The Kolmogorov-Arnold representation theorem revisited

TL;DR

This paper revisits the Kolmogorov-Arnold representation theorem, proposing modifications that better align with neural network structures and clarify the role of multiple hidden layers in function approximation.

Contribution

It introduces modifications to the theorem that transfer smoothness properties and align the representation with deep neural network architectures.

Findings

Modified the representation to transfer smoothness to the outer function

Outer function can be well approximated by ReLU networks

Supports view of deep networks as natural extensions of the theorem

Abstract

There is a longstanding debate whether the Kolmogorov-Arnold representation theorem can explain the use of more than one hidden layer in neural networks. The Kolmogorov-Arnold representation decomposes a multivariate function into an interior and an outer function and therefore has indeed a similar structure as a neural network with two hidden layers. But there are distinctive differences. One of the main obstacles is that the outer function depends on the represented function and can be wildly varying even if the represented function is smooth. We derive modifications of the Kolmogorov-Arnold representation that transfer smoothness properties of the represented function to the outer function and can be well approximated by ReLU networks. It appears that instead of two hidden layers, a more natural interpretation of the Kolmogorov-Arnold representation is that of a deep neural network where most of the layers are required to approximate the interior function.