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) = xy / 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

Further reading

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. 1 2 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)
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