{"id":"f10037b5-c4fc-47a9-9b94-302ef0dbb566","arxiv_id":"2412.18515","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A subsample-and-average pipeline with rejection sampling and Procrustes alignment yields density-robust circular coordinates from persistent cohomology, validated on synthetic and C. elegans data.","lead":"This paper introduces an algorithm that computes circular coordinates for recurrent time series data, correcting for uneven sampling density by subsampling, aligning, and averaging. The authors show the approach improves coordinates on synthetic circles and on C. elegans neuronal recordings, while running faster than the standard method.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The only real-data validation is the KSG mutual-information metric, and Remark 3.7 explicitly suggests tuning the density bandwidth by maximizing that same metric; without disclosing ε or ruling out MI-based selection, the claimed 'higher MI in every case' is not a fair comparison.","rationale":"The reader identified the hill-climbing convergence of the Procrustes alignment as the weakest assumption, and the authors themselves admit in §6 that they do not understand the optimization landscape. That is a genuine limitation, but it is not the most load-bearing for the central claim: even a locally optimal alignment might yield coordinates that beat the uncorrected baseline, and the synthetic experiments with ground truth provide some evidence that the pipeline works. The MI-tuning issue is more directly fatal to the strongest claim because it attacks the validity of the only quantitative real-data evidence. The reader did note in the rationale that 'the evaluation metric is suggested as a tuning target in Remark 3.7', but did not make it the central load-bearing concern; hence partial agreement. My verdict remains CONDITIONAL/UNCHANGED because the issue is addressable by reporting hyperparameter choices and rerunning without MI-based selection. The concrete test above would settle whether the MI advantage is genuine. I am not accusing the authors of tuning; the point is that the manuscript does not rule it out, and the central claim is presented as a universal empirical fact. Independent support in the paper includes the synthetic unbalanced-circle experiment (where MI is computed against the true angle, not a tuning target), which is good and partially mitigates the concern. However, the real-data claim is the one highlighted in the strongest claim, and it lacks this independent ground truth.","tokens_in":17484,"tokens_out":6886,"duration_ms":67490,"concrete_test":"Re-run the Kato et al. and Yemini et al. analyses under three protocols: (i) ε fixed by Scott's rule as in §5.1; (ii) ε chosen by the MI-maximization suggestion in Remark 3.7 on the same data; (iii) ε chosen by an independent criterion such as cross-validated density log-likelihood. Report the corrected-vs-uncorrected MI difference for each protocol. If the corrected coordinate fails to beat the uncorrected coordinate under protocol (i) or (iii), the 'in every case' claim is an artifact of MI-based tuning. If it still beats it, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central experimental claim in §5.3 is that the corrected coordinate achieves higher mutual information than the uncorrected coordinate in every case, on both Kato et al. and Yemini et al. data. But the only quantitative validation on real data is the KSG MI estimate, and Remark 3.7 suggests choosing the density-estimation bandwidth ε by 'maximizing the mutual information metric we describe in Section 5.3'. If ε (or related hyperparameters such as the Gaussian kernel width β in Eq. 4.2) was selected to maximize this same metric, then the comparison in §5.3 is biased: the corrected pipeline has an extra tunable knob that can inflate MI, while the uncorrected baseline has no analogous tuning. The paper does not report the ε values used for the C. elegans experiments, nor whether any MI-based selection was performed. The abstract's 'more robust coordinate than other approaches' is therefore supported, on real data, only by a metric that the manuscript itself suggests as a tuning target. This does not necessarily invalidate the method, but it makes the headline empirical claim unfalsifiable as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":17734,"tokens_out":4991,"duration_ms":45715,"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":[{"comment":"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.","section":"Remark 3.7 and Section 5.3"},{"comment":"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.","section":"Section 5.3 and Section 1"},{"comment":"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.","section":"Section 4, Step 3, and Section 6"},{"comment":"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.","section":"Section 5.4"}],"minor_comments":[{"comment":"There is a typo in the abstract: “C. elegansthat” should be “C. elegans that.”","section":"Abstract"},{"comment":"The ellipse validation is only qualitative; please report quantitative errors or correlations with the arc-length parametrization, as is done for the circle example.","section":"Section 5.1"},{"comment":"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.","section":"Section 5.2.1"},{"comment":"Reference [38] contains a typo: “Desnity” should be “Density.”","section":"References"},{"comment":"The word “proccesses” in the final paragraph should be “processes.”","section":"Section 6"}],"recommendation":"major_revision","confidential_remarks":"The core idea is promising and the theoretical parts are plausible, but the empirical validation needs substantial strengthening before publication. The most pressing issue is the potential circularity between Remark 3.7's suggestion to tune ε by the mutual-information metric and the use of that same metric as the sole quantitative real-data evaluation. I would also strongly encourage the authors to include at least one comparison with existing density-robust methods, since the current baseline does not support the comparative wording in the abstract. The runtime and optimization-convergence claims also need more evidence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the rejection-sampling + Procrustes-averaging pipeline for circular coordinates is a real contribution, and the paper is honestly written. The main problem is that the quantitative evidence for 'more robust' is thinner than the abstract suggests, and the MI-based evaluation has a circular-tuning pitfall that needs to be closed.\n\nWhat's new: subsample the data to equalize density, compute standard circular coordinates per subsample, then align and average using O(2) Procrustes with hill climbing. This is a clean extension of their Euclidean resampling work, and it does address a known weakness of de Silva-Morozov-Vejdemo-Johansson coordinates. Prop 3.3 (density estimator consistency) and Prop 4.4 (O(2) approximation to circle Procrustes) are modest but correct and useful. The synthetic examples show real improvement over the uncorrected coordinate, and the C. elegans results are visually convincing and a nice demonstration.\n\nSoft spots, in order of importance. First, the abstract claims a 'more robust coordinate than other approaches,' but no experiment compares against the existing density-robust methods: Rybakken et al.'s weighted harmonic representatives, Perea's sparse coordinates, or Paik-Park's circular coordinates. The only baseline is the uncorrected coordinate. Second, Remark 3.7 explicitly says epsilon can be tuned by maximizing the same KSG mutual information used as the evaluation metric in Section 5.3. The paper never reports the epsilon used on the real data, nor rules out MI-based selection. That makes 'higher MI in every case' vulnerable to tuning bias. This is not fatal, but the headline empirical claim is unfalsifiable as written. Third, the runtime comparison reports the minimum over 20 repeats, which flatters the method; median would be more standard. Fourth, no code or data are provided, so the worm numbers can't be checked. Finally, the hill-climbing convergence is admittedly not understood; that is fine for a heuristic, but Prop 4.4 only bounds the seed when the optimal loss is small, and no such smallness is demonstrated for real data.\n\nBottom line: the methodological core is sound and likely useful. The evaluation needs transparency (disclose epsilon, use median runtime, add at least one existing method as baseline). I'd send it to peer review with major-revision expectations.","headline":"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.","tokens_in":18276,"tokens_out":3188,"would_cite":true,"duration_ms":28046,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N31","62R40"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["circular coordinates","persistent cohomology","rejection sampling","Procrustes alignment","density estimation","recurrent time series","topological data analysis","C. elegans neuronal recordings"],"falsifier":"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.","tokens_in":17247,"feed_emoji":"🧠","tokens_out":11068,"duration_ms":91983,"temperature":0.7,"pith_summary":"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.","feed_headline":"Corrected circular coordinates beat standard pipeline on every test","feed_subtitle":"On synthetic loops and worm brain recordings, the resampled coordinate is more informative and up to 23x faster to compute.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the persistent-cohomology method for turning a 1-dimensional cohomology class into a circular coordinate, which the paper's pipeline uses on each subsample.","marker":"[42]"},{"why":"Identified the uneven-sampling distortion in circular coordinates and proposed a renormalization fix; it is the problem the paper's rejection-sampling step is designed to solve.","marker":"[39]"},{"why":"The authors' prior work on aligning and averaging coordinates in manifold learning; the circular Procrustes alignment and averaging step adapts this framework.","marker":"[7]"},{"why":"Provides the orthogonal Procrustes algorithm used to solve the O(2) alignment problem in R^2, producing the seed for the hill-climbing refinement.","marker":"[4]"},{"why":"Supplies the nearest-neighbor mutual-information estimator used to compare corrected and uncorrected coordinates without ground truth.","marker":"[23]"},{"why":"Provides the C. elegans whole-brain calcium-imaging recordings and discrete behavioral-state labels used for the first real-world validation.","marker":"[22]"},{"why":"Provides the second, larger set of C. elegans whole-brain recordings used to test the method across multiple animals.","marker":"[46]"}],"fun_headline_variants":["Resampling fixes circular coordinates, beats old method on all tests","Density-corrected circular coordinates: more robust and up to 23x faster","New algorithm makes circular coordinates robust to uneven sampling","Worm brain loops mapped with faster, more robust circular coordinates","Align-and-average trick yields better circular coordinates for recurrent data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Resampling fixes circular coordinates, beats old method on all tests","Density-corrected circular coordinates: more robust and up to 23x faster","New algorithm makes circular coordinates robust to uneven sampling","Worm brain loops mapped with faster, more robust circular coordinates","Align-and-average trick yields better circular coordinates for recurrent data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000248,"raw_usage":{"total_tokens":1551,"prompt_tokens":955,"completion_tokens":596,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":571,"completion_tokens_details":{"reasoning_tokens":509}},"tokens_in":571,"tokens_out":596,"duration_ms":5141,"temperature":1.0,"reasoning_tokens":509,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T04:42:36.357433+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Resampling and averaging coordinates on data","cited_arxiv_id":"2408.01379","evidence_quote":"The authors' prior work on aligning and averaging coordinates in manifold learning; the circular Procrustes alignment and averaging step adapts this framework."},{"cited_title":"Orthogonal Procrustes rotation for two or more matri- ces","cited_arxiv_id":null,"evidence_quote":"Provides the orthogonal Procrustes algorithm used to solve the O(2) alignment problem in R^2, producing the seed for the hill-climbing refinement."},{"cited_title":"Estimating mutual information","cited_arxiv_id":null,"evidence_quote":"Supplies the nearest-neighbor mutual-information estimator used to compare corrected and uncorrected coordinates without ground truth."},{"cited_title":"Global Brain Dynamics Embed the Motor Command Sequence of Caenorhabditis elegans","cited_arxiv_id":null,"evidence_quote":"Provides the C. elegans whole-brain calcium-imaging recordings and discrete behavioral-state labels used for the first real-world validation."},{"cited_title":"NeuroPAL: A Multicolor Atlas for Whole-Brain Neuronal Identification in C. elegans","cited_arxiv_id":null,"evidence_quote":"Provides the second, larger set of C. elegans whole-brain recordings used to test the method across multiple animals."}],"review_version":1}