Theorem

Suppose ff is multivariate and continuous on a bounded domain, then it may be written as a finite sum of continuous univariate functions. More specifically for continuous and smooth f:[0,1]n→ℝf: [0,1]^n \to \mathbb{R},

f(𝐱)=f(x1,...,xn)=∑q=12n+1Φq(∑p=1nϕq,p(xp))f(\mathbf{x}) = f(x_1,...,x_n) = \sum_{q=1}^{2n+1} \Phi_q \left(\sum_{p=1}^n \phi_{q,p}(x_p)\right)

where ϕq,p:[0,1]→ℝ\phi_{q,p}: [0,1] \to \mathbb{R} are univariate functions, and Φq:ℝ→ℝ\Phi_q : \mathbb{R} \to \mathbb{R} composes these to reconstruct f(𝐱)f(\mathbf{x}).

Hilbert's 13th problem

#incomplete

Application: KANs

see Kolmogorov-Arnold network

See also

References

  1. https://en.wikipedia.org/wiki/Kolmogorov–Arnold_representation_theorem
  2. Z. Liu, Y. Wang, S. Vaidya, F. Ruehle, J. Halverson, M. Soljačić, T. Y. Hou, and M. Tegmark, “KAN: Kolmogorov-Arnold Networks,” Feb. 09, 2025, arXiv: arXiv:2404.19756. doi: 10.48550/arXiv.2404.19756.
  3. A. N. Kolmogorov, “On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition”, Dokl. Akad. Nauk SSSR, 114:5 (1957), 953–956. http://mi.mathnet.ru/dan22050
  4. R. Hecht-Nielsen, “Kolmogorov’s mapping neural network existence theorem,” in Proceedings of the international conference on neural networks, IEEE press New York, NY, USA, 1987, pp. 11–14. [Online]. Available: https://cs.uwaterloo.ca/~y328yu/classics/Hecht-Nielsen.pdf
  5. https://kindxiaoming.github.io/pykan/intro.html
  6. https://www.reddit.com/r/MachineLearning/comments/1clcu5i/d_kolmogorovarnold_network_is_just_an_mlp/
  7. https://blog.csdn.net/qq_44648285/article/details/143316549