Related papers: On the approximation by weighted ridge functions

We present two approaches for computing rational approximations to multivariate functions, motivated by their effectiveness as surrogate models for high-energy physics (HEP) applications. Our first approach builds on the Stieltjes process…

Universal approximation theorem suggests that a shallow neural network can approximate any function. The input to neurons at each layer is a weighted sum of previous layer neurons and then an activation is applied. These activation…

Using recently developed algorithms, we compute and compare best $L^2$ and $L^\infty$ rational approximations of analytic functions on the unit disk. Although there is some theory for these problems going back decades, this may be the first…

We consider the problem of approximating a function from $L^2$ by an element of a given $m$-dimensional space $V_m$, associated with some feature map $\boldsymbol{\varphi}$, using evaluations of the function at random points $x_1,…

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…

We consider the piecewise linear approximation of saddle functions of the form $f(x,y)=ax^2-by^2$ under the L-infinity error norm. We show that interpolating approximations are not optimal. One can get slightly smaller errors by allowing…

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…

In the first part we have shown that, for $L_2$-approximation of functions from a separable Hilbert space in the worst-case setting, linear algorithms based on function values are almost as powerful as arbitrary linear algorithms if the…

In this paper we focus on the linear functionals defining an approximate version of the gradient of a function. These functionals are often used when dealing with optimization problems where the computation of the gradient of the objective…

We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also find the order of this approximation by using…

In this work, approximations for real two variables function $f$ which has continuous partial $(n-1)$-derivatives $(n \ge 1)$ and has the $n$--th partial derivative of bounded bivariation or absolutely continuous are established. Explicit…

We consider in this paper the optimal approximations of convex univariate functions with feed-forward Relu neural networks. We are interested in the following question: what is the minimal approximation error given the number of…

The direct and inverse theorems are established for the best approximation in the weighted $L^p$ space on the unit sphere of $\RR^{d+1}$, in which the weight functions are invariant under finite reflection groups. The theorems are stated…

We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some linear methods of this approximation for univariate functions in the class induced by the convolution…

A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal…

We obtain closed expressions for weighted orthogonal polynomials and optimal approximants associated with the function $f(z)=1-\frac{1}{\sqrt{2}}(z_1+z_2)$ and a scale of Hilbert function spaces in the unit $2$-ball having reproducing…

We present one of the approaches to find the best approximation of the given function by trigonometric polynomials in $L^1$ metric and applied it to find the optimal constants in the Nikolsky's type inequality, concerning approximation of…

We propose a randomized lattice algorithm for approximating multivariate periodic functions over the $d$-dimensional unit cube from the weighted Korobov space with mixed smoothness $\alpha > 1/2$ and product weights…

The paper develops applications of symmetric orbit functions, known from irreducible representations of simple Lie groups, in numerical analysis. It is shown that these functions have remarkable properties which yield to cubature formulas,…

Functions with singularities are notoriously difficult to approximate with conventional approximation schemes. In computational applications, they are often resolved with low-order piecewise polynomials, multilevel schemes, or other types…