Tensor-based multivariate function approximation: methods benchmarking and comparison

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.