The Kolmogorov-Arnold Theorem

Any function which is continuous can be represented as the sum of single variable functions within a bounded region.

Multiplication of two variables can be broken in to a sum of smaller functions.

"Phi" functions are added to transform a variable prior to summing it: t1 = phi(x) + phi(y)

Summed phis become "terms" of the Kolmogorov function.

Each Kolmogorov term is wrapped in a separate "capital phi" function, as in Phi(t1) + Phi(t2)

f(x,y) =

Every Kolmogorov function has 2n+1 Kolmogorov terms, where n is the number of variables in the function.

Kolmogorov functions can be drawn out as a graph similar to a neural network; demonstrating that Kolmogorov approximates using only two layers.

Also some basic discourse on how this is important to KAN networks.