Variable decoupling and the Kolmogorov Superposition Theorem for rational functions

TL;DR

This paper demonstrates that multivariate rational functions can be decoupled into single-variable functions using the Kolmogorov Superposition Theorem without computation, leveraging the Loewner Framework.

Contribution

It shows that variable decoupling for rational functions can be achieved by inspection, simplifying the process and enabling efficient approximation of complex functions.

Findings

Decoupling of rational multivariate functions is straightforward and explicit.

The Loewner Framework facilitates approximation of non-rational functions by rational and polynomial functions.

No computational effort is needed for variable decoupling in the rational case.

Abstract

This work shows that for rational multivariate functions, the Kolmogorov Superposition Theorem (KST) involves several single-variable functions, which can be written down by inspection. In other words, no computation is required for decoupling the variables of multivariate rational functions. The key tool for this development is the Loewner Framework for multivariate functions. Applications of this result involve approximating multivariate non-rational functions by low-complexity multivariate rational and polynomial functions.