Related papers: Approximation of high-dimensional periodic functio…
Decomposition theorems in classical Fourier analysis enable us to express a bounded function in terms of few linear phases with large Fourier coefficients plus a part that is pseudorandom with respect to linear phases. The Goldreich-Levin…
Consider an $s$-dimensional function being evaluated at $n$ points of a low discrepancy sequence (LDS), where the objective is to approximate the one-dimensional functions that result from integrating out $(s-1)$ variables. Here, the…
The paper develops a method for discrete computational Fourier analysis of functions defined on quasicrystals and other almost periodic sets. A key point is to build the analysis around the emerging theory of quasicrystals and diffraction…
This paper presents a quadratic formula-based nonlinear representation for a given single-variable function f(x), $-1 \leq x \leq 1$. First, we construct the explicit polynomial coefficient functions a(x), b(x), and c(x) using a…
The reconstruction of high-dimensional sparse signals is a challenging task in a wide range of applications. In order to deal with high-dimensional problems, efficient sparse fast Fourier transform algorithms are essential tools. The second…
Approximating the action of a matrix function $f(\mathbf{A})$ on a vector $\mathbf{b}$ is an increasingly important primitive in machine learning, data science, and statistics, with applications such as sampling high dimensional Gaussians,…
We propose a new cross-conv algorithm for approximate computation of convolution in different low-rank tensor formats (tensor train, Tucker, Hierarchical Tucker). It has better complexity with respect to the tensor rank than previous…
A key problem in approximation theory is the recovery of high-dimensional functions from samples. In many cases, the functions of interest exhibit anisotropic smoothness, and, in many practical settings, the nature of this anisotropy may be…
We develop new approximation algorithms and data structures for representing and computing with multivariate functions using the functional tensor-train (FT), a continuous extension of the tensor-train (TT) decomposition. The FT represents…
For the Poisson equation posed in a domain containing a large number of polygonal perforations, we propose a low-dimensional coarse approximation space based on a coarse polygonal partitioning of the domain. Similarly to other multiscale…
The present article is concerned scattered data approximation for higher dimensional data sets which exhibit an anisotropic behavior in the different dimensions. Tailoring sparse polynomial interpolation to this specific situation, we…
We present a method to approximate functionals $\text{Tr} \, f(A)$ of very high-dimensional hermitian matrices $A$ represented as Matrix Product Operators (MPOs). Our method is based on a reformulation of a block Lanczos algorithm in tensor…
We exploit the truncated singular value decomposition and the recently proposed circulant decomposition for an efficient first-order approximation of the multiplication of large dense matrices. A decomposition of each matrix into a sum of a…
We present a method for constructing superoscillatory functions the superoscillatory part of which approximates a given polynomial with arbitrarily small error in a fixed interval. These functions are obtained as the product of the…
Matrix factorization methods are important tools in data mining and analysis. They can be used for many tasks, ranging from dimensionality reduction to visualization. In this paper we concentrate on the use of matrix factorizations for…
In this paper, we introduce a method for multivariate function approximation using function evaluations, Chebyshev polynomials, and tensor-based compression techniques via the Tucker format. We develop novel randomized techniques to…
We present a constructive approximation framework for analyzing the expressive power of Fourier residual networks in approximating a broad class of one-dimensional functions. Our study covers both piecewise continuous functions -- including…
We propose algorithms for approximate filtering and smoothing in high-dimensional Factorial hidden Markov models. The approximation involves discarding, in a principled way, likelihood factors according to a notion of locality in a factor…
This work provides a systematic recipe for computing accurate high order Fourier expansions of quasiperiodic invariant circles in area preserving maps. The recipe requires only a finite data set sampled from the quasiperiodic circle. Our…
We introduce a numerical method for the approximation of functions which are analytic on compact intervals, except at the endpoints. This method is based on variable transforms using particular parametrized exponential and…