Contributions claim

We claim the following innovations

1. We propose a generalized realization form for rational functions in n-variables (for any "n"), which are described in the Lagrange basis;

2. We show that the n-dimensional Loewner matrix can be written as the solution of a series of cascaded Sylvester equations;

3. We demonstrate that the required variables to be determined, i.e. the barycentric coefficients, can be computed using a sequence of small-scale 1-dimensional Loewner matrices instead of the large-scale (NxN) n-dimensional one, therefore drastically reducing the both computational effort and memory needs, and improving accuracy;

4. We show that this decomposition achieves variables decoupling; thus connecting the Loewner framework for rational interpolation of multivariate functions and the Kolmogorov Superposition Theorem (KST), restricted to rational functions. The result is the formulation of KST for the special case of rational functions;

5. Connections with KAN neural nets follows (detailed in future work).

About, and benchmark description

This page provides supplementary materials to the paper entitled "On the Loewner framework, the Kolmogorov superposition theorem, and the curse of dimensionality", by A.C. Antoulas, I-V. Gosea and C. Poussot-Vassal (available as arXiv).

We provide additional (regularly updated) numerical benches and results, grounded on the above contribution. Next, we treat problems of three different nature:

• Approximation from an available  function (set #1):
  the multi-variables function  H(s1,s2,...sn) - either rational or non-rational / complex or real-valued - is available.

• Approximation from an available tensor data (set #2):
  the (large-scale) n-D tensor data is available. In this configuration, the multi-variables function H(s1,s2,...sn) is unknown, but its evaluation at some points has been computed, and is provided. The n-D tensor denotes the evaluation of H at some points, thus leading to a n-dimensional matrix.

• Nonlinear eigenvalue approximation (set #3): 

The objective, based on one of the above ingredients, is to construct a rational model that matches / approximates the data.