The key contributions of this work are:
A generalized realization form for rational functions in n variables, described in the Lagrange basis. This allows controlling the complexity of the realization.
The n-dimensional Loewner matrix is shown to be the solution of a cascaded Sylvester equation set.
The barycentric coefficients can be obtained using a sequence of 1-dimensional Loewner matrices instead of the large-scale n-dimensional one, drastically reducing the computational complexity from O(N^3) to about O(N^1.4) and taming the curse of dimensionality.
Two algorithms are proposed for the direct and iterative construction of multivariate (or parametric) realizations ensuring (approximate) interpolation.
The method provides a solution to the tensor approximation problem by approximating any tensorized data set with a rational function, while taming the curse of dimensionality. It also enables the multi-linearization of underlying nonlinear eigenvalue problems through an interpolatory approach.
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by Athanasios C... at arxiv.org 05-02-2024
https://arxiv.org/pdf/2405.00495.pdfDeeper Inquiries