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Finite-dimensional approximations of push-forwards on loc...
Isao Ishikaw · 2026-04-22 · via cs.LG updates on arXiv.org

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Abstract: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.
Comments: Revised version. Removed one auxiliary lemma, corrected finite-dimensional matrix conventions and the Hermitian moment-matrix formulation, revised the statements and estimates in the push-forward approximation, analytic-map reconstruction/least-squares approximation, and vector-field reconstruction results, and fixed references and typographical errors
Subjects: Numerical Analysis (math.NA); Machine Learning (cs.LG); Complex Variables (math.CV); Dynamical Systems (math.DS); Functional Analysis (math.FA)
MSC classes: Primary 37C30, Secondary 32E30, 30H20, 41A25, 46F15
Cite as: arXiv:2404.10769 [math.NA]
  (or arXiv:2404.10769v4 [math.NA] for this version)
  https://doi.org/10.48550/arXiv.2404.10769

arXiv-issued DOI via DataCite

Submission history

From: Isao Ishikawa [view email]
[v1] Tue, 16 Apr 2024 17:53:59 UTC (552 KB)
[v2] Sun, 1 Sep 2024 11:26:11 UTC (548 KB)
[v3] Sun, 23 Nov 2025 06:45:39 UTC (29 KB)
[v4] Tue, 21 Apr 2026 04:19:43 UTC (30 KB)