Kolmogorov–Arnold representation theorem

In real analysis and approximation theory, the Kolmogorov–Arnold representation theorem (or superposition theorem) states that every multivariable continuous function can be represented as a superposition of continuous functions of two variables. It solved a version of Hilbert's thirteenth problem.^[1][2]

The works of Kolmogorov and Arnold 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.^[3]

More specifically

$f(\vec x) = \sum_{q=0}^{2n} \Phi_{q}(\sum_{p=1}^{n} \phi_{q,p}(x_{p}))$

Constructive proofs, and even more specific constructions can be found in ^[4]

For example:^[3]

f(x,y) = xy can be written as f(x,y) = exp(log x + log y)

f(x,y,z) = x^y / z can be written as f(x, y, z) = exp(exp(log y + log log x) + (−log z))

In a sense, they showed that the only true multivariate function is the sum, since every other function can be written using univariate functions and summing.^[5]

Original references

• A. N. Kolmogorov, "On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables", Proceedings of the USSR Academy of Sciences, 108 (1956), pp. 179–182; English translation: Amer. Math. Soc. Transl., 17 (1961), pp. 369–373.
• V. I. Arnold, "On functions of three variables", Proceedings of the USSR Academy of Sciences, 114 (1957), pp. 679–681; English translation: Amer. Math. Soc. Transl., 28 (1963), pp. 51–54.

Further reading

• S. Ya. Khavinson, Best Approximation by Linear Superpositions (Approximate Nomography), AMS Translations of Mathematical Monographs (1997)

References

1. Boris A. Khesin; Serge L. Tabachnikov (2014). Arnold: Swimming Against the Tide. American Mathematical Society. p. 165. ISBN 978-1-4704-1699-7.
2. Shigeo Akashi (2001). "Application of ϵ-entropy theory to Kolmogorov—Arnold representation theorem", Reports on Mathematical Physics, v. 48, pp. 19–26 doi:10.1016/S0034-4877(01)80060-4
3. Dror Bar-Natan, Dessert: Hilbert's 13th Problem, in Full Colour (link)
4. Braun and Griebel. "On a constructive proof of Kolmogorov’s superposition theorem", http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.91.5436&rep=rep1&type=pdf
5. Persi Diaconis and Mehrdad Shahshahani, On Linear Functions of Linear Combinations (1984) p. 180 (link)