# A structured proof of Kolmogorov's Superposition Theorem

arXiv: https://arxiv.org/abs/2105.00408 · HTML: https://mlsanity.com/paper/2105.00408/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 9 references that are in the map.
- First posted on arXiv 2 May 2021.

## Builds on (most cited first, 1 of 1)

- [The Kolmogorov-Arnold representation theorem revisited](https://mlsanity.com/paper/2007.15884/) (2020)

## Built on by (most cited first, 3 of 3)

- [fKAN: Fractional Kolmogorov-Arnold Networks with trainable Jacobi basis functions](https://mlsanity.com/paper/2406.07456/) (2024)
- [rKAN: Rational Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.14495/) (2024)
- [Kolmogorov-Arnold Energy Models: Fast, Interpretable Generative Modeling](https://mlsanity.com/paper/2506.14167/) (2025)

## Abstract (arXiv)

We present a well-structured detailed exposition of a well-known proof of the following celebrated result solving Hilbert's 13th problem on superpositions. For functions of 2 variables the statement is as follows. Kolmogorov Theorem. There are continuous functions varphi1,ldots,varphi5 : [,0, 1,]→ [,0,1,] such that for any continuous function f: [,0,1,]2→mathbb R there is a continuous function h: [,0,3,]→mathbb R such that for any x,y∈ [,0, 1,] we have f(x,y)=sumlimitsk=15 hleft(varphik(x)+√2,varphik(y)right). The proof is accessible to non-specialists, in particular, to students familiar with only basic properties of continuous functions.