mLF: multivariate Loewner Framework
Overview
The Multivariate Loewner Framework is introduced by A.C. Antoulas, I-V. Gosea and C. Poussot-Vassal in "On the Loewner framework, the Kolmogorov superposition theorem, and the curse of dimensionality", in SIAM Review (Research Spotlight), Vol. 67(4), pp. 737-770, November 2025.
It allows constructing a $n$-variate rational function approximating a $n$-dimensional tensor (either real or complex valued).
It is suited to approximate, from any $n$-dimensional tensor, $n$-variable static functions and $n$-variable (parametric) dynamical systems.
Contributions claim
- We propose a generalized realization form for rational functions in $n$-variables (for any $n$), which are described in the Lagrange basis;
- We show that the $n$-dimensional Loewner matrix can be written as the solution of a series of cascaded Sylvester equations;
- 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 ($Q\times K, Q \geq K$) $n$-dimensional one, therefore drastically taming the curse of dimensionality, i.e. reducing both the computational effort and the memory needs, and, in addition improving accuracy;
- 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;
- Connections with KAN neural nets follows (detailed in future work).