Addressing common misinterpretations of KART and UAT in neural network literature

TL;DR

This paper clarifies common misunderstandings of KART and UAT in neural network literature and shows that the neuron count for exact KART representation also applies to universal approximation in standard networks.

Contribution

It provides a clearer interpretation of KART and UAT and demonstrates the equivalence in neuron requirements for exact representation and approximation.

Findings

Clarifies misconceptions about KART and UAT

Shows neuron count for exact representation applies to approximation

Supports accurate understanding among neural network researchers

Abstract

This note addresses the Kolmogorov-Arnold Representation Theorem (KART) and the Universal Approximation Theorem (UAT), focusing on their frequent misinterpretations found in the neural network literature. Our remarks aim to support a more accurate understanding of KART and UAT among neural network specialists. In addition, we explore the minimal number of neurons required for universal approximation, showing that the same number of neurons needed for exact representation of functions in KART-based networks also suffices for standard multilayer perceptrons in the context of approximation.