An Algorithm for Computing Lipschitz Inner Functions in Kolmogorov's Superposition Theorem

Understand

Kolmogorov famously proved that multivariate continuous functions can be represented as a superposition of a small number of univariate continuous functions, $$ f(x_1,\dots,x_n) = \sum_{q=0}^{2n+1} \chi^q \left( \sum_{p=1}^n \psi^{pq}(x_p) \right).$$ Fridman \cite{fridman} posed the best smoothness bound for the functions $\psi^{pq}$, that such functions can be constructed to be Lipschitz continuous with constant 1.

• Previous algorithms to describe these inner functions have only been H\"older continuous, such as those proposed by K\"oppen and Braun and Griebel.
• This is problematic, as pointed out by Griebel, in that non-smooth functions have very high storage/evaluation complexity, and this makes Kolmogorov's representation (KR) impractical using the standard definition of the inner functions.
• To date, no one has presented a method to compute a Lipschitz continuous inner function.