Related papers: On approximating the shape of one dimensional func…
In the paper we consider the problem of multivariate function approximation in polynomial basis. In order to solve this problem, we adjust the least squares method (LSM) by adding information about derivatives of the function. This…
Most approximation methods in high dimensions exploit smoothness of the function being approximated. These methods provide poor convergence results for non-smooth functions with kinks. For example, such kinks can arise in the uncertainty…
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…
The Linear Smoothing (LS) scheme \cite{francisa.ortiz-bernardin2017} ameliorates linear and quadratic approximations over convex polytopes by employing a three-point integration scheme. In this work, we propose a linearly consistent one…
Sleeve functions are generalizations of the well-established ridge functions that play a major role in the theory of partial differential equation, medical imaging, statistics, and neural networks. Where ridge functions are non-linear,…
Our main interest in this paper is to study some approximation problems for classes of functions with mixed smoothness. We use technique, based on a combination of results from hyperbolic cross approximation, which were obtained in 1980s --…
$\renewcommand{\Re}{\mathbb{R}}\newcommand{\eps}{{\varepsilon}}\newcommand{\poly}{\mathrm{poly}} $In this paper, we study the problem of $L_1$-fitting a shape to a set of $n$ points in $\Re^d$ (where $d$ is a fixed constant), where the…
The Active Subspace (AS) method is a widely used technique for identifying the most influential directions in high-dimensional input spaces that affect the output of a computational model. The standard AS algorithm requires a sufficient…
The objective of this paper is to design an embedding method that maps local features describing an image (e.g. SIFT) to a higher dimensional representation useful for the image retrieval problem. First, motivated by the relationship…
Learning approximations to smooth target functions of many variables from finite sets of pointwise samples is an important task in scientific computing and its many applications in computational science and engineering. Despite well over…
This study presents a generalised least squares based method for fitting polygons and ellipses to data points. The method is based on a trigonometric fitness function that approximates a unit shape accurately, making it applicable to…
Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously.…
One-dimensional function approximation is a fundamental problem in scientific computing and engineering applications. While neural networks possess powerful universal approximation capabilities, their optimization process is often hindered…
In this paper, we consider the fundamental problem of approximation of functions on a low-dimensional manifold embedded in a high-dimensional space, with noise affecting both in the data and values of the functions. Due to the curse of…
In order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes…
We present an algorithm for approximating a function defined over a $d$-dimensional manifold utilizing only noisy function values at locations sampled from the manifold with noise. To produce the approximation we do not require any…
In recent years, the use of sparse recovery techniques in the approximation of high-dimensional functions has garnered increasing interest. In this work we present a survey of recent progress in this emerging topic. Our main focus is on the…
We consider the problem of approximating a smooth function from finitely-many pointwise samples using $\ell^1$ minimization techniques. In the first part of this paper, we introduce an infinite-dimensional approach to this problem. Three…
We study stochastic gradient descent (SGD) with gradient clipping on convex functions under a generalized smoothness assumption called $(L_0,L_1)$-smoothness. Using gradient clipping, we establish a high probability convergence rate that…
In this paper, we introduce a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in $d$ dimensions, where $d$ can be arbitrary. This method is simple, and relies only…