Extension of localised approximation by neural networks

Approximation of smooth functions by neural networks

We review some aspects of our recent work on the approximation of functions by neural and generalized translation networks.

Geometric Rates of Approximation by Neural Networks

Model complexity of feedforward neural networks is studied in terms of rates of variable-basis approximation. Sets of functions, for which the errors in approximation by neural networks with n hidden units converge to zero geometrically fast with increasing number n, are described. However, the geometric speed of convergence depends on parameters, which are specific for each function to be appr...

Point Convolutional Neural Networks by Extension Operators

This paper presents Point Convolutional Neural Networks (PCNN): a novel framework for applying convolutional neural networks to point clouds. The framework consists of two operators: extension and restriction, mapping point cloud functions to volumetric functions and viseversa. A point cloud convolution is defined by pull-back of the Euclidean volumetric convolution via an extensionrestriction ...

Intelligent Systems: Approximation by Artificial Neural Networks

Approximation by Ridge Functions and Neural Networks

We investigate the efficiency of approximation by linear combinations of ridge functions in the metric of L2(B ) with Bd the unit ball in Rd. If Xn is an n-dimensional linear space of univariate functions in L2(I), I = [−1, 1], and Ω is a subset of the unit sphere Sd−1 in Rd of cardinality m, then the space Yn := span{r(x · ξ) : r ∈ Xn, ω ∈ Ω} is a linear space of ridge functions of dimension ≤...