Year: 2026
Type: article
Abstract: . We consider the problem of approximating a subset [Formula: see text] of a Hilbert space [Formula: see text] by a low-dimensional manifold [Formula: see text], using samples from [Formula: see text]. We propose a nonlinear approximation method where [Formula: see text] is defined as the range of a smooth nonlinear decoder [Formula: see text] defined on [Formula: see text] with values in a possibly high-dimensional linear space [Formula: see text] and a linear encoder [Formula: see text] which associates to an element from [Formula: see text] its coefficients [Formula: see text] on a basis of an [Formula: see text]-dimensional subspace [Formula: see text], where [Formula: see text] is an optimal or near to optimal linear space, depending on the selected error measure. The linearity of the encoder allows us to easily obtain the parameters [Formula: see text] associated with a given element [Formula: see text] in [Formula: see text]. The proposed decoder is a polynomial map from [Formula: see text] to [Formula: see text] which is obtained by a tree-structured composition of polynomial maps, estimated sequentially from samples in [Formula: see text]. Rigorous error and stability analyses are provided, as well as an adaptive strategy for constructing the subspace [Formula: see text], and a decoder that guarantees an approximation of the set [Formula: see text] with controlled mean-squared or worst-case errors, and a controlled stability (Lipschitz continuity) of the encoder and decoder pair. We demonstrate the performance of our method through numerical experiments. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/JoelSOFFO/CPN_MOR and in the supplementary materials ( CPN_MOR.zip [977KB], CPN_MOR-main.zip [162KB], Dataset.zip [8.69MB]). [Formula: see text]
DOI: 10.1137/25m1734002
Authors: Antoine Bensalah, Anthony Nouy, Joel Soffo
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4407310247