Related papers: Fourier Extension Based on Weighted Generalized In…
For multi-variable finite measure spaces, we present in this paper a new framework for non-orthogonal $L^2$ Fourier expansions. Our results hold for probability measures $\mu$ with finite support in $\mathbb{R}^d$ that satisfy a certain…
Standard sparse pseudo-input approximations to the Gaussian process (GP) cannot handle complex functions well. Sparse spectrum alternatives attempt to answer this but are known to over-fit. We suggest the use of variational inference for…
We present an algorithm for the forward propagation of intervals through the discrete Fourier transform. The algorithm yields best-possible bounds when computing the amplitude of the Fourier transform for real and complex valued sequences.…
Variational methods have become an important kind of methods in signal and image restoration - a typical inverse problem. One important minimization model consists of the squared $\ell_2$ data fidelity (corresponding to Gaussian noise) and…
We study generative compressed sensing when the measurement matrix is randomly subsampled from a unitary matrix (with the DFT as an important special case). It was recently shown that $\textit{O}(kdn\| \boldsymbol{\alpha}\|_{\infty}^{2})$…
Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. The second paper is concerned with simultaneous approximation to functions and their…
The FC-Gram algorithm approximates non-periodic functions to high order by constructing a periodic extension with controlled boundary behavior and applying trigonometric interpolation. In this paper we introduce a generalized FC-Gram…
Traditional finite element approaches are well-known to introduce spurious oscillations when applied to advection-dominated problems. We explore alleviation of this issue from the perspective of a generalized finite element formulation,…
We consider the problem of recovering a compactly-supported function from a finite collection of pointwise samples of its Fourier transform taking nonuniformly. First, we show that under suitable conditions on the sampling frequencies -…
This paper presents a novel systematic methodology to obtain new simple and tight approximations, lower bounds, and upper bounds for the Gaussian Q-function, and functions thereof, in the form of a weighted sum of exponential functions.…
To address the common problem of high dimensionality in tensor regressions, we introduce a generalized tensor random projection method that embeds high-dimensional tensor-valued covariates into low-dimensional subspaces with minimal loss of…
This paper presents a Carleman-Fourier linearization method for nonlinear dynamical systems with periodic vector fields involving multiple fundamental frequencies. By employing Fourier basis functions, the nonlinear dynamical system is…
Image smoothing is a fundamental procedure in applications of both computer vision and graphics. The required smoothing properties can be different or even contradictive among different tasks. Nevertheless, the inherent smoothing nature of…
The main goal of this paper is to propose a new quaternion total variation regularization model for solving linear ill-posed quaternion inverse problems, which arise from three-dimensional signal filtering or color image processing. The…
Total generalization variation (TGV) is a very powerful and important regularization for various inverse problems and computer vision tasks. In this paper, we proposed a semismooth Newton based augmented Lagrangian method to solve this…
The efficient and reliable approximation of convection-dominated problems continues to remain a challenging task. To overcome the difficulties associated with the discretization of convection-dominated equations, stabilization techniques…
We provide faster algorithms for the problem of Gaussian summation, which occurs in many machine learning methods. We develop two new extensions - an O(Dp) Taylor expansion for the Gaussian kernel with rigorous error bounds and a new error…
We propose a novel sparse spectrum approximation of Gaussian process (GP) tailored for Bayesian optimization. Whilst the current sparse spectrum methods provide desired approximations for regression problems, it is observed that this…
This paper introduces a new functional expansion framework that extends classical ideas beyond the Taylor series. Unlike traditional Taylor expansions based on local polynomial approximations, the proposed approach arises from exact…
This paper concerns the inverse source scattering problems of recovering random sources for acoustic and elastic waves. The underlying sources are assumed to be random functions driven by an additive white noise. The inversion process aims…