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Neural implicit surfaces are a promising tool for geometry processing that represent a solid object as the zero level set of a neural network. Usually trained to approximate a signed distance function of the considered object, these methods…

We propose a local Legendre frame (LLF) method for function approximation from equispaced data on a finite interval. Motivated by the difficulty of stable high-order polynomial approximation at equispaced points, especially in the presence…

Scientists and engineers rely on accurate mathematical models to quantify the objects of their studies, which are often high-dimensional. Unfortunately, high-dimensional models are inherently difficult, i.e. when observations are sparse or…

Multivariate functions encountered in high-dimensional uncertainty quantification problems often vary most strongly along a few dominant directions in the input parameter space. We propose a gradient-based method for detecting these…

Motivated by conforming finite element methods for elliptic problems of second order, we analyze the approximation of the gradient of a target function by continuous piecewise polynomial functions over a simplicial mesh. The main result is…

Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…

This paper demonstrates that the space of piecewise smooth functions can be well approximated by the space of functions defined by a set of simple (non-linear) operations on smooth uniform splines. The examples include bivariate functions…

Function approximation based on data drawn randomly from an unknown distribution is an important problem in machine learning. The manifold hypothesis assumes that the data is sampled from an unknown submanifold of a high dimensional…

Single hidden layer feedforward neural networks can represent multivariate functions that are sums of ridge functions. These ridge functions are defined via an activation function and customizable weights. The paper deals with best…

We study universal approximation of continuous functionals on compact subsets of products of Hilbert spaces. We prove that any such functional can be uniformly approximated by models that first take finitely many continuous linear…

Generic approximation of entire functions by their Pad\'{e} approximants has been achieved in the past (\cite{3}). In the present article we obtain generic approximation of holomorphic functions on arbitrary open sets by sequences of their…

In this paper, we present a sharper version of the results in the paper Dimension independent bounds for general shallow networks; Neural Networks, \textbf{123} (2020), 142-152. Let $\mathbb{X}$ and $\mathbb{Y}$ be compact metric spaces. We…

Several important algorithms for machine learning and data analysis use pairwise distances as input. On Riemannian manifolds these distances may be prohibitively costly to compute, in particular for large datasets. To tackle this problem,…

The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space $\mathbb{R}^n$. All continuous functions that vanish at infinity can be uniformly approximated by neural networks with one hidden…

Functions on a bounded domain in scientific computing are often approximated using piecewise polynomial approximations on meshes that adapt to the shape of the geometry. We study the problem of function approximation using splines on a…

Analyzing high-dimensional data with manifold learning algorithms often requires searching for the nearest neighbors of all observations. This presents a computational bottleneck in statistical manifold learning when observations of…

We study the approximation of measurable functions on the hypercube by functions arising from affine neural networks. Our main achievement is an approximation of any measurable function $f \colon W_n \to [-1,1]$ up to a prescribed precision…

We prove that every nonnegative continuous real-valued function on a given compact metric space is the uniform limit of some increasing sequence of nonnegative simple functions being linear combinations of indicators of open sets; here the…

We study the best uniform approximation by polynomials of fixed degree of the function sgn(x) on the union of two intervals symmetric with respect to the origin. We obtain precise asymptotics, with explicit constants, for the error of the…

We describe a method for approximating a single-variable function $f$ using persistence diagrams of sublevel sets of $f$ from height functions in different directions. We provide algorithms for the piecewise linear case and for the smooth…