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arxiv: 2404.10769 · v4 · submitted 2024-04-16 · 🧮 math.NA · cs.LG· cs.NA· math.CV· math.DS· math.FA

Finite-dimensional approximations of push-forwards on locally analytic functionals

Isao Ishikawa This is my paper

Pith reviewed 2026-05-24 02:19 UTC · model grok-4.3

classification 🧮 math.NA cs.LGcs.NAmath.CVmath.DSmath.FA
keywords push-forward approximationlocally analytic functionalsFourier-Borel transformHankel moment matricesvector field reconstructiondata-driven ODE recoveryfinite-dimensional operatorsanalytic maps
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The pith

Finite-dimensional approximations of push-forwards on locally analytic functionals allow linear algebraic reconstruction of analytic vector fields from discrete trajectory data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets up a functional-analytic framework that approximates the push-forward operator induced by an analytic map, working on the space of locally analytic functionals rather than the map itself. Identification via the Fourier-Borel transform converts the problem into operations on entire functions of exponential type, producing finite-dimensional matrix approximations together with explicit error bounds from the smallest eigenvalues of associated Hankel moment matrices. Sample-complexity results are obtained for i.i.d. data, and the approximations support reconstruction of the underlying analytic vector field. The central payoff is a convergent data-driven procedure that recovers the vector field of an ordinary differential equation directly from finite-time flow-map observations.

Core claim

The push-forward induced by an analytic map acts on locally analytic functionals and, via the Fourier-Borel transform, corresponds to an operator on entire functions of exponential type. Finite-dimensional approximations of this operator are constructed from finitely many samples, equipped with error bounds controlled by the smallest eigenvalues of Hankel moment matrices, and shown to yield convergent reconstruction of the generating analytic vector field when linear algebra is performed on the resulting matrices. In particular, the method converges for recovery of the vector field of an ODE from finite-time flow-map data under general analyticity assumptions.

What carries the argument

The Fourier-Borel transform that identifies the push-forward on the space of locally analytic functionals with an operator on entire functions of exponential type, enabling finite-dimensional approximations controlled by Hankel moment matrices.

If this is right

  • Explicit error bounds for the finite-dimensional approximations are given directly in terms of the smallest eigenvalues of certain Hankel moment matrices.
  • Sample complexity bounds hold for the approximation when data are drawn i.i.d.
  • Linear algebraic operations performed on the finite-dimensional approximations reconstruct the analytic vector field from discrete trajectory data.
  • The data-driven recovery procedure for the vector field of an ODE from finite-time flow-map data converges under the stated analyticity conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same finite-dimensional operator approximations could be applied to learn maps in other settings where analyticity is known or approximately satisfied.
  • Numerical implementation would amount to forming and eigendecomposing Hankel matrices built from observed flow-map samples.
  • The framework supplies a route to rigorous error control for data-driven dynamical-system identification that does not presuppose a particular basis expansion.

Load-bearing premise

The underlying map is analytic, so that the push-forward on locally analytic functionals is faithfully represented by an operator on entire functions of exponential type via the Fourier-Borel transform.

What would settle it

For a concrete analytic map, compute the finite-dimensional approximations from increasing numbers of trajectory samples and observe whether the linearly reconstructed vector field fails to converge to the true field in a suitable norm.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper develops a functional-analytic framework for finite-dimensional approximations of the push-forward induced by an analytic map, acting on locally analytic functionals. It identifies this operator via the Fourier-Borel transform with an operator on entire functions of exponential type, derives explicit error bounds in terms of the smallest eigenvalues of associated Hankel moment matrices, obtains sample-complexity results from i.i.d. data, and applies the construction to prove convergence of a data-driven method for recovering analytic vector fields from finite-time flow-map samples.

Significance. If the identification and error bounds hold without hidden circularity or commutation gaps, the work supplies a rigorous bridge between functional analysis and data-driven dynamical systems, with explicit, eigenvalue-controlled approximation guarantees that could support reliable vector-field reconstruction under analyticity assumptions. The emphasis on linear-algebraic operations on the finite-dimensional approximations and the avoidance of free parameters in the bounds are potential strengths.

major comments (1)
  1. [Abstract] Abstract, paragraph 2: the central claim that linear algebraic operations on the finite-dimensional push-forward approximations recover the analytic vector field rests on the Fourier-Borel identification being faithful and commuting with the Hankel truncation. The skeptic note correctly flags that if truncation error interacts with the exponential-type growth, the explicit eigenvalue bounds may not transfer to convergence of the reconstructed generator; this step requires explicit verification in the derivations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to confirm that the Fourier-Borel identification commutes appropriately with the Hankel truncation so that the eigenvalue bounds transfer to the reconstructed generator. We address this point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 2: the central claim that linear algebraic operations on the finite-dimensional push-forward approximations recover the analytic vector field rests on the Fourier-Borel identification being faithful and commuting with the Hankel truncation. The skeptic note correctly flags that if truncation error interacts with the exponential-type growth, the explicit eigenvalue bounds may not transfer to convergence of the reconstructed generator; this step requires explicit verification in the derivations.

    Authors: The Fourier-Borel transform is shown to be a topological isomorphism in Theorem 2.3, mapping locally analytic functionals onto entire functions of exponential type while preserving the relevant seminorms. The push-forward operator is transferred to this space in Section 3, where Proposition 3.2 establishes that it commutes with the finite-rank Hankel projections up to an error controlled by the exponential-type radius. The explicit eigenvalue bounds in Theorem 4.1 are obtained after this commutation and already incorporate the interaction between truncation and growth via a uniform estimate on the remainder terms (see the proof of Lemma 4.3). Consequently, the linear-algebraic reconstruction of the vector field in Section 5 inherits the same bounds, yielding the stated convergence. We are prepared to insert a short clarifying paragraph after Proposition 3.2 that isolates the commutation identity if the referee finds the current placement insufficiently prominent. revision: partial

Circularity Check

0 steps flagged

Derivation chain is self-contained; error bounds and convergence are independent of target reconstruction

full rationale

The abstract and description present a functional-analytic framework that identifies the push-forward operator via the Fourier-Borel transform, then derives finite-dimensional approximations together with explicit error bounds stated in terms of the smallest eigenvalues of Hankel moment matrices and sample-complexity estimates from i.i.d. data. These quantities are introduced as direct consequences of the analyticity assumption and the transform identification, not as quantities fitted to or defined by the downstream vector-field recovery task. The claimed convergence of the data-driven reconstruction method is asserted to follow from the preceding bounds under general conditions, without any quoted reduction that makes the bounds or the identification equivalent to the reconstruction result by construction. No self-citation load-bearing steps, ansatzes smuggled via prior work, or renaming of known results appear in the supplied text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard properties of the Fourier-Borel transform and the topology of locally analytic functionals; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The Fourier-Borel transform identifies the push-forward on locally analytic functionals with an operator on entire functions of exponential type.
    Invoked in abstract paragraph 2 as the key identification step.

pith-pipeline@v0.9.0 · 5684 in / 1264 out tokens · 21343 ms · 2026-05-24T02:19:06.075059+00:00 · methodology

discussion (0)

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Reference graph

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