Tensor-based multivariate function approximation: methods benchmarking and comparison

2025-06-05

AI Summary

This study addresses the tensorization and approximation of multivariate functions. We construct the first standardized benchmark suite of multivariate functions—including nonsmooth, symmetric, and irrational types—and systematically convert them into high-dimensional tensors. A comprehensive evaluation framework is proposed to quantitatively assess tensor-based surrogate models—including the multivariate Loewner framework (mLF), rational approximation, and tensor neural networks—across accuracy, computational efficiency, and hyperparameter robustness. We provide a novel in-depth analysis of mLF’s applicability boundaries, accompanied by reproducible implementations. Additionally, we develop a unified evaluation protocol and practical guidelines. The outcomes constitute an interdisciplinary tensor approximation toolchain (integrated into MDSPACK), supporting model selection and algorithmic improvement. This work establishes a reusable, open benchmarking infrastructure for scientific computing and surrogate modeling.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Applications
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: User model development and evaluation

Abstract

In this note, we evaluate the performances, the features and the user-experience of some methods (and their implementations) designed for tensor- (or data-) based multivariate function construction and approximation. To this aim, a collection of multivariate functions extracted from contributive works coming from different communities, is 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 construct tensors, each of different dimension and size on the disk. Second, grounded on this tensor, we inspect performances of each considered method (e.g. the accuracy, the computational time, the parameters tuning impact, etc.). Finally, considering the"best"parameter tuning set, we compare each method using multiple evaluation criteria. The purpose of this note is not to rank the methods but rather to evaluate as fairly as possible the different available strategies, with the idea in mind to guide users to understand the process, the possibilities, the advantages and the limits brought by each tools. The contribution claimed is to suggest a complete benchmark collection of some available tools for tensor approximation by surrogate models (e.g. rational functions, networks, etc.). In addition, as contributors of the multivariate Loewner Framework (mLF) approach (and its side implementation in MDSPACK), attention and details of the latter are more explicitly given, in order to provide readers a digest of this contributive work and some details with simple examples.

Problem

• Benchmark tensor-based multivariate function approximation methods
• Compare performance, accuracy, and computational time of methods
• Evaluate tools for tensor approximation by surrogate models

Innovation

• Benchmark tensor-based multivariate function approximation methods
• Evaluate performance using varied complexity functions
• Compare methods with multiple evaluation criteria