---
date: '2024-12-17'
description: also known as the superposition theorem.
id: Kolmogorov-Arnold representation theorem
modified: 2026-06-05 15:08:05 GMT-04:00
tags:
  - math
title: Kolmogorov–Arnold representation theorem
created: '2024-12-17'
published: '2024-12-17'
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---
_every multivariate continuous function_ $f: [0,1]^n \to \mathbb{R}$ can be <mark>represented</mark> as superposition of continuous single-variable functions

> \[!definition\] Definition 1.
>
> if $f$ is a multivariate continuous function, then $f$ can be written as a finite composition of continuous single-variable functions with binary operation of addition. Formally:
>
> $$
> \displaystyle f(\mathbf {x} )=f(x_{1},\ldots ,x_{n})=\sum _{q=0}^{2n}\Phi _{q}\!\left(\sum _{p=1}^{n}\phi_{q,p}(x_{p})\right)
> $$
>
> where $\phi_{q,p} : [0,1] \to \mathbb{R}$ and $\Phi_q: \mathbb{R} \to \mathbb{R}$

See also [[thoughts/FFN#universal approximation theorem]]