Related papers: Spectral algorithms in higher-order Fourier analys…
The nonlinear Fourier transform (NFT) has recently gained significant attention in fiber optic communications and other engineering fields. Although several numerical algorithms for computing the NFT have been published, the design of…
We offer a spectral analysis for a class of transfer operators. These transfer operators arise for a wide range of stochastic processes, ranging from random walks on infinite graphs to the processes that govern signals and recursive wavelet…
Derivatives and integration operators are well-studied examples of linear operators that commute with scaling up to a fixed multiplicative factor; i.e., they are scale-invariant. Fractional order derivatives (integration operators) also…
We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. We interpret this…
Amplitude estimation algorithms are based on Grover's algorithm: alternating reflections about the input state and the desired outcome. But what if we are given the ability to perform arbitrary rotations, instead of just reflections? In…
In this paper, we establish a condition on the coefficients of differential operators generated in the space of square-integrable functions on the entire real line by an ordinary differential expression with periodic, complex-valued…
For smooth random dynamical systems we consider the quenched linear and higher-order response of equivariant physical measures to perturbations of the random dynamics. We show that the spectral perturbation theory of Gou\"ezel, Keller, and…
Many computational algorithms applied to geometry operate on discrete representations of shape. It is sometimes necessary to first simplify, or coarsen, representations found in modern datasets for practicable or expedited processing. The…
A plentitude of applications in scientific computing requires the approximation of mappings between Banach spaces. Recently introduced Fourier Neural Operator (FNO) and Deep Operator Network (DeepONet) can provide this functionality. For…
We introduce a simple, general, and convergent scheme to compute generalized eigenfunctions of self-adjoint operators with continuous spectra on rigged Hilbert spaces. Our approach does not require prior knowledge about the eigenfunctions,…
The work reported in this article presents a high-order, stable, and efficient Gegenbauer pseudospectral method to solve numerically a wide variety of mathematical models. The proposed numerical scheme exploits the stability and the…
We address the problem of causal effect estimation in the presence of hidden confounders, using nonparametric instrumental variable (IV) regression. A leading strategy employs spectral features - that is, learned features spanning the top…
Our goal is to provide a review of deep learning methods which provide insight into structured high-dimensional data. Rather than using shallow additive architectures common to most statistical models, deep learning uses layers of…
Neural operators have emerged as powerful surrogates for modeling complex physical problems. However, they suffer from spectral bias making them oblivious to high-frequency modes, which are present in multiscale physical systems. Therefore,…
Ergodic theory, Higher order Fourier analysis and the hyper graph regularity method are three possible approaches to Szemer\'edi type theorems in abelian groups. In this paper we develop an algebraic theory that creates a connection between…
The aim of this paper is to give an overview of the spectral theories associated with the notions of holomorphicity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is…
The main goal of this paper is to propose an approach to inverse spectral problems for functional-differential operators (FDO) with involution. For definiteness, we focus on the second-order FDO with involution-reflection. Our approach is…
A sequential quadratic optimization algorithm for minimizing an objective function defined by an expectation subject to nonlinear inequality and equality constraints is proposed, analyzed, and tested. The context of interest is when it is…
The goal of this paper is to combine ideas from the theory of mixed spectral problems for differential operators with new results in the area of the Uncertainty Principle in Harmonic Analysis (UP). Using recent solutions of Gap and Type…
This paper introduces a novel deep neural network architecture for solving the inverse scattering problem in frequency domain with wide-band data, by directly approximating the inverse map, thus avoiding the expensive optimization loop of…