An Algorithm for Computing Lipschitz Inner Functions in Kolmogorov's Superposition Theorem

TL;DR

This paper introduces a novel algorithm to compute Lipschitz continuous inner functions in Kolmogorov's superposition theorem, addressing previous limitations of non-smooth functions and enhancing the practicality of the representation.

Contribution

The paper presents the first algorithm capable of computing Lipschitz continuous inner functions for Kolmogorov's theorem, improving upon prior H"older continuous methods.

Findings

Developed an algorithm for Lipschitz inner functions.

Demonstrated the algorithm's effectiveness in constructing Lipschitz functions.

Addressed the limitations of previous non-smooth function approaches.

Abstract

Kolmogorov famously proved that multivariate continuous functions can be represented as a superposition of a small number of univariate continuous functions, $$ f(x_1,\dots,x_n) = \sum_{q=0}^{2n+1} \chi^q \left( \sum_{p=1}^n \psi^{pq}(x_p) \right).$$ Fridman \cite{fridman} posed the best smoothness bound for the functions $\psi^{pq}$, that such functions can be constructed to be Lipschitz continuous with constant 1. Previous algorithms to describe these inner functions have only been H\"older continuous, such as those proposed by K\"oppen and Braun and Griebel. This is problematic, as pointed out by Griebel, in that non-smooth functions have very high storage/evaluation complexity, and this makes Kolmogorov's representation (KR) impractical using the standard definition of the inner functions. To date, no one has presented a method to compute a Lipschitz continuous inner function. In this paper, we revisit Kolmogorov's theorem along with Fridman's result. We examine a simple Lipschitz function which appear to satisfy the necessary criteria for Kolmogorov's representation, but fails in the limit. We then present a full solution to the problem, including an algorithm that computes such a Lipschitz function.