Kolmogorov–Arnold representation theorem
Multivariate functions can be written using univariate functions and summing / From Wikipedia, the free encyclopedia
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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 the two-argument addition of continuous functions of one variable. It solved a more constrained form of Hilbert's thirteenth problem, so the original Hilbert's thirteenth problem is a corollary.[1][2][3]
The works of Vladimir Arnold and Andrey Kolmogorov 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.[4] More specifically,
- .
where and .
There are proofs with specific constructions.[5]
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.[6]: 180