An Explicit Neural Network Construction for Piecewise Constant Function Approximation

TL;DR

This paper introduces an explicit two-hidden-layer neural network that constructs piecewise constant approximations of multivariate functions without numerical training, using Voronoi tessellation for practical applications.

Contribution

The paper proposes a novel explicit neural network design that avoids numerical optimization and tensor structures, enabling practical piecewise constant function approximation.

Findings

The network automatically creates Voronoi tessellations based on data.

The construction does not require training via optimization.

Numerical examples validate the theoretical properties.

Abstract

We present an explicit construction for feedforward neural network (FNN), which provides a piecewise constant approximation for multivariate functions. The proposed FNN has two hidden layers, where the weights and thresholds are explicitly defined and do not require numerical optimization for training. Unlike most of the existing work on explicit FNN construction, the proposed FNN does not rely on tensor structure in multiple dimensions. Instead, it automatically creates Voronoi tessellation of the domain, based on the given data of the target function, and piecewise constant approximation of the function. This makes the construction more practical for applications. We present both theoretical analysis and numerical examples to demonstrate its properties.