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Subsampling, aligning, and averaging to find circular coordinates in recurrent time series

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Circular coordinates from persistent cohomology can be made insensitive to uneven sampling by subsampling each point cloud, computing a coordinate per subsample, and Procrustes-aligning and averaging the results, yielding more informative…

desk verdict A solid, useful method paper whose central empirical claim is weakened by a self-referential tuning option and a missing comparison to existing density-robust baselines. read the letter →

arxiv 2412.18515 v1 pith:ZOJO2YTS submitted 2024-12-24 stat.ML cs.CGcs.LGmath.AT

Andrew J. Blumberg , Mathieu Carrière , Jun Hou Fung , Michael A. Mandell This is my paper · ORCID
classification stat.MLcs.CGcs.LGmath.AT MSC 55N3162R40
keywords circularcoordinatespersistentcohomologyrejectionsamplingProcrustesalignmentdensityestimationrecurrenttimeseriestopologicaldataanalysisC.elegansneuronalrecordings
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

Persistent-cohomology circular coordinates are a principled way to assign a phase to recurrent time series, but they distort when the data are sampled unevenly: dense regions compress the coordinate's angular change. This paper claims that the distortion can be removed without ad hoc reweighting by a subsample-and-average scheme: estimate local density, reject points to create several nearly uniform subsamples, extract a circular coordinate from each via persistent cohomology, and then align and average those coordinates with a Procrustes-style procedure. The authors report that the resulting corrected coordinate achieves higher mutual information with the original data than the uncorrected coordinate in every experiment, on both synthetic unbalanced circles and ellipses and on whole-brain C. elegans recordings, and that it is much cheaper to compute because persistent cohomology runs on small subsamples rather than the full dataset. If this holds, the method gives a practical, density-robust coordinate for exploratory analysis of recurrent neural and behavioral data.

What carries the argument

The object that carries the argument is the corrected circular coordinate, defined as the averaged output of an alignment step on subsample coordinates. The machinery has four parts: (1) a count-based density estimator $\hat{\rho}_\epsilon(x) = \#(X \cap B_\epsilon(x))$ with a bandwidth chosen by a multivariate heuristic, used in rejection sampling to equalize density; (2) per-subsample circular coordinates obtained from persistent cohomology; (3) extension of each subsample coordinate to the whole dataset by circular-mean interpolation with a Gaussian kernel; and (4) a generalized Procrustes alignment on the circle, solved approximately by an O(2) Procrustes problem in $\mathbb{R}^2$ and refined by hill climbing on the circular loss. Proposition 4.4 bounds the distance between the O(2) seed and the circular optimum by $(1+\pi)\sqrt{L^*}$, where $L^*$ is the optimal circular Procrustes loss, which justifies using the seed when the coordinates already nearly agree.

What would settle it

Generate a synthetic circle whose sampling density has two well-separated modes instead of one, run the paper's algorithm with its default settings (thirty subsamples of expected size fifty), and compare the normalized mutual information of the corrected and uncorrected coordinates against the true angle. If the corrected coordinate does not beat the uncorrected one across twenty replicates, the claim of uniform improvement fails in a regime where the optimal alignment loss is not small.

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Extended reading notes

Core claim

The central discovery is that density distortion in circular coordinates can be corrected by replacing a single global coordinate with an ensemble: after evaluating a count-based density estimator at each point, the algorithm accepts points with probability inversely proportional to local density, producing subsamples that are nearly uniform. Each subsample yields a circular coordinate from its first persistent cohomology class, those coordinates are extended to the full dataset by circular-mean interpolation, and the extensions are aligned and averaged by minimizing a circular Procrustes loss, seeded by an O(2) Procrustes solution in $R^{2}$. On the datasets tested, the aligned average retains more structure of the original point cloud than the uncorrected coordinate, as measured by normalized nearest-neighbor mutual information, and the subsampled computation is faster by a factor of 2.16 on the synthetic circle and 23.0 on the worm dataset. The paper also presents the resulting loop-based picture of C. elegans locomotory brain dynamics, in which specific behavioral states occupy specific regions of the circular coordinate.

Load-bearing premise

The method assumes the alignment step, which rotates and reflects the subsample coordinates into agreement, finds the best possible agreement instead of getting stuck in a bad local pattern; the paper proves this only when the subsample coordinates already nearly agree, and real data are not guaranteed to satisfy that.

Editorial extensions

If this is right

  • Persistent-cohomology circular coordinates on a full point cloud can be replaced by the subsample-and-average version without losing fidelity, and with runtime gains that grow with dataset size.
  • The corrected coordinate is defined on every point of the dataset, not just points in a subsample, so downstream analyses such as phase plots, tangent vectors, and behavioral segmentation can use it directly.
  • On C. elegans recordings, the corrected coordinate recovers the loop structure of locomotory brain dynamics and brings out a small ventral-turn loop that the uncorrected coordinate obscures.
  • The normalized nearest-neighbor mutual-information score gives a ground-truth-free criterion for comparing circular coordinates, applicable to any recurrent dataset.
  • Because persistent cohomology is computed on small subsamples, the approach can handle datasets too large for a single full persistence computation.

Reading between the lines

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

  • Inference: a multi-start variant of the hill-climbing alignment, running from several O(2) seeds and keeping the best circular Procrustes loss, would supply a practical stability check for datasets where the optimal loss is not known to be small.
  • Inference: the same subsample, align, and average recipe could be applied to other coordinate constructions, for instance principal-component or diffusion-map phases, since the density-distortion mechanism is not specific to persistent cohomology.
  • Inference: the mutual-information criterion suggests an automated rule for choosing the density-estimation bandwidth: select epsilon that maximizes the normalized mutual information between the coordinate and the original data, a procedure the paper only mentions as manual tuning.
  • Inference: the approach extends naturally to multiple simultaneous circular coordinates (tori) by aligning subsample coordinates under the corresponding product of rotations and reflections, a case the paper leaves to future work.
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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

4 major / 5 minor

Summary. The paper proposes an algorithm for computing circular coordinates on recurrent time-series data, intended to address the sensitivity of the standard persistent-cohomology pipeline to uneven sampling density. The method has three stages: rejection sampling to approximately uniformize the sampling density, computation of circular coordinates on each subsample via persistent cohomology, and alignment and averaging of these coordinates using O(2) Procrustes matching seeded by a ten Berge solver and refined by hill climbing. The authors prove a consistency result for a count-based density estimator (Proposition 3.3) and a bound relating the O(2) Procrustes solution to the circle Procrustes solution (Proposition 4.4). They validate on synthetic unbalanced circles and ellipses and on two C. elegans whole-brain calcium-imaging datasets, using normalized KSG mutual information as a quantitative metric and reporting runtime comparisons.

Significance. If the empirical claims hold, the algorithm would be a useful contribution to applied topological data analysis: it is conceptually simple, parallelizable, and directly addresses a known failure mode of circular-coordinate extraction. The theoretical sections are a genuine strength: Proposition 3.3 is a clean consistency statement with explicit rate conditions, and Proposition 4.4 gives a nontrivial geometric bound connecting the Euclidean and circular Procrustes problems. The use of subsampling also targets an important practical bottleneck, since persistent cohomology on the full Rips complex is costly. However, the significance of the paper currently rests on the empirical comparison, and that comparison has important gaps: the real-data evaluation uses only the uncorrected coordinate as a baseline, and the paper itself, in Remark 3.7, suggests tuning a key hyperparameter by the same mutual-information metric used for evaluation. The efficiency claim, while plausible, is supported by a very limited runtime experiment. These issues do not invalidate the method, but they need to be addressed before the advertised robustness and efficiency advantages can be accepted.

major comments (4)
  1. [Remark 3.7 and Section 5.3] The only quantitative real-data validation is the KSG mutual-information comparison in Section 5.3, and the paper's headline claim is that the corrected coordinate yields higher mutual information in every case. However, Remark 3.7 explicitly proposes tuning the density-estimation bandwidth ε by “maximizing the mutual information metric we describe in Section 5.3.” If ε, or the Gaussian kernel width β in Eq. (4.2), was selected this way, the comparison against the uncorrected coordinate is biased, because the corrected pipeline has an extra tunable knob while the baseline has no analogous tuning. The manuscript does not report the ε values used for the C. elegans experiments or state whether any MI-based selection was performed. Please report all hyperparameter choices, avoid using the evaluation metric for tuning, and provide a sensitivity analysis over ε and β; without this, the claim “higher mutual information values than the uncorrected coordinate in every case” is not a fair test.
  2. [Section 5.3 and Section 1] The experiments compare the corrected coordinate only with the uncorrected coordinate. The abstract claims a “more robust coordinate than other approaches,” and the introduction cites existing density-robust methods by Rybakken, Baas, and Dunn, Perea, and Paik and Park. To support the comparative claim, the paper should benchmark against at least one of these existing methods on the unbalanced circle and ellipse examples and, if feasible, on the C. elegans data, using the same evaluation metric. Without such a baseline, the contribution is established only relative to the standard uncorrected pipeline, which is a weaker statement than the paper makes.
  3. [Section 4, Step 3, and Section 6] The hill-climbing procedure is central to the averaging step, but the authors state in Section 6 that they “do not understand the local convexity and global properties of the optimization problem.” Proposition 4.4 only bounds the O(2) seed's distance to the circle-optimal centroid when the optimal Procrustes loss L* is small, and no evidence is given that L* is small for the real datasets. Please report the achieved loss values for all experiments and test sensitivity of the final coordinate to random restarts or alternative seeds. Without this, the corrected coordinate is not guaranteed to be a well-defined, reproducible output of the algorithm, which weakens the claim that the method is robust.
  4. [Section 5.4] The runtime comparison is presented only as minimum wall-clock times over 20 replicates, with no software/hardware details, no variance or error bars, and no breakdown of preprocessing, subsampling, persistent cohomology, alignment, and extension costs. Since “better efficiency” is a stated contribution in the abstract, please provide fuller experimental detail: implementation language and version, hardware, timing methodology, and ideally scaling behavior as dataset size increases. Also clarify whether the reported times include the same coordinate-extraction code path for both methods, since the corrected method involves repeated subsample computations.
minor comments (5)
  1. [Abstract] There is a typo in the abstract: “C. elegansthat” should be “C. elegans that.”
  2. [Section 5.1] The ellipse validation is only qualitative; please report quantitative errors or correlations with the arc-length parametrization, as is done for the circle example.
  3. [Section 5.2.1] The comparison with the Kato et al. labels in Figure 5 is described qualitatively. A quantitative agreement measure, such as a contingency-table statistic or adjusted mutual information between the discrete labels and the coordinate-derived states, would strengthen the claim.
  4. [References] Reference [38] contains a typo: “Desnity” should be “Density.”
  5. [Section 6] The word “proccesses” in the final paragraph should be “processes.”

Circularity Check

1 steps flagged · score 4.0 of 10

Derivation is self-contained, but the real-data MI validation is potentially circular because Remark 3.7 suggests tuning the density bandwidth by the same MI metric used in Section 5.3.

  1. other [Remark 3.7; Section 5.3; Section 4 (β ≈ ε)]
    "Alternatively, the value of the hyperparameter ϵ can be manually tuned, for example by maximizing the mutual information metric we describe in Section 5.3."

    Section 5.3 uses the KSG mutual-information estimate as the sole quantitative validation of the headline claim that 'the corrected coordinate from our approach yields higher mutual information values than the uncorrected coordinate in every case.' Remark 3.7 explicitly proposes tuning the density-estimation bandwidth ε, which controls the rejection-sampling correction and, via 'β ≈ ε' in Eq. 4.2, the coordinate extension, by maximizing that same MI metric. If that suggestion was followed, the corrected coordinate's MI advantage is partly by construction, because ε is a free parameter of the corrected pipeline that can be optimized to inflate the evaluation metric, while the uncorrected baseline has no analogous tunable knob.

full rationale

The core algorithm is self-contained: circular coordinates are produced from persistent cohomology on subsamples, with density equalization by rejection sampling and O(2)/circular Procrustes alignment, and the synthetic validation uses Scott's rule for ε with ground-truth angles. The self-citation to [7] supplies the alignment framework but is not used to justify the target claim, and Proposition 4.4 is a new bound rather than an imported uniqueness result. The only circularity risk is in the real-data validation: Remark 3.7 suggests tuning the density bandwidth by the same KSG mutual-information metric that Section 5.3 uses to support the headline claim of universal MI improvement. Because ε (and the coupled extension bandwidth β ≈ ε) is a free parameter of the corrected pipeline, and the paper neither reports ε values nor rules out MI-based selection, the real-data MI comparison is not demonstrably independent. This is a conditional/partial circularity rather than an observed fit, so the score is moderate rather than severe.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The algorithm's central claim depends on several unproven practical assumptions: the data manifold model, the subsample homology consistency, the adequacy of the density bandwidth, and the convergence of the alignment optimization. The latter is explicitly disclaimed by the authors.

free parameters (5)
  • epsilon (density estimation bandwidth) = chosen via Scott's rule or manual tuning (Remark 3.7)
    Controls the ball radius in the count-based density estimator (Eq. 3.6) and thus the rejection sampling correction; also sets beta approximately equal to epsilon for the extension kernel.
  • k (number of subsamples) = 30 for synthetic circle; unspecified for real data
    Number of subsamples averaged; affects statistical stability and runtime.
  • expected subsample size = 50 for synthetic circle; unspecified for real data
    Set via the acceptance constant m in Section 3, Step 1; determines how many points remain after rejection sampling.
  • beta (Gaussian kernel width in circular mean extension) = taken approximately equal to epsilon in experiments
    Kernel width in Eq. 4.2 for extending subsample coordinates to the full data set.
  • delay embedding parameters (d, tau, PCA dimension) = d=4, tau=20 frames, PCA dim=5 for Yemini data
    Preprocessing choices for constructing the phase space; results stated to be robust but no sensitivity analysis shown.
assumptions (6)
  • domain assumption Data lies on a compact smooth submanifold M of R^N with inherited volume form
    Invoked at the start of Section 3 to justify density estimation and rejection sampling; real data may not satisfy this.
  • domain assumption Samples are iid from a density rho supported on all of M
    Required for Proposition 3.3 consistency; time series samples are temporally correlated, not iid.
  • domain assumption The long bar in PH1 of each subsample corresponds to the same circular feature
    The algorithm assumes the cohomology class selected in each subsample represents the same loop; alignment and averaging are otherwise meaningless.
  • domain assumption The circular mean extension (Eq. 4.2) yields a well-defined coordinate on the whole data set
    Used to extend subsample coordinates; unstable when the weighted sum is near zero.
  • ad hoc to paper Hill climbing converges to a near-global optimum of the circular Procrustes loss
    Admitted in Section 6: no formal guarantees on the optimization problem.
  • ad hoc to paper Intrinsic dimension d is small enough that Scott's rule yields a valid bandwidth
    Remark 3.7 notes the asymptotic conditions may fail for larger d.

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Cite this review

Pith. "Pith review of Subsampling, aligning, and averaging to find circular coordinates in recurrent time series." pith.science (2026). https://pith.science/paper/ZOJO2YTS

@misc{pith2026241218515,
  author       = {Pith},
  title        = {Pith review of: Subsampling, aligning, and averaging to find circular coordinates in recurrent time series},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZOJO2YTS}},
  note         = {Machine review of arXiv:2412.18515}
}
read the original abstract

We introduce a new algorithm for finding robust circular coordinates on data that is expected to exhibit recurrence, such as that which appears in neuronal recordings of C. elegans. Techniques exist to create circular coordinates on a simplicial complex from a dimension 1 cohomology class, and these can be applied to the Rips complex of a dataset when it has a prominent class in its dimension 1 cohomology. However, it is known this approach is extremely sensitive to uneven sampling density. Our algorithm comes with a new method to correct for uneven sampling density, adapting our prior work on averaging coordinates in manifold learning. We use rejection sampling to correct for inhomogeneous sampling and then apply Procrustes matching to align and average the subsamples. In addition to providing a more robust coordinate than other approaches, this subsampling and averaging approach has better efficiency. We validate our technique on both synthetic data sets and neuronal activity recordings. Our results reveal a topological model of neuronal trajectories for C. elegans that is constructed from loops in which different regions of the brain state space can be mapped to specific and interpretable macroscopic behaviors in the worm.

Figures

Figures reproduced from arXiv: 2412.18515 by the authors.

Figure 1
Figure 1. Top: An unbalanced circle dataset colored by the “true” angle (left) or the uncorrected coordinate inferred from per￾sistent cohomology (center). A plot of the uncorrected coordinate against the true coordinate (right); the dotted diagonal line indi￾cates equality. Bottom: Examples of the subsampled unbalanced circle and their inferred phases (left), the final corrected coordi￾nate obtained by aligning the coordinat… view at source ↗
Figure 2
Figure 2. Top: An unbalanced ellipse dataset colored by arc length (left) or the uncorrected coordinate inferred from persistent cohomology (center). A plot of the uncorrected coordinate against the arc length parametrization (right). Bottom: Examples of the subsampled unbalanced ellipse and their inferred phases (left), the final corrected coordinate (center), and a plot of the corrected co￾ordinate against the arc length pa… view at source ↗
Figure 3
Figure 3. The projection of the neuronal trajectories into PCA space (left) and the persistence diagram (right) for two different worms. We reproduced the dimensionally-reduced version of the phase space from [22]. More specifically, by phase space here we mean a combination of all the available ac￾tivity traces and their derivatives obtained through total-variation regularization. Then, we examined the persistent cohomology … view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The neuronal manifold of Caenorhabditis elegans global brain dynamics, coordinatizing a cyclic locomotory gait, col￾ored by (a) the uncorrected coordinate or (b) the corrected coor￾dinate. For the phase space of the first worm above with the long-lived cohomology class…
Figure 5
Figure 5. Figure 5: Comparison between the labels from [22] and our inferred coordinates, either (a) uncorrected or (b) corrected. Points are colored according to the discrete categories of the pro￾vided labels: forward (FWD), reversal (REV), sustained reversal (REVSUS), dorsal turn (DT),…
Figure 6
Figure 6. Figure 6: C. elegans neuronal manifolds constructed for six worms from [46], colored by (a) the uncorrected coordinate and (b) the corrected coordinate. data sets, we do not typically have access to “ground truth”. We could compare our results to expert-annotated labels as we di…
Figure 7
Figure 7. Figure 7: Estimates of mutual information for (a) the unbalanced circle, (b) the worm from Kato et al., and (c) the six worms from Yemini et al. Where displayed, the p-values are calculated from an one-sided paired t-test. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: Time taken to compute the uncorrected and corrected circular coordinates respectively, with 20 replicates in two datasets. To demonstrate this in practice, we timed the execution of the direct approach of computing the uncorrected and the corrected coordinate on two of…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Density-Robust Spherical Coordinates from Persistent Cohomology

    math.AT 2026-07 conditional novelty 6.0 of 10

    Density-robust S² coordinates are obtained by rejection-sampling uniform subsamples, computing classical spherical coordinates on each, and aligning them via a spherical Procrustes problem with a proven Euclidean relaxation.

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Pith tools

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