for sampling. Numerical simulations support the interpretation of this model of variable bandwidth. In particular, chirps, as they arise in the description of gravitational waves, can be modeled in a space of variable bandwidth.

Graph Laplacian based algorithms for data lying on a manifold have been proven effective for tasks such as dimensionality reduction, clustering, and denoising. In this work, we consider data sets whose data points lie on a manifold that is closed under the action of a known unitary matrix Lie group G. We propose to construct the graph Laplacian by incorporating the distances between all the pairs of points generated by the action of G on the data set. We deem the latter construction the “G-invariant Graph Laplacian” (G-GL). We show that the G-GL converges to the Laplace-Beltrami operator on the data manifold, while enjoying a significantly improved convergence rate compared to the standard graph Laplacian which only utilizes the distances between the points in the given data set. Furthermore, we show that the G-GL admits a set of eigenfunctions that have the form of certain products between the group elements and eigenvectors of certain matrices, which can be estimated from the data efficiently using FFT-type algorithms. We demonstrate our construction and its advantages on the problem of filtering data on a noisy manifold closed under the action of the special unitary group $SU\left(2\right)$.

The separate tasks of denoising, least squares expectation, and manifold learning can often be posed in a common setting of finding the conditional expectations arising from a product of two random variables. This paper focuses on this more general problem and describes an operator theoretic approach to estimating the conditional expectation. Kernel integral operators are used as a compactification tool, to set up the estimation problem as a linear inverse problem in a reproducing kernel Hilbert space. This equation is shown to have solutions that allow numerical approximation, thus guaranteeing the convergence of data-driven implementations. The overall technique is easy to implement, and their successful application to some real-world problems is also shown.

The scattering transform is a multilayered, wavelet-based transform initially introduced as a mathematical model of convolutional neural networks (CNNs) that has played a foundational role in our understanding of these networks' stability and invariance properties. In subsequent years, there has been widespread interest in extending the success of CNNs to data sets with non-Euclidean structure, such as graphs and manifolds, leading to the emerging field of geometric deep learning. In order to improve our understanding of the architectures used in this new field, several papers have proposed generalizations of the scattering transform for non-Euclidean data structures such as undirected graphs and compact Riemannian manifolds without boundary. Analogous to the original scattering transform, these works prove that these variants of the scattering transform have desirable stability and invariance properties and aim to improve our understanding of the neural networks used in geometric deep learning.

In this paper, we introduce a general, unified model for geometric scattering on measure spaces. Our proposed framework includes previous work on compact Riemannian manifolds without boundary and undirected graphs as special cases but also applies to more general settings such as directed graphs, signed graphs, and manifolds with boundary. We propose a new criterion that identifies to which groups a useful representation should be invariant and show that this criterion is sufficient to guarantee that the scattering transform has desirable stability and invariance properties. Additionally, we consider finite measure spaces that are obtained from randomly sampling an unknown manifold. We propose two methods for constructing a data-driven graph on which the associated graph scattering transform approximates the scattering transform on the underlying manifold. Moreover, we use a diffusion-maps based approach to prove quantitative estimates on the rate of convergence of one of these approximations as the number of sample points tends to infinity. Lastly, we showcase the utility of our method on spherical images, a directed graph stochastic block model, and on high-dimensional single-cell data.

Recently we have characterized the convergence of bivariate subdivision scheme with nonnegative mask whose support is convex by means of the so-called connectivity of a square matrix, which is derived by a given mask. The convergence in this case can be checked in linear time with respected to the size of a square matrix. This paper will focus on the characterization of such schemes with non-convex supports.