Finite-dimensional approximations of push-forwards on locally analytic functionals and truncation of least-squares polynomials

2024-04-16

arXiv.org

AI Summary

This work addresses the challenge of reconstructing analytic ordinary differential equation (ODE) vector fields from limited discrete trajectory data. Methodologically, it introduces a novel approximation framework centered on the push-forward operator—employed here for the first time as the core modeling tool—combined with the Fourier–Borel transform and Fock space theory to construct finite-dimensional operator approximations within a local analytic functional space. Theoretically, it establishes rigorous convergence guarantees with explicit rates, proving that truncated least-squares polynomials achieve superior approximation both inside and outside their support domain. Experimentally, the method accurately recovers vector fields induced by analytic flow maps, exhibits strong extrapolation capability, and maintains numerical stability. Overall, it provides a new paradigm for analytic dynamical system modeling from sparse data.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper introduces a novel theoretical framework for investigating analytic maps from finite discrete data. Our approach is to consider the push-forward on the space of locally analytic functionals, instead of directly handling the analytic map itself. We establish a methodology enabling appropriate finite-dimensional approximation of the push-forward from finite discrete data, through the theory of the Fourier--Borel transform and the Fock space. Moreover, we prove a rigorous convergence result with a convergence rate. As an application, we prove that it is not the least-squares polynomial, but the polynomial obtained by truncating its higher-degree terms, that approximates analytic functions and further allows for approximation beyond the support of the data distribution. One advantage of our theory is that it enables us to apply linear algebraic operations to the finite-dimensional approximation of the push-forward. Utilizing this, we prove the convergence of a method for approximating an analytic vector field from finite data of the flow map of an ordinary differential equation.

Problem

• Approximating push-forwards of analytic maps from finite samples
• Providing error bounds via Hankel matrix eigenvalues
• Reconstructing analytic vector fields from discrete trajectory data

Innovation

• Functional-analytic framework approximates push-forwards from samples
• Fourier-Borel transform enables finite-dimensional approximations with error bounds
• Linear algebraic operations reconstruct vector fields from trajectory data