Related papers: A Local Fourier Extension Method for Function Appr…
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…
Let us assume that $f$ is a continuous function defined on the unit ball of $\mathbb R^d$, of the form $f(x) = g (A x)$, where $A$ is a $k \times d$ matrix and $g$ is a function of $k$ variables for $k \ll d$. We are given a budget $m \in…
This work introduces a reduced order modeling (ROM) framework for the solution of parameterized second-order linear elliptic partial differential equations formulated on unfitted geometries. The goal is to construct efficient…
Sparse variational approximations are popular methods for scaling up inference and learning in Gaussian processes to larger datasets. For $N$ training points, exact inference has $O(N^3)$ cost; with $M \ll N$ features, state of the art…
We study the random sampling of the short-time Fourier transform of functions that are localized in a compact region in the time-frequency plane. We follow the approach introduced by Bass and Gr\"ochenig for band-limited functions, and show…
In this article, we develop comprehensive frequency domain methods for estimating and inferring the second-order structure of spatial point processes. The main element here is on utilizing the discrete Fourier transform (DFT) of the point…
We suggest a new method, called Functional Additive Regression, or FAR, for efficiently performing high-dimensional functional regression. FAR extends the usual linear regression model involving a functional predictor, $X(t)$, and a scalar…
We present a closed-form finite-dimensional projection method for regularizing a function defined by a discrete set of measurement data, which have been contaminated by random, zero mean errors, and for estimating the derivative and…
This paper introduces a new method for semi-supervised learning on high dimensional nonlinear manifolds, which includes a phase of unsupervised basis learning and a phase of supervised function learning. The learned bases provide a set of…
This paper presents a new approach in application of the Fourier transform to the complex error function resulting in an efficient rational approximation. Specifically, the computational test shows that with only $17$ summation terms the…
We introduce an efficient algorithm, called partition of unity extension or PUX, to construct an extension of desired regularity of a function given on a complex multiply connected domain in $2D$. Function extension plays a fundamental role…
Physical laws governing population dynamics are generally expressed as differential equations. Research in recent decades has incorporated fractional-order (non-integer) derivatives into differential models of natural phenomena, such as…
The recent development of spectral method has been praised for its high-order convergence in simulating complex physical problems. The combination of embedded boundary method and spectral method becomes a mainstream way to tackle…
Modern deep neural networks suffer from performance degradation when evaluated on testing data under different distributions from training data. Domain generalization aims at tackling this problem by learning transferable knowledge from…
A method for numerical approximation of a new class of fractional parabolic stochastic evolution equations is introduced and analysed. This class of equations has recently been proposed as a space-time extension of the SPDE-method in…
In this article we propose a new adaptive numerical quadrature procedure which includes both local subdivision of the integration domain, as well as local variation of the number of quadrature points employed on each subinterval. In this…
We consider the problem of approximating a truncated Gaussian kernel using Fourier (trigonometric) functions. The computation-intensive bilateral filter can be expressed using fast convolutions by applying such an approximation to its range…
This article introduces the Generalized Fourier Series (GFS), a novel spectral method that extends the clas- sical Fourier series to non-periodic functions. GFS addresses key challenges such as the Gibbs phenomenon and poor convergence in…
Most commonly used \emph{adaptive} algorithms for univariate real-valued function approximation and global minimization lack theoretical guarantees. Our new locally adaptive algorithms are guaranteed to provide answers that satisfy a…
We introduce a framework for subspace methods which approximate the spectra of self-adjoint, unbounded operators in a local region. Using the projection-valued measure, we derive integrated spectral inequalities that also apply to unbounded…