Speaker
Description
We extend the Loewner framework to multivariate (static and dynamic) functions with an arbitrary number $n$ of variables [1]. We present the following facts:
(i) That $n$-variable rational functions (and realization), described in the barycentric basis, can be constructed to interpolate and/or approximate/compress any tensorized $n$-D data or $n$-variate function;
(ii) That these $n$-variable rational functions can be obtained thanks to a sequence of small-scale single-variable interpolation (performed with Loewner matrices), leading to drastically taming the curse of dimensionality (both in memory and computational effort);
(iii) That such sequence results in the variables decoupling, providing a numerically robust solution to the Kolmogorov Superposition Theorem (KST), restricted to rational functions;
(iv) That the Loewner framework bridges "Approximation theory" (both functions and tensors) with "Systems theory", and provides connections with Kolmogorov Arnold Networks (KAN).
A collection of numerical examples and method comparison illustrates the effectiveness and scalability of the proposed method and its ability to tame the curse of dimensionality [2]. Issues and outlook will be specifically addressed. Some research codes are also presented [3].
[1] https://doi.org/10.1137/24M1656657
[2] https://arxiv.org/abs/2506.04791
[3] https://github.com/cpoussot/mLF
This works is carried out in collaboration with A.C. Antoulas, I-V. Gosea, C. Poussot-Vassal and P. Vuillemin.