Kolmogorov–Arnold representation theorem
can be represented as a superposition of continuous single-variable functions.
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.
[2] It solved a more constrained form of Hilbert's thirteenth problem, so the original Hilbert's thirteenth problem is a corollary.
[6]: 180 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.
[7] 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 can be translated to the form
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".
In this context, it has stimulated many studies in the theory of functions and other related problems by different authors.
[8] A variant of Kolmogorov's theorem that reduces the number of outer functions
More precisely, Lorentz proved the existence of functions
, such that Phillip A. Ostrand [11] generalized the Kolmogorov superposition theorem to compact metric spaces.
be compact metric spaces of finite dimension
[12] The theorem does not hold in general for complex multi-variate functions, as discussed here.
[4] Furthermore, the non-smoothness of the inner functions and their "wild behavior" has limited the practical use of the representation,[13] although there is some debate on this.
[14] In the field of machine learning, there have been various attempts to use neural networks modeled on the Kolmogorov–Arnold representation.
[21] A proof for the case of functions depending on two variables is given, as the generalization is immediate.
In the notation, we have the following: Theorem — The Kolmogorov–Arnold tuples make up an open and dense subset of
into an overlapping system of small squares, each with a unique address, and define
By adding a small and linearly independent irrational number (the construction is similar to that of the Hamel basis) to each of
By plotting out the entire grid system, one can see that every point in
It remains to show this construction has the desired properties.
Iterating the above construction, then applying the Baire category theorem, we find that the following kind of 5-tuples are open and dense in
interlocking grid systems, such that each point in the cube is on
A relatively short proof is given in [23] via dimension theory.
In another direction of generality, more conditions can be imposed on the Kolmogorov–Arnold tuples.
Theorem — There exists a Kolmogorov–Arnold tuple where each function is strictly monotonically increasing.
[24] (Vituškin, 1954)[25] showed that the theorem is false if we require all functions
