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Tensor-based multivariate function approximation: methods benchmarking and comparison

Charles Poussot-Vassal, Ion Victor Gosea, Pierre Vuillemin, Athanasios C. Antoulas

TL;DR

A collection of multivariate functions and an evaluation methodology are suggested to evaluate the different available strategies to guide users on the prospects, advantages, and limits of the various tools.

Abstract

We evaluate some methods designed for tensor- (or data-) based multivariate model construction (approximation and compression). To this aim, a collection of multivariate functions and an evaluation methodology are suggested. First, these functions, with varying complexity (e.g., number and degree of the variables) and nature (e.g., rational, irrational, differentiable or not, symmetric, etc.) are used to build $n$-dimensional tensors, each of different dimension and memory size. Second, grounded on this tensor, we evaluate the performances of different methods and implementations leading to different types of surrogate models (e.g., rational functions, networks). The accuracy, the computational time, the parameter tuning impact, etc. are monitored and reported. One objective is to evaluate the different available strategies to guide users on the prospects, advantages, and limits of the various tools. The contributions are twofold: (i) to suggest a comprehensive benchmark collection together with a methodology for tensor approximation with a surrogate model and, in addition, (ii) to provide a digest and additional details of the multivariate Loewner Framework (mLF) approach [Antoulas et al., 2025], as well as detailed examples and code.

Tensor-based multivariate function approximation: methods benchmarking and comparison

TL;DR

A collection of multivariate functions and an evaluation methodology are suggested to evaluate the different available strategies to guide users on the prospects, advantages, and limits of the various tools.

Abstract

We evaluate some methods designed for tensor- (or data-) based multivariate model construction (approximation and compression). To this aim, a collection of multivariate functions and an evaluation methodology are suggested. First, these functions, with varying complexity (e.g., number and degree of the variables) and nature (e.g., rational, irrational, differentiable or not, symmetric, etc.) are used to build -dimensional tensors, each of different dimension and memory size. Second, grounded on this tensor, we evaluate the performances of different methods and implementations leading to different types of surrogate models (e.g., rational functions, networks). The accuracy, the computational time, the parameter tuning impact, etc. are monitored and reported. One objective is to evaluate the different available strategies to guide users on the prospects, advantages, and limits of the various tools. The contributions are twofold: (i) to suggest a comprehensive benchmark collection together with a methodology for tensor approximation with a surrogate model and, in addition, (ii) to provide a digest and additional details of the multivariate Loewner Framework (mLF) approach [Antoulas et al., 2025], as well as detailed examples and code.

Paper Structure

This paper contains 45 sections, 4 theorems, 27 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Forewords
  3. Problem description
  4. Multivariate functions
  5. From multivariate functions to tensors (with grid structure)
  6. Tensor-based model approximation: context and motivation
  7. Tensors and the curse of dimensionality (C-o-D)
  8. Contribution and structure
  9. Reproducibility
  10. Benchmarks and evaluation methodology
  11. Overview of the evaluated methods
  12. M1 - Method 1 Antoulas:2025
  13. M2 - Method 2 Antoulas:2025
  14. M3 - Method 3 mdspack
  15. M4 - Method 4 Poluektov:2025
  16. ...and 30 more sections

Key Result

Theorem 1

Given the data $P_c^{({n})}$ and $P_r^{({n})}$ in response of the ${n}$-variable ${\mathbf H}({x}_{1},\cdots,{x}_{{n}})$ function, the null space ${\mathbf c}_{n}$ of the corresponding ${n}$-D Loenwer matrix ${\mathbb L}_{n}$, is spanned by where (i) ${\mathbf c}_1^{({{{\lambda}_{2}({{k_2}})}},{{{\lambda}_{3}({{k_3}})}},\cdots,{{{\lambda}_{{n}}({{k_{n}}})}})}$ spans ${\cal N}({\mathbb L}_1^{({{{\

Figures (12)

  • Figure 1: Illustration of the data to tensor construction (via ${\mathbf H}$). On the left-hand side are the discrete data along each variables, while on the right-hand side is the tensor with grid structure (here, graphical representation limited to ${n}=6$).
  • Figure 2: The Loewner framework aims at bridging approximation and control theory.
  • Figure 3: flop comparison: cascaded/recursive ${n}$-D Loewner worst-case upper bounds $\overline{\text{{\rm flop}}_{1}}$\ref{['eq:flop_worst']} for varying number of variables ${n}$, while the full ${n}$-D Loewner is ${\cal O}(N^3)$ (black dashed); comparison with ${\cal O}(N^2)$ and ${\cal O}(N \log(N))$ references are shown in dash-dotted and dotted black lines.
  • Figure 4: Single variable Loewner matrices normalized singular values (order detection).
  • Figure 5: Equivalent NN representation of the denominator ${\mathbf d}_{\textrm{lag}}$.
  • ...and 7 more figures

Theorems & Definitions (10)

  • Remark 1: Domain restriction
  • Remark 2: The case of dynamical systems
  • Remark 3: Acknowledgements & third party software
  • Remark 4: Computational setup
  • Remark 5: Additional details in Section \ref{['sec:ex_details']}
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 6: Additional functions