What is the Kolmogorov theorem?
A page from the diary of Andrey Kolmogorov. “Two little discoveries today”
April 25, 2023, marked the 120th birth anniversary of the legendary mathematician Andrei Nikolaevich Kolmogorov. His groundbreaking work in the mid-20th century profoundly influenced mathematical theory and practice.
Kolmogorov’s Theorems and David Hilbert’s Thirteenth Problem
In 1956, Kolmogorov made significant strides towards resolving David Hilbert’s thirteenth problem. He demonstrated three pivotal theorems that described the representation of real continuous functions with n independent variables. Furthermore, these theorems showed that these functions could be expressed through sums and superpositions of functions with fewer variables, inching closer to a solution for Hilbert’s problem.
Vladimir Igorevich Arnold’s Revelation
The following year, Vladimir Igorevich Arnold, a promising student of Kolmogorov, made a remarkable discovery. He established a theorem that effectively proved Hilbert’s assumptions on the matter were incorrect, a significant development in the field.
Kolmogorov and Arnold’s Application of Theories
Closeup of the Mandelbrot set centered at (0.282, -0.01) at magnification 195.3125 times along the Y-Axis using 5000 iterations, an example of Kolmogorov complexity.
Kolmogorov and Arnold utilized Alexander Semenovich Kronrod’s theory, which created a unique correspondence between multi-variable functions and trees. In this process, Kolmogorov employed Karl Menger’s universal tree concept, enhancing the depth of their research.
The works of Vladimir Arnold and Andrey Kolmogorov established that if f is a multivariate continuous function, then f can be written as a finite composition of continuous functions of a single variable and the binary operation of addition.
Kolmogorov’s Remarkable Theorem of 1957
In a significant achievement, Kolmogorov proved a theorem on the representation of real continuous functions within an n-dimensional unit cube. He expressed these functions using only a combination of outer functions, chi(y), and inner functions, psi(xp), as a result, elucidating a complex mathematical concept with elegance and precision.
The Proofs and Non-Constructive Nature
Kolmogorov’s proof of the Main Theorem (KMT) notably did not utilize trees. Instead, it established the existence of certain lambda numbers and psi(xp) functions through three lemmas. However, the proof remained non-constructive as it did not explicitly present these functions for the defined intervals and cubes. This left a gap in the literature, requiring further suggestions and constructions for these lambdas and inner functions.
Developments and Constructions Post-Kolmogorov
Several mathematicians took up the challenge of constructing these functions in the following years. In 1963, Vladimir Mikhailovich Tikhomirov proposed a construction for psi(xp) in the case of n=2, diverging from Kolmogorov’s original intervals. Similarly, David Sprecher, since 1965, and others continued to build functions that deviated from Kolmogorov’s initial framework.
Renewed Interest and Modern Applications
The interest in KMT saw a resurgence in the late 20th century, particularly due to its relevance to neural networks (NN) and machine learning (ML). The concepts of summation and superpositions, central to KMT, are foundational elements in NN, which have become integral to ML. George Cybenko, in 1989, further contributed useful theorems in this area.
21st Century Perspectives and Research by Valerii Salov
Continuing into the 21st century, the intrigue around KMT persists, especially with the boom in NN and ML technologies.
Valerii Salov’s research, notably his paper ‘The Price of My House. Kolmogorov’s Theorem,’ takes an innovative approach!
Salov explores the practical application of NN in estimating house prices, linking them to the superpositions of functions as per Kolmogorov’s theorem. His work includes matrix and vector representations related to NN, network diagrams for various superposition theorems, and programming of inner functions and lambdas in C++.