Efficient Multivariate Rational Function Approximation: Taming the Curse of Dimensionality

Core Concepts

The Loewner framework is extended to efficiently construct data-driven multivariate rational models, taming the curse of dimensionality by avoiding the explicit construction of large-scale n-dimensional Loewner matrices.

Abstract

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.

Stats

The complexity of the multivariate rational function H(1s, 2s, ..., ns) is denoted as (d1, d2, ..., dn), where dj is the highest degree in which the variable js occurs. The realization dimension m is given by m = 2ℓ + κ - 1, where ℓ = Πn j=k+1 nj and κ = Πk j=1 nj, and k is the number of column (right) variables.

Quotes

"The principal result of this work is to show how the null space of the n-dimensional Loewner matrix can be computed using a sequence of 1-dimensional Loewner matrices, leading to a drastic computational burden reduction." "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."

Key Insights Distilled From

The Loewner framework for parametric systems: Taming the curse of dimensionality

by Athanasios C... at arxiv.org 05-02-2024

https://arxiv.org/pdf/2405.00495.pdf

Deeper Inquiries

How can the proposed approach be extended to handle more general forms of multivariate functions beyond rational functions

The proposed approach can be extended to handle more general forms of multivariate functions beyond rational functions by incorporating different basis functions for interpolation. While the current framework is based on Lagrange basis functions, other basis functions such as Chebyshev polynomials, Legendre polynomials, or even neural networks can be utilized to approximate a wider range of multivariate functions. By adapting the realization structure to accommodate different basis functions, the approach can effectively handle more complex and diverse multivariate functions.

What are the theoretical guarantees on the approximation accuracy and stability of the constructed multivariate models

The theoretical guarantees on the approximation accuracy and stability of the constructed multivariate models can be established through rigorous analysis of the interpolation error and the properties of the chosen basis functions. By ensuring that the chosen basis functions have good approximation properties and that the interpolation points are strategically selected, the accuracy of the model can be controlled. Additionally, stability analysis can be performed by examining the behavior of the system under perturbations in the data or parameters, ensuring that the model remains stable and reliable in practical applications.

Can the ideas of tensor decomposition be further integrated with the Loewner framework to achieve even more efficient multivariate function approximation

The ideas of tensor decomposition can be further integrated with the Loewner framework to achieve even more efficient multivariate function approximation. By leveraging tensor decomposition techniques such as Canonical Polyadic Decomposition (CPD), Tucker Decomposition, or Tensor Train Decomposition, the high-dimensional data structures can be efficiently represented and manipulated. This integration can help in reducing the computational complexity of handling large-scale multivariate functions, making the approximation process more scalable and efficient. Additionally, tensor decomposition can provide insights into the underlying structure of the multivariate functions, leading to improved modeling and analysis capabilities.