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On the Loewner framework, the Kolmogorov superposition theorem, and the curse of dimensionality

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

TL;DR

The paper develops a scalable, data-driven framework to interpolate and realize multivariate parametric linear systems by extending the Loewner framework to n dimensions. It introduces an n-D Loewner matrix defined via cascaded Sylvester equations and shows that its null space can be computed through a sequence of 1-D Loewner matrices, enabling decoupling of variables and substantial reductions in computational complexity and memory usage. By embedding the multivariate rational function in a barycentric Lagrange form, the authors derive a generalized realization H = C Φ^{-1} B with a dimension m that grows modestly with the chosen variable partition, effectively taming the curse of dimensionality. The work also establishes a link to the Kolmogorov superposition theorem, showing that multivariate rational functions can be built from univariate components, and provides two data-driven algorithms for directly constructing parametric realizations from data. Numerical experiments across synthetic and industrial-like scenarios demonstrate strong interpolation accuracy and impressive scalability to high dimensionality.

Abstract

The Loewner framework is an interpolatory approach for the approximation of linear and nonlinear systems. The purpose here is to extend this framework to linear parametric systems with an arbitrary number n of parameters. To achieve this, a new generalized multivariate rational function realization is proposed. Then, we introduce the n-dimensional multivariate Loewner matrices and show that they can be computed by solving a set of coupled Sylvester equations. The null space of these Loewner matrices allows the construction of the multivariate barycentric rational function. 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 reduction of the computational burden. Equally importantly, this burden is alleviated by avoiding the explicit construction of large-scale n-dimensional Loewner matrices of size $N \times N$. Instead, the proposed methodology achieves decoupling of variables, leading to (i) a complexity reduction from $O(N^3)$ to below $O(N^{1.5})$ when $n > 5$ and (ii) to memory storage bounded by the largest variable dimension rather than their product, thus taming the curse of dimensionality and making the solution scalable to very large data sets. This decoupling of the variables leads to a result similar to the Kolmogorov superposition theorem for rational functions. Thus, making use of barycentric representations, every multivariate rational function can be computed using the composition and superposition of single-variable functions. Finally, we suggest two algorithms (one direct and one iterative) to construct, directly from data, multivariate (or parametric) realizations ensuring (approximate) interpolation. Numerical examples highlight the effectiveness and scalability of the method.

On the Loewner framework, the Kolmogorov superposition theorem, and the curse of dimensionality

TL;DR

The paper develops a scalable, data-driven framework to interpolate and realize multivariate parametric linear systems by extending the Loewner framework to n dimensions. It introduces an n-D Loewner matrix defined via cascaded Sylvester equations and shows that its null space can be computed through a sequence of 1-D Loewner matrices, enabling decoupling of variables and substantial reductions in computational complexity and memory usage. By embedding the multivariate rational function in a barycentric Lagrange form, the authors derive a generalized realization H = C Φ^{-1} B with a dimension m that grows modestly with the chosen variable partition, effectively taming the curse of dimensionality. The work also establishes a link to the Kolmogorov superposition theorem, showing that multivariate rational functions can be built from univariate components, and provides two data-driven algorithms for directly constructing parametric realizations from data. Numerical experiments across synthetic and industrial-like scenarios demonstrate strong interpolation accuracy and impressive scalability to high dimensionality.

Abstract

The Loewner framework is an interpolatory approach for the approximation of linear and nonlinear systems. The purpose here is to extend this framework to linear parametric systems with an arbitrary number n of parameters. To achieve this, a new generalized multivariate rational function realization is proposed. Then, we introduce the n-dimensional multivariate Loewner matrices and show that they can be computed by solving a set of coupled Sylvester equations. The null space of these Loewner matrices allows the construction of the multivariate barycentric rational function. 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 reduction of the computational burden. Equally importantly, this burden is alleviated by avoiding the explicit construction of large-scale n-dimensional Loewner matrices of size . Instead, the proposed methodology achieves decoupling of variables, leading to (i) a complexity reduction from to below when and (ii) to memory storage bounded by the largest variable dimension rather than their product, thus taming the curse of dimensionality and making the solution scalable to very large data sets. This decoupling of the variables leads to a result similar to the Kolmogorov superposition theorem for rational functions. Thus, making use of barycentric representations, every multivariate rational function can be computed using the composition and superposition of single-variable functions. Finally, we suggest two algorithms (one direct and one iterative) to construct, directly from data, multivariate (or parametric) realizations ensuring (approximate) interpolation. Numerical examples highlight the effectiveness and scalability of the method.
Paper Structure (49 sections, 19 theorems, 64 equations, 4 figures, 5 tables, 2 algorithms)

This paper contains 49 sections, 19 theorems, 64 equations, 4 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation and context: non-intrusive data-driven model construction
  3. Literature overview on reduced-order modeling (ROM)
  4. Connection with the Kolmogorov Superposition Theorem
  5. Connection to other fields
  6. Problem statement
  7. Historical notes
  8. Contribution and paper organization
  9. Realizations of multivariate rational functions
  10. Preliminaries
  11. Pseudo-companion Lagrange matrix
  12. Multi-row/multi-column indices and the coefficient matrices
  13. The coefficient matrices
  14. Characteristic polynomial in the barycentric representation
  15. The multivariate realization in the Lagrange basis
  16. ...and 34 more sections

Key Result

Proposition 2.1

Given the polynomial ${\mathbf p}(s)$ of degree less than or equal to $n$, expressed in a Lagrange basis as ${\mathbf p}(s)={\mathbf \pi}\left(\frac{\alpha_1}{{\mathbf x}_1}+\cdots+\frac{\alpha_{n+1}}{{\mathbf x}_{n+1}}\right)$ where ${\mathbf \pi}=\prod_{i=1}^{n+1}{\mathbf x}_i$. It follows that $\

Figures (4)

  • Figure 1: flop comparison: cascaded ${n}$-D Loewner worst-case upper bounds 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 2: 2-D simple synthetic model: Algorithm \ref{['algo:LL_nD']} normalized singular values of each 1-D (left) and the 2-D (right) Loewner matrices.
  • Figure 3: 2-D simple model: frequency responses (left) and errors (right); original compared to Algorithm \ref{['algo:LL_nD']} (black lines), and to Algorithm \ref{['algo:LL_nD_adaptive']} (orange dots and dashed lines).
  • Figure 4: 3-D flutter model: 3-D Loewner matrix singular values (left) and frequency responses (right). Original (solid colored) and pROM (black dotted).

Theorems & Definitions (51)

  • Remark 1.1: Taming the C-o-D
  • Remark 1.2: Connection to tensors
  • Remark 1.3: Connections to NEPs
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.1
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • ...and 41 more