Related papers: Dimension lower bounds for linear approaches to fu…
Model order reduction has been extensively studied over the last two decades. Projection-based methods such as the Proper Orthogonal Decomposition and the Reduced Basis Method enjoy the important advantages of Galerkin methods in the…
Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of $d$-dimensional second-order elliptic PDEs…
Many interesting functions arising in applications map into Riemannian manifolds. We present an algorithm, using the manifold exponential and logarithm, for approximating such functions. Our approach extends approximation techniques for…
Many interesting machine learning problems are best posed by considering instances that are distributions, or sample sets drawn from distributions. Previous work devoted to machine learning tasks with distributional inputs has done so…
Over the last decade, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. This is primarily due to the…
We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when…
Let $N(d,d^\perp)$ denote the minimum length $n$ of a linear code $C$ with $d$ and $d^{\bot}$, where $d$ is the minimum Hamming distance of $C$ and $d^{\bot}$ is the minimum Hamming distance of $C^{\bot}$. In this paper, we show a lower…
This article surveys nonlinear model reduction methods that remain effective in regimes where linear reduced-space approximations are intrinsically inefficient, such as transport-dominated problems with wave-like phenomena and moving…
In this paper, we focus on nonlinear infinite-norm minimization problems that have many applications, especially in computer science and operations research. We set a reliable Lagrangian dual aproach for solving this kind of problems in…
This paper presents generalizations of semidefinite programming formulations of 1-norm optimization problems over infinite dictionaries of vectors of complex exponentials, which were recently proposed for superresolution, gridless…
A basic problem of approximation theory, the approximation of functions from the Sobolev space W_p^r([0,1]^d) in the norm of L_q([0,1]^d), is considered from the point of view of quantum computation. We determine the quantum query…
Recently, it has been proposed in the literature to employ deep neural networks (DNNs) together with stochastic gradient descent methods to approximate solutions of PDEs. There are also a few results in the literature which prove that DNNs…
Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks as the curse of dimensionality (CoD) cannot be evaded when trying to approximate even a single ReLU neuron…
Dimension reduction is a technique used to transform data from a high-dimensional space into a lower-dimensional space, aiming to retain as much of the original information as possible. This approach is crucial in many disciplines like…
This paper generalizes recent advances on quadratic manifold (QM) dimensionality reduction by developing kernel methods-based nonlinear-augmentation dimensionality reduction. QMs, and more generally feature map-based nonlinear corrections,…
As demonstrated in many areas of real-life applications, neural networks have the capability of dealing with high dimensional data. In the fields of optimal control and dynamical systems, the same capability was studied and verified in many…
In this paper, we analyze the number of neurons and training parameters that a neural networks needs to approximate multivariate functions of bounded second mixed derivatives -- Korobov functions. We prove upper bounds on these quantities…
We propose a new technique for constructing low-rank approximations of matrices that arise in kernel methods for machine learning. Our approach pairs a novel automatically constructed analytic expansion of the underlying kernel function…
This paper deals with the problem of increasing the minimum distance of a linear code by adding one or more columns to the generator matrix. Several methods to compute extensions of linear codes are presented. Many codes improving the…
We establish a quantitative lower bound on the reach of flat norm minimizers for boundaries in $\mathbb{R}^2$.