Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.10769 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917423736684544 |
|---|---|
| author | Ishikawa, Isao |
| author_facet | Ishikawa, Isao |
| contents | This paper develops a functional-analytic framework for approximating the push-forward induced by an analytic map from finitely many samples. Instead of working directly with the map, we study the push-forward on the space of locally analytic functionals and identify it, via the Fourier--Borel transform, with an operator on the space of entire functions of exponential type. This yields finite-dimensional approximations of the push-forward together with explicit error bounds expressed in terms of the smallest eigenvalues of certain Hankel moment matrices. Moreover, we obtain sample complexity bounds for the approximation from i.i.d.~sampled data. As a consequence, we show that linear algebraic operations on the finite-dimensional approximations can be used to reconstruct analytic vector fields from discrete trajectory data. In particular, we prove convergence of a data-driven method for recovering the vector field of an ordinary differential equation from finite-time flow map data under fairly general conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10769 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite-dimensional approximations of push-forwards on locally analytic functionals Ishikawa, Isao Numerical Analysis Machine Learning Complex Variables Dynamical Systems Functional Analysis Primary 37C30, Secondary 32E30, 30H20, 41A25, 46F15 This paper develops a functional-analytic framework for approximating the push-forward induced by an analytic map from finitely many samples. Instead of working directly with the map, we study the push-forward on the space of locally analytic functionals and identify it, via the Fourier--Borel transform, with an operator on the space of entire functions of exponential type. This yields finite-dimensional approximations of the push-forward together with explicit error bounds expressed in terms of the smallest eigenvalues of certain Hankel moment matrices. Moreover, we obtain sample complexity bounds for the approximation from i.i.d.~sampled data. As a consequence, we show that linear algebraic operations on the finite-dimensional approximations can be used to reconstruct analytic vector fields from discrete trajectory data. In particular, we prove convergence of a data-driven method for recovering the vector field of an ordinary differential equation from finite-time flow map data under fairly general conditions. |
| title | Finite-dimensional approximations of push-forwards on locally analytic functionals |
| topic | Numerical Analysis Machine Learning Complex Variables Dynamical Systems Functional Analysis Primary 37C30, Secondary 32E30, 30H20, 41A25, 46F15 |
| url | https://arxiv.org/abs/2404.10769 |