On the representation by bivariate ridge functions

TL;DR

This paper investigates how bivariate functions can be represented as sums of ridge functions, demonstrating that finite sums of arbitrary ridge functions can be replaced by sums of ridge functions with the same smoothness, with applications to PDEs.

Contribution

It proves that functions represented by finite sums of arbitrary ridge functions can also be represented by sums of ridge functions with the same smoothness class, extending the understanding of function representation.

Findings

Representation equivalence for smooth functions and ridge functions.

Application to homogeneous constant coefficient PDEs.

Extension of ridge function representation theory.

Abstract

We consider the problem of representation of a bivariate function by sums of ridge functions. We show that if a function of a certain smoothness class is represented by a sum of finitely many, arbitrarily behaved ridge functions, then it can also be represented by a sum of ridge functions of the same smoothness class. As an example, this result is applied to a homogeneous constant coefficient partial differential equation.