Kolmogorov–Arnold representation theorem

In real analysis and approximation theory, the Kolmogorov–Arnold representation theorem (or superposition theorem) states that every multivariate continuous function can be represented as a superposition of continuous functions of one variable. It solved a more constrained, yet more general form of Hilbert's thirteenth problem.[1][2]

The works of Andrey Kolmogorov and Vladimir 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,

${\displaystyle f(\mathbf {x} )=f(x_{1},\ldots ,x_{n})=\sum _{q=0}^{2n}\Phi _{q}\left(\sum _{p=1}^{n}\phi _{q,p}(x_{p})\right)}$.

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

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]

History

The Kolmogorov–Arnold representation theorem is closely related to Hilbert's 13th problem. In his Paris lecture at the International Congress of Mathematicians in 1900, David Hilbert formulated 23 problems which in his opinion were important for the further development of mathematics.[6] The 13th of these problems dealt with the solution of general equations of higher degrees. It is known that for algebraic equations of degree 4 the solution can be computed by formulae that only contain radicals and arithmetic operations. For higher orders, Galois theory shows us that the solutions of algebraic equations cannot be expressed in terms of basic algebraic operations. It follows from the so called Tschirnhaus transformation that the general algebraic equation

${\displaystyle x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}=0}$

can be translated to the form ${\displaystyle y^{n}+b_{n-4}y^{n-4}+\cdots +b_{1}y+1=0}$. The Tschirnhaus transformation is given by a formula containing only radicals and arithmetic operations and transforms. Therefore, the solution of an algebraic equation of degree ${\displaystyle n}$ can be represented as a superposition of functions of two variables if ${\displaystyle n<7}$ and as a superposition of functions of ${\displaystyle n-4}$ variables if ${\displaystyle n\geq 7}$. For ${\displaystyle n=7}$ the solution is a superposition of arithmetic operations, radicals, and the solution of the equation ${\displaystyle y^{7}+b_{3}y^{3}+b_{2}y^{2}+b_{1}y+1=0}$.

A further simplification with algebraic transformations seems to be impossible which led to Hilbert's conjecture that "A solution of the general equation of degree 7 cannot be represented as a superposition of continuous functions of two variables". This explains the relation of Hilbert's thirteenth problem to the representation of a higher-dimensional function as superposition of lower-dimensional functions. In this context, it has stimulated many studies in the theory of functions and other related problems by different authors.[7]

Variants

A variant of Kolmogorov's theorem that reduces the number of outer functions ${\displaystyle \Phi _{q}}$ is due to George Lorentz.[8] He showed in 1962 that the outer functions ${\displaystyle \Phi _{q}}$ can be replaced by a single function ${\displaystyle \Phi }$. More precisely, Lorentz proved the existence of functions ${\displaystyle \phi _{q,p}}$, ${\displaystyle q=0,1,\ldots ,2n}$, ${\displaystyle p=1,\ldots ,n,}$ such that

${\displaystyle f(\mathbf {x} )=\sum _{q=0}^{2n}\Phi \left(\sum _{p=1}^{n}\phi _{q,p}(x_{p})\right)}$.

David Sprecher[9] replaced the inner functions ${\displaystyle \phi _{q,p}}$ by one single inner function with an appropriate shift in its argument. He proved that there exist real values ${\displaystyle \eta ,\lambda _{1},\ldots ,\lambda _{n}}$, a continuous function ${\displaystyle \Phi \colon \mathbb {R} \rightarrow \mathbb {R} }$, and a real increasing continuous function ${\displaystyle \phi \colon [0,1]\rightarrow [0,1]}$ with ${\displaystyle \phi \in \operatorname {Lip} (\ln 2/\ln(2N+2))}$, for ${\displaystyle N\geq n\geq 2}$, such that

${\displaystyle f(\mathbf {x} )=\sum _{q=0}^{2n}\Phi \left(\sum _{p=1}^{n}\lambda _{p}\phi (x_{p}+\eta q)+q\right)}$.

Phillip A. Ostrand [10] generalized the Kolmogorov superposition theorem to compact metric spaces. For ${\displaystyle p=1,\ldots ,m}$ let ${\displaystyle X_{p}}$ be compact metric spaces of finite dimension ${\displaystyle n_{p}}$ and let ${\displaystyle n=\sum _{p=1}^{m}n_{p}}$. Then there exists continuous functions ${\displaystyle \phi _{q,p}\colon X_{p}\rightarrow [0,1],q=0,\ldots ,2n,p=1,\ldots ,m}$ and continuous functions ${\displaystyle G_{q}\colon [0,1]\rightarrow \mathbb {R} ,q=0,\ldots ,2n}$ such that any continuous function ${\displaystyle f\colon X_{1}\times \dots \times X_{m}\rightarrow \mathbb {R} }$ is representable in the form

${\displaystyle f(x_{1},\ldots ,x_{m})=\sum _{q=0}^{2n}G_{q}\left(\sum _{p=1}^{m}\phi _{q,p}(x_{p})\right)}$.

See also

• Universal approximation theorem

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. Bar-Natan, Dror. "Dessert: Hilbert's 13th Problem, in Full Colour".
4. Jürgen Braun and Michael 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)
6. Hilbert, David (1902). "Mathematical problems". Bulletin of the American Mathematical Society. 8 (10): 461–462. doi:10.1090/S0002-9904-1902-00923-3.
7. Jürgen Braun, On Kolmogorov's Superposition Theorem and Its Applications, SVH Verlag, 2010, 192 pp.
8. Lorentz, G. G. (1962). "Metric entropy, widths, and superpositions of functions". American Mathematical Monthly. 69 (6): 469–485. doi:10.1080/00029890.1962.11989915.
9. David A. Sprecher, On the structure of continuous functions of several variables, Transactions of the American Mathematical Society, 115 (1965), pp. 340–355.
10. Ostrand, Phillip A. (1965). "Dimension of metric spaces and Hilbert's problem 13". Bulletin of the American Mathematical Society. 71 (4): 619–622. doi:10.1090/s0002-9904-1965-11363-5.

Sources

• Andrey 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.
• Vladimir 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)

External links

• A deep machine learning algorithm for construction of the Kolmogorov-Arnold representation.