Explicit Construction of Approximate Kolmogorov Superpositions with C2 Smoothness

TL;DR

This paper presents an explicit construction of approximate Kolmogorov superpositions with C2 smoothness, enabling accurate approximation of Holder continuous functions while addressing previous pathological issues.

Contribution

It introduces a novel explicit method for constructing smooth Kolmogorov superpositions with practical approximation capabilities and neural network applications.

Findings

Achieves approximation accuracy of N^(-alpha) for alpha-Hölder functions.

Constructs inner functions via translations and dilations of piecewise C2 functions.

Outer functions are designed through piecewise C2 interpolation with new shape functions.

Abstract

We explicitly construct an approximate version of the Kolmogorov superpositions, which is composed of C2-inner and outer functions, and can approximate an arbitrary alpha Holder continuous function with accuracy of N to the power -alpha, where N denotes the number of outer summations. The inner functions are generated by applying suitable translations and dilations to a piecewise C2, strictly increasing function, while the outer functions are constructed rowwise through piecewise C2 interpolation using newly designed shape functions. This novel variant of Kolmogorov superpositions overcomes the wild and pathological behaviors of the inherent single variable functions, but retains the essence of Kolmogorov strategy of exact representation-an objective that Sprecher (Neural Netw. 144(2021)438-442) has actively pursued. We also discuss the implications of this new construction and demonstrate its applicability to related neural networks.