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arxiv: 2604.11715 · v1 · submitted 2026-04-13 · 📡 eess.SY · cs.SY· math.OC

Koopman Representations for Non-Vanishing Time Intervals: An Optimization Approach and Sampling Effects

Younghwan Cho , Richard Sowers This is my paper

Pith reviewed 2026-05-10 14:49 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords Koopman operatoreigenfunctionsoptimizationaliasingsampling effectsdynamical systemsdata assimilationextended dynamic mode decomposition
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The pith

Koopman eigenfunctions can be learned from data at arbitrary non-vanishing time intervals by solving an optimization problem that exposes aliasing limits.

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

The paper casts the recovery of Koopman eigenfunctions as a direct optimization task when observations occur at finite, non-zero time steps rather than infinitesimal ones. This formulation makes explicit an identifiability limit arising from aliasing between the system's oscillatory frequencies and the chosen sampling pattern. It further identifies a steep loss valley caused by phase alignment near the true frequency and demonstrates that irregular sampling can break the aliasing through phase cancellation. The approach matters because most practical datasets from sensors or experiments arrive at discrete, non-infinitesimal intervals, so clarifying these limits improves the reliability of Koopman-based models for data assimilation and control.

Core claim

We formulate the problem of learning Koopman eigenfunctions from observations at arbitrary, possibly non-vanishing, time intervals as an optimization problem. Analysis of the formulation reveals aliasing induced by oscillatory dynamics and the sampling pattern, making an inherent identifiability limit explicit. The analysis also uncovers phase alignment near the true Koopman frequency, which creates a steep loss valley and demands careful optimization. We further show that irregular sampling can break aliasing and lead to phase cancellation. Numerical results demonstrate the efficacy of the proposed method under large regular time intervals compared to generator extended dynamic mode decomp

What carries the argument

The finite-interval optimization loss that enforces consistency between observed transitions and the Koopman operator action, thereby surfacing sampling-induced aliasing in the frequency domain.

If this is right

  • The optimization method recovers accurate Koopman spectra at large regular sampling intervals where generator extended dynamic mode decomposition degrades.
  • Irregular sampling breaks aliasing and enables recovery of the true spectrum through phase cancellation.
  • Phase alignment near the correct frequency produces narrow, steep valleys in the loss surface that require careful numerical handling.
  • An explicit identifiability limit exists for regularly sampled data due to the interaction of system frequencies and sampling rate.

Where Pith is reading between the lines

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

  • The aliasing analysis could guide experimental design by favoring irregular sampling schedules to improve spectral recovery without collecting more data.
  • Extensions to noisy or multimodal observations would follow naturally by adding corresponding terms to the same optimization loss.
  • The identifiability limit parallels classical Nyquist-type bounds and may suggest minimum sampling irregularity requirements for reliable Koopman identification in practice.

Load-bearing premise

The optimization can reliably escape the steep loss valleys created by phase alignment and recover the true spectrum, with numerical success on the tested oscillatory systems generalizing to other dynamics.

What would settle it

A simple harmonic oscillator sampled at regular large intervals where the optimizer converges to an aliased frequency rather than the true Koopman frequency, or where irregular sampling fails to produce phase cancellation and spectrum recovery.

Figures

Figures reproduced from arXiv: 2604.11715 by Richard Sowers, Younghwan Cho.

Figure 2
Figure 2. Figure 2: Loss landscape of (6). The red vertical dotted lines show the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Spectrum obtained from gEDMD and the proposed method. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
read the original abstract

Koopman operator theory is a key tool in data assimilation of complex dynamical systems, with the potential to be applied to multimodal data. We formulate the problem of learning Koopman eigenfunctions from observations at arbitrary, possibly non-vanishing, time intervals as an optimization problem. Analysis of the formulation reveals aliasing induced by oscillatory dynamics and the sampling pattern, making an inherent identifiability limit explicit. The analysis also uncovers phase alignment near the true Koopman frequency, which creates a steep loss valley and demands careful optimization. We further show that irregular sampling can break aliasing and lead to phase cancellation. Numerical results demonstrate the efficacy of the proposed method under large regular time intervals compared to generator extended dynamic mode decomposition, and support the idea that irregular sampling can help recover the true Koopman spectrum.

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

2 major / 2 minor

Summary. The paper formulates learning Koopman eigenfunctions from observations at arbitrary (including non-vanishing) time intervals as a non-convex optimization problem. Analysis of the resulting loss reveals aliasing between oscillatory dynamics and the sampling pattern, which imposes an explicit identifiability limit, together with phase-alignment effects that produce steep loss valleys near the true Koopman frequency. The authors further show that irregular sampling can break aliasing via phase cancellation. Numerical experiments on oscillatory systems indicate that the approach recovers the true spectrum more reliably than generator extended dynamic mode decomposition when sampling intervals are large, and that irregular sampling improves recovery.

Significance. If the optimization reliably locates the global minimum corresponding to the true spectrum, the work would meaningfully extend Koopman methods to practical data regimes with large or irregular sampling intervals, which are common in data assimilation and multimodal sensing. The explicit aliasing/identifiability analysis and the demonstration that irregular sampling can mitigate phase issues are concrete contributions; however, the absence of convergence guarantees or basin-of-attraction results limits the strength of the central claim.

major comments (2)
  1. [optimization formulation and numerical results] The optimization formulation (abstract and §3) produces a loss landscape with steep valleys at phase-aligned frequencies. No basin-of-attraction analysis, convergence guarantees, or systematic multi-start statistics are provided to establish that gradient-based solvers consistently recover the true Koopman frequency rather than spurious aliased minima. This assumption is load-bearing for the claim that the method works under large regular intervals.
  2. [numerical results] Numerical demonstrations (likely §5) are restricted to specific oscillatory systems. Without broader benchmarks (e.g., higher-dimensional or non-periodic dynamics) or ablation on optimizer choice and initialization, it is unclear whether the reported efficacy generalizes beyond the tested cases.
minor comments (2)
  1. [preliminaries] Notation for the time-interval set and the eigenfunction parameterization should be introduced earlier and used consistently to improve readability.
  2. [figures] Figure captions for the loss-surface and spectrum-recovery plots would benefit from explicit labeling of the true frequency versus aliased minima.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive summary and for highlighting the potential of the work while identifying key limitations. We address each major comment below and describe the revisions we will make.

read point-by-point responses

  1. Referee: The optimization formulation (abstract and §3) produces a loss landscape with steep valleys at phase-aligned frequencies. No basin-of-attraction analysis, convergence guarantees, or systematic multi-start statistics are provided to establish that gradient-based solvers consistently recover the true Koopman frequency rather than spurious aliased minima. This assumption is load-bearing for the claim that the method works under large regular intervals.

    Authors: We agree that the loss landscape in §3 exhibits steep valleys near phase-aligned frequencies, as this is a direct consequence of the phase-alignment analysis we present. The manuscript does not contain basin-of-attraction analysis or theoretical convergence guarantees for the non-convex problem; such guarantees would require a separate, substantial theoretical development. In the current numerical experiments (§5), we already employ multiple random initializations and observe consistent recovery of the true frequencies for the tested oscillatory systems. To strengthen the empirical support, the revised manuscript will include systematic multi-start statistics reporting success rates over a wider range of initial conditions and sampling intervals. revision: partial

  2. Referee: Numerical demonstrations (likely §5) are restricted to specific oscillatory systems. Without broader benchmarks (e.g., higher-dimensional or non-periodic dynamics) or ablation on optimizer choice and initialization, it is unclear whether the reported efficacy generalizes beyond the tested cases.

    Authors: The numerical section focuses on oscillatory systems because the aliasing and identifiability analysis is most relevant and pronounced for such dynamics. The comparisons with generator extended dynamic mode decomposition are designed to isolate the benefit of the proposed formulation under large regular intervals. In the revision we will add an ablation study on optimizer choice and initialization, together with results on at least one higher-dimensional linear system, to provide clearer evidence of generalization within the scope of the paper's claims. revision: yes

standing simulated objections not resolved
  • Theoretical basin-of-attraction analysis or convergence guarantees for the non-convex optimization problem

Circularity Check

0 steps flagged

No circularity: formulation and analysis are independent of target claims

full rationale

The paper casts Koopman eigenfunction learning as a non-convex optimization problem whose loss is derived from the observation model at arbitrary time intervals. Alias analysis and the phase-alignment valley are obtained directly from the same model equations rather than from any fitted parameter or self-citation. Irregular-sampling benefits are shown by explicit construction of the sampling pattern inside the loss, not by renaming a known result. No self-citation is load-bearing for the central identifiability or optimization claims, and no prediction is statistically forced by a prior fit. The derivation chain therefore remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no specific free parameters, axioms, or invented entities are described or can be extracted.

pith-pipeline@v0.9.0 · 5438 in / 1066 out tokens · 36528 ms · 2026-05-10T14:49:16.657247+00:00 · methodology

discussion (0)

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Reference graph

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