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A Modified Bisecting K-Means for Approximating Transfer Operators: Application to the Lorenz Equations

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Pith's one-line read This paper shows that a piecewise-constant dictionary built by modified bisecting k-means, scaled to over one million cells, makes data-driven transfer operators converge on the Lorenz attractor's invariant statistics, autocorrelations…

desk verdict Useful scaling study of a k-means partition-based transfer operator, but the convergence claims are measured against a reference trajectory short enough to produce the observed saturation itself. read the letter →

arxiv 2412.03734 v1 pith:GOEXFT47 submitted 2024-12-04 physics.comp-ph

Andre N. Souza , Simone Silvestri This is my paper · ORCID
classification physics.comp-ph MSC 68Q2568R1068U05
keywords modifiedbisectingk-meanstransferoperatorKoopmaneigenfunctionPerron-FrobeniusextendeddynamicmodedecompositionLorenzequationsinvariantmeasurepiecewise-constantdictionary
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper is trying to establish that the extended dynamic mode decomposition, supplied with a state-space partition built by a modified bisecting k-means algorithm, can approximate the transfer operator of a chaotic system well enough to recover its steady-state statistics, temporal autocorrelations, and a distinctive Koopman eigenfunction. The test case is the Lorenz equations, with dictionaries of piecewise-constant functions up to roughly one million cells constructed from a single trajectory of length $T=10^5$. The authors report first-order convergence of invariant cumulants that saturates near $10^5$ cells, convergence of the $z$-autocorrelation toward the time-series value, and a quasi-invariant Koopman eigenfunction that becomes increasingly intermittent as resolution increases. The practical stake is that this partition-based approach can be pushed to very large dictionaries on a single CPU core, so the limits appear to be set by available data rather than by the cost of the dictionary.

What carries the argument

The central object is a partition of state space into leaf cells of nearly equal probability, produced by a modified bisecting k-means algorithm with a splitting threshold $p_{\min}$: a cell is split only if its probability exceeds the threshold, yielding near-uniform entropy and no user-specified cluster count. Each cell is an element of the piecewise-constant dictionary for the extended dynamic mode decomposition, and the classifier maps each state to an integer cell index. From the resulting cell-index time series, the paper builds sparse matrix representations of the infinitesimal generator $Q$ and the Perron-Frobenius operator, then computes eigenvalues and eigenvectors by inverse iteration. The near-uniform entropy property ensures every column of the operator is estimated from sufficiently many samples, while the hierarchical tree provides consistent coarse-to-fine partitions and makes classification of new data cheap.

What would settle it

Recompute the same cumulant and autocorrelation errors against a second, much longer trajectory, say $T=10^6$ or $10^7$, generated independently; if the saturation near $10^5$ cells persists, the paper's convergence picture holds, and if the error drops substantially, the earlier saturation was reference-sampling noise. A simpler check is to plot log error versus log cell count and test that the slope is approximately one only below the resolution limit set by the decorrelated sample count.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that a data-driven discretization of the continuity equation for a noiseless chaotic system can be refined to over a million piecewise-constant cells and, in that limit, reproduces the invariant measure, the autocorrelation of the $z$ variable, and the quasi-invariant Koopman eigenfunction. The cumulants converge at first order in cell count until roughly $10^5$ cells, after which the error saturates at about three digits, consistent with the finite number of decorrelated samples in the reference trajectory. The $z$-autocorrelation computed from the operator approaches the temporal average as resolution increases, and this holds both for the infinitesimal generator and for Perron-Frobenius operators built at several timescales. The associated Koopman eigenfunction develops finer-scale, increasingly intermittent structure with resolution, and the authors conclude that while statistics converge, the eigenfunctions themselves are harder to declare convergent. The paper also finds that the generator is more dissipative than Perron-Frobenius operators constructed at larger timescales, but that Perron-Frobenius operators at timescales much longer than the target eigenvalue lose meaning.

Load-bearing premise

The reference statistics come from a single trajectory of length $T=10^5$, which the authors estimate contains only $10^3$ to $10^4$ effectively independent samples; if those reference values are noisier than the reported three digits, the observed saturation near $10^5$ cells could be an artifact of the reference rather than a property of the operator approximation.

Editorial extensions

If this is right

  • For the Lorenz test case, invariant cumulants converge at first order as the partition is refined, saturating near $10^5$ cells for a trajectory of length $T=10^5$.
  • The $z$-autocorrelation recovered from the operator converges to the temporal average as resolution increases, for both the generator and Perron-Frobenius constructions.
  • The quasi-invariant Koopman eigenfunction develops finer and more intermittent structure with increasing resolution, so individual eigenfunctions are harder to call converged even while statistical quantities converge.
  • The infinitesimal generator is more dissipative than Perron-Frobenius operators built at larger timescales, but very large timescales skip the target eigenvalue; the paper recommends an iterative strategy of estimating timescales from the generator first and then constructing a Perron-Frobenius operator at an appropriate timescale.
  • The full computation runs on a single CPU core, showing that the modified bisecting k-means dictionary scales to over a million terms without specialized hardware.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • I infer that the near-uniform-entropy splitting rule effectively makes the hierarchy an adaptive quadrature for state space, and the same splitting criterion could be replaced by local error indicators if a different observable family were the target.
  • I infer that if a much longer reference trajectory were used, the saturation near $10^5$ cells would likely move outward, which would confirm that the accuracy ceiling is data-limited rather than dictionary-limited.
  • I infer that the increasingly intermittent Koopman eigenfunction means early-warning indicators built from such eigenfunctions should be reported together with the partition resolution, since their spike structure changes qualitatively as the cell count grows.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a modified bisecting k-means algorithm for generating hierarchical, nearly uniform partitions of state space, and uses these partitions as piecewise-constant dictionaries in an extended dynamic mode decomposition (EDMD) of the transfer operator. The method is applied to the Lorenz equations with up to O(10^6) cells. The central numerical claims are: (i) invariant-measure cumulants converge at roughly first order in the number of cells and saturate near 10^5 cells (Figs. 3.3 and 3.4); (ii) the z-autocorrelation computed from the operator converges to the temporal autocorrelation as resolution increases, with the Perron-Frobenius operator at larger timescales converging fastest (Fig. 3.5); and (iii) a quasi-invariant-set Koopman eigenfunction becomes increasingly intermittent as resolution increases (Figs. 3.6 and 3.7). The authors are explicit that the eigenfunction convergence itself is difficult to assess and that the temporal reference is limited to 10^3–10^4 decorrelated samples.

Significance. If the convergence claims hold, the paper demonstrates a practical, single-CPU method for constructing transfer-operator approximations with dictionary sizes that are unusually large for EDMD-type methods, while keeping matrix entries well-sampled through nearly uniform hierarchical partitions. The paper is also honest about the limitations of the eigenfunction interpretation. Strengths include: a clear, reproducible experimental setup (initial condition, timestep, threshold parameter p = 1.4e-6 are specified); a parameter-free construction in the sense that no constants are fitted to the reported statistics; and consistent use of the hierarchical partition to define a meaningful refinement path from four cells to over one million cells. The main weakness is that all error measurements compare the operator against temporal statistics computed from the same trajectory that generated the operator, and the reference uncertainty is acknowledged but never quantified.

major comments (3)
  1. [§3.1, Figs. 3.3–3.5] The central convergence and saturation claims are measured entirely against temporal averages from the same T = 10^5 trajectory that was used to construct the operator (Eqs. 3.1–3.4). As the authors state, this trajectory contains only 10^3–10^4 decorrelated samples, which implies a standard error on the order of 1–3% for the reference statistics. The saturation seen near 10^5 cells at relative errors around 10^-3 is therefore exactly what one would expect from reference sampling noise. The paper should quantify this uncertainty directly, for example by showing bootstrap confidence intervals for the temporal averages, or by validating against an independently generated longer trajectory, before asserting that the saturation and the first-order slope are properties of the operator approximation.
  2. [§3.1, Figs. 3.3–3.4] The claimed 'first-order convergence rate' is supported only by a dashed guide line drawn through the points; no regression slope, confidence interval, or quantitative fit is reported. Given that the saturation may be a reference-noise artifact, the paper should either provide a formal fit with uncertainty or soften the first-order claim to a qualitative observation. This is load-bearing because the first-order convergence is a headline result in the abstract and conclusions.
  3. [§3.1, Fig. 3.7] The claim that the Koopman eigenfunction 'becomes increasingly intermittent as resolution increases' is based on visual inspection of three resolutions. No quantitative intermittency metric, norm, or convergence diagnostic is provided, and the authors themselves note that it is unclear in what sense the eigenfunctions converge. If this eigenfunction is a central object of the paper, the claim should be supported by a reproducible measure, such as a scale-dependent variance or a Sobolev-type norm; otherwise, the statement should be presented as a qualitative observation rather than a finding.
minor comments (5)
  1. [§3.1, figure captions] The captions and axis labels of Figs. 3.3–3.5 contain corrupted glyphs such as '/uni2212', '/g22', and '/g24', which make the figures unintelligible in the current version. These need to be repaired before publication.
  2. [§2, clustering threshold] The definitions of Nmin, Nthreshold, and pmin are confusingly written. In particular, pmin is introduced as a 'minimum probability' but then used in expressions where it must be interpreted as a percentage or a scaled quantity (e.g., Nmin = floor(100/pmin) and Nthreshold = ceil(pmin Nmin)). The units and scaling should be made explicit.
  3. [Figure 3.1 caption] The caption says the hierarchy starts 'with four cells (top right)' while the main text says 'four partitions (top left)'; these should be reconciled.
  4. [References] References [1] and [2] are the same paper, and references [8] and [9] are also duplicates. These should be consolidated.
  5. [General] The paper would benefit from a data/code availability statement, since the numerical experiments are a central contribution and the exact algorithm parameters are only partially specified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the operator is constructed from the same trajectory but the convergence claims are measured against that trajectory as ground truth, and no parameter is fitted to the target outputs.

full rationale

The paper's derivation chain is: integrate the Lorenz equations (Section 3), partition state space with modified bisecting k-means, build the transfer operator using the data-driven construction from the same author's prior work [34,35], and compare ensemble statistics (3.3)-(3.4) with temporal statistics (3.1)-(3.2). No parameter is fitted to the target statistics: the partition threshold p = 1.4e-6 is fixed a priori, and the operator construction is a published Ulam-type method. The agreement in Figures 3.3-3.5 is an in-sample consistency check, but it is not circular: a finer partition is not forced by construction to reproduce the temporal cumulants or the full lag-τ autocorrelation, because a generator estimated from one-step transitions does not encode empirical lag-τ pair statistics. The paper explicitly flags the finite-sample limitation ('we have between 10^3 and 10^4 decorrelated in time samples... empirical error... should begin dominating the inaccuracy at around three digits', Section 3.1); this is a statistical robustness concern, not a definitional equivalence. The self-citations [34,35] supply transfer-operator formulas that are restated in Equations (3.1)-(3.4) and do not themselves contain the Lorenz convergence results, so they are not load-bearing in a circular sense. The near-unity scaled entropy in Figure 3.2 is a designed property of the splitting criterion described in Section 2, and the paper presents it as such rather than as an independent prediction. Overall, no step in the derivation reduces to its own inputs by definition or by fitted parameter.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central numerical claims rest on a resolution parameter p, a heuristic 100-point threshold, the ergodicity of the Lorenz system, and the validity of the author's own transfer-operator formulas from [34,35]. No new physical entities are postulated.

free parameters (2)
  • probability threshold p = 1.4e-6
    Sets the target cell probability and hence the finest resolution (k about 1.5/p, roughly 10^6 cells). Chosen to explore convergence, not fitted to target outputs.
  • minimum points per cluster constant = 100
    Nmin = floor(100/pmin) sets the cluster splitting threshold; the value 100 is a heuristic with no independent justification (Section 2, paragraph 4).
assumptions (3)
  • domain assumption Temporal averages over a single long trajectory equal the invariant measure of the Lorenz system (ergodicity).
    Used to define the ground-truth statistics in equations (3.1) to (3.4) and to interpret relative errors; the paper provides only a sampling-based argument, not a proof or quantitative error estimate (Section 3).
  • domain assumption The partition-based data-driven generator Q, constructed via the formulas in [34,35], converges to the true generator / Perron-Frobenius operator as the partition is refined and the data length grows.
    This is the theoretical foundation of the study; the formulas are imported from self-cited references [34,35] and not derived in this paper (Section 3, paragraph 2).
  • domain assumption The Lorenz system is symmetric under (x,y,z) -> (-x,-y,z), so appending the mirrored trajectory doubles the data without biasing the operator.
    Used to build the data matrix X in Section 3, first paragraph; correct for the standard Lorenz equations but an assumption about the system.

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Pith. "Pith review of A Modified Bisecting K-Means for Approximating Transfer Operators: Application to the Lorenz Equations." pith.science (2026). https://pith.science/paper/GOEXFT47

@misc{pith2026241203734,
  author       = {Pith},
  title        = {Pith review of: A Modified Bisecting K-Means for Approximating Transfer Operators: Application to the Lorenz Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GOEXFT47}},
  note         = {Machine review of arXiv:2412.03734}
}
read the original abstract

We investigate the convergence behavior of the extended dynamic mode decomposition for constructing a discretization of the continuity equation associated with the Lorenz equations using a nonlinear dictionary of over 1,000,000 terms. The primary objective is to analyze the resulting operator by varying the number of terms in the dictionary and the timescale. We examine what happens when the number of terms of the nonlinear dictionary is varied with respect to its ability to represent the invariant measure, Koopman eigenfunctions, and temporal autocorrelations. The dictionary comprises piecewise constant functions through a modified bisecting k-means algorithm and can efficiently scale to higher-dimensional systems.

Figures

Figures reproduced from arXiv: 2412.03734 by the authors.

Figure 2
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Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png]
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Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png]
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Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png]
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Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png]

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Forward citations

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.