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On the Kolmogorov Superposition Theorem and Regular Means

Miguel de Carvalho

TL;DR

The paper shows that Kolmogorov's regular means, in the form $M_g(oldsymbol{x}) = g^{-1}igl( rac{1}{n} extstyle rac{1}{n}igr)$, can be derived from the Kolmogorov superposition theorem, bridging axiomatic mean theory with a constructive function representation. It establishes a universal central limit theorem for all regular means, namely $ obreak\sqrt{n}igl(M_g(oldsymbol{X})-E_g(X)igr) o Nigl(0, rac{varigl(g(X)igr)}{(g'(E_g(X)))^2}igr)$, and analyzes convergence via Edgeworth expansions alongside numerical experiments. It also proves a continuity/stability result: $M_g$ depends continuously on the generator $g$ under mild Lipschitz conditions, ensuring robust performance under small changes to $g$. The work highlights the conceptual link between a classical axiomatic notion of mean and modern function-approximation perspectives, suggesting broader implications for statistical modeling (e.g., additive representations) and potential connections to neural architectures that leverage univariate inner/outer function combinations.

Abstract

While Kolmogorov's probability axioms are widely recognized, it is less well known that in an often-overlooked 1930 note, Kolmogorov proposed an axiomatic framework for a unifying concept of the mean -- referred to as regular means. This framework yields a well-defined functional form encompassing the arithmetic, geometric, and harmonic means, among others. In this article, we uncover an elegant connection between two key results of Kolmogorov by showing that the class of regular means can be derived directly from the Kolmogorov superposition theorem. This connection is conceptually appealing and illustrates that the superposition theorem deserves wider recognition in Statistics -- not only because of its link to regular means as shown here, but also due to its influence on the development of neural models and its potential connections with other statistical frameworks. In addition, we establish a stability property of regular means, showing that they vary smoothly under small perturbations of the generator. Finally, we provide insights into a recent universal central limit theorem that applies to the broad class of regular means.

On the Kolmogorov Superposition Theorem and Regular Means

TL;DR

The paper shows that Kolmogorov's regular means, in the form , can be derived from the Kolmogorov superposition theorem, bridging axiomatic mean theory with a constructive function representation. It establishes a universal central limit theorem for all regular means, namely , and analyzes convergence via Edgeworth expansions alongside numerical experiments. It also proves a continuity/stability result: depends continuously on the generator under mild Lipschitz conditions, ensuring robust performance under small changes to . The work highlights the conceptual link between a classical axiomatic notion of mean and modern function-approximation perspectives, suggesting broader implications for statistical modeling (e.g., additive representations) and potential connections to neural architectures that leverage univariate inner/outer function combinations.

Abstract

While Kolmogorov's probability axioms are widely recognized, it is less well known that in an often-overlooked 1930 note, Kolmogorov proposed an axiomatic framework for a unifying concept of the mean -- referred to as regular means. This framework yields a well-defined functional form encompassing the arithmetic, geometric, and harmonic means, among others. In this article, we uncover an elegant connection between two key results of Kolmogorov by showing that the class of regular means can be derived directly from the Kolmogorov superposition theorem. This connection is conceptually appealing and illustrates that the superposition theorem deserves wider recognition in Statistics -- not only because of its link to regular means as shown here, but also due to its influence on the development of neural models and its potential connections with other statistical frameworks. In addition, we establish a stability property of regular means, showing that they vary smoothly under small perturbations of the generator. Finally, we provide insights into a recent universal central limit theorem that applies to the broad class of regular means.
Paper Structure (16 sections, 5 theorems, 27 equations, 2 figures, 1 table)

This paper contains 16 sections, 5 theorems, 27 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Preparations on Regular Means
  4. Outline
  5. On Regular Means as Superpositions
  6. Means are Superpositions
  7. Regular Means as Simple Neural Models?
  8. On the Continuity of Regular Means
  9. On a Universal Central Limit Theorem
  10. Large Sample Behavior of Regular Means
  11. Numerical Experiments
  12. Proofs and Technical Details
  13. Proof of Corollary \ref{['KLATB']}
  14. Proof of Corollary \ref{['KLATC']}
  15. Proof of Theorem \ref{['cont']}
  16. ...and 1 more sections

Key Result

Theorem 1

Let $f:A \rightarrow \mathbb{R}$ be a multivariate continuous function over a compact set $A \subset \mathbb{R}^n$. Then, it has the representation with continuous one–dimensional outer and inner functions $l$ and $g_{j}$, where $g_{j}$ is increasing, and $\sum_{i = 1}^n \lambda_i \leq 1$ with $\lambda_i \geq 0$, for all $i$.

Figures (2)

  • Figure 1: Monte Carlo simulation results for the universal central limit theorem with $n = 1\,000$.
  • Figure 2: Additional simulation results with $n = 1\,000$ from a $\text{LogNormal}(\mu, \sigma^2)$ distribution with $\mu = 2$ and $\sigma = 2.5$. As anticipated by the Edgeworth expansion in Section \ref{['ls']}, convergence is slow for the arithmetic mean but fast for the geometric mean.

Theorems & Definitions (6)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Theorem 4
  • Example 1: Portfolio theory
  • Theorem 5: Universal Central Limit Theorem