{"id":"d7983901-0dce-497e-994a-6393d2c58179","arxiv_id":"2505.10004","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"The paper proposes three persistent-homology methods that estimate recurrence times in multi-variate time series, with stability proofs and a new industrial benchmark dataset.","lead":"This paper introduces a persistent homology framework that estimates when a multi-variate time series repeats itself, handling strict periodic, repetitive, and recurring behaviors with three methods. It also contributes a new benchmark dataset from a partially simulated injection molding machine and reports cycle-time estimation errors.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'provably stable' claim overreaches: Appendix A bounds only persistence diagrams, while the thresholding step that maps to recurrence times has no stability guarantee.","rationale":"I read Appendix A in good faith: the inequalities (2), (3), and (4) are correct as statements about persistence diagrams of the constructed surrogate functions. The load-bearing weakness is the discontinuity of the thresholding and time-mapping step, exactly the gap the reader identified as the weakest assumption. The paper's own text says that selecting points with birth < epsilon and persistence > delta is the mechanism by which recurrence times are obtained, and Appendix A does not analyze the stability of that selection. The evaluation's exclusion of failed cycle counts in Section 4.3 makes the practical validation non-conservative, but the formal gap alone is sufficient to block the unqualified 'provably stable' claim. My conclusion therefore matches the reader's CONDITIONAL verdict: the framework is promising and the Appendix proofs are valid for what they prove, but the stability claim needs to be either weakened or supplemented with a margin condition and a stability proof for the thresholded output, and the evaluation should report failure rates rather than omitting them.","tokens_in":11369,"tokens_out":5080,"duration_ms":56008,"concrete_test":"Construct a synthetic 2D time series family x_alpha(t) with one true recurrence at time T and an additional candidate local minimum of v_x whose persistence point lies exactly on the boundary of the selection region at alpha=0, e.g., birth = epsilon - eta/2 and persistence = delta + eta/2, and moves outside the region at alpha=eta, e.g., birth = epsilon + eta/2 and persistence = delta - eta/2. For a fine sweep of alpha, run the full Method 1 pipeline: compute v_x, build the sublevel-set persistence diagram, apply the epsilon/delta threshold, and map selected points back to times. Check whether the number of estimated cycles jumps at alpha=0 while ||x_alpha - x_0||_infinity < eta and the bottleneck distance between D(v_x_alpha) and D(v_x_0) is below eta.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, repeated in the abstract and in Section 5, is that the three methods are provably stable. Appendix A proves correct Lipschitz bounds for the maps x -> D(v_x) in bottleneck distance, namely inequalities (2), (3), and (4), but the methods do not output a persistence diagram. They apply fixed thresholds, selecting points with birth < epsilon and persistence > delta in Section 3.1, and then map the selected persistence points back to time-domain recurrence times. This map from diagrams to selected recurrence times is not continuous in bottleneck distance. A point with birth just below epsilon and persistence just above delta can cross either threshold under an arbitrarily small L-infinity perturbation of the input series, so the selected set, its cardinality, and the resulting cycle boundaries can change discontinuously even while the full persistence diagram moves by less than eta. The proofs in Appendix A therefore bound only the input to the thresholding step, not the output of the algorithm. No quantitative separation or margin assumption is stated or proven that would make the selection stable. The evaluation in Section 4.3 compounds the issue: sections where a method identified the wrong number of cycles are omitted from Table 2 and the boxplots in Figure 3, so the reported MAE/MARE values are conditional on successful counting and do not reflect the method's behavior on failures. Consequently, the 'provably stable' and 'stability guarantees of all three methods' statements are not established by the provided arguments.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a persistent homology framework for estimating recurrence times in multi-variate time series, covering three increasingly constrained notions of cyclic behavior: recurring, repetitive, and periodic. Three methods are presented: Method 1 applies a sublevel-set filtration to the distance from the starting point; Method 2 first forms a delay embedding and then applies the same distance-based filtration; Method 3 filters the diagonal average of pairwise distances. The authors claim all three methods are provably stable under perturbations and validate them on a new injection molding benchmark dataset with annotated cycles, reporting MAE and MARE per dataset section. Appendix A derives Lipschitz bounds on the bottleneck distance between the persistence diagrams of the surrogate functions for the three methods.","tokens_in":11706,"tokens_out":4107,"duration_ms":44190,"significance":"If the claims were fully supported, the framework would be a useful contribution: it addresses a real gap in period/repetition estimation for unevenly sampled and multi-variate data, and the public benchmark dataset is a valuable asset for future comparison. The three-level taxonomy (periodic/repetitive/recurring) is a sensible structuring device, and the computational complexity (O(n log n)) is attractive. The stability proofs in Appendix A are correct as far as they go, and the experiments cover a realistic industrial scenario. However, the central 'provably stable' claim overreaches: the proofs bound only the persistence diagrams, not the thresholded selection procedure that actually produces the estimated recurrence times. In addition, the empirical evaluation omits exactly the cases where a method failed to count cycles correctly, so the reported accuracies are conditional on success. The underlying ideas remain defensible, and both issues are fixable in revision, but the current manuscript does not substantiate its headline claims.","major_comments":[{"comment":"The paper's central 'provably stable' claim (abstract and Section 5) is not supported by Appendix A. The proofs in Eqs. (2)-(4) bound only the bottleneck distance between the full persistence diagrams, but the methods do not output a persistence diagram: they select points with birth < epsilon and persistence > delta (Section 3.1) and map those selected points back to time-domain recurrence times. This selection map is not continuous in the bottleneck distance; a point with birth just below epsilon or persistence just above delta can cross either threshold under an arbitrarily small L_infinity perturbation of the time series, so the selected set, its cardinality, and the resulting cycle boundaries can change discontinuously. The manuscript provides no margin or separation assumption that would make the thresholded output stable. Please either add a quantitative separation condition and prove stability of the thresholded selection, or revise the abstract and Section 5 to state that only the persistence diagrams, not the estimated recurrence times, are stable.","section":"Section 3.1 and Appendix A, Eq. (2)"},{"comment":"The evaluation masks failures. The text states, 'For cases where a method failed to identify the correct number of cycles, results are omitted (marked with dashes in the table and excluded from the boxplots).' This makes every reported MAE and MARE conditional on the method having counted the cycles correctly. A method that frequently returns the wrong number of cycles can therefore receive an arbitrarily good conditional error. The conclusion that 'the experimental results validate their practical applicability' is not supported without reporting, at minimum, the number and rate of failures per section, and ideally an unconditional error measure that also penalizes mis-counted cycles.","section":"Section 4.3, Table 2, and Figure 3"},{"comment":"The proof of Eq. (4) as written is invalid. The equality |v_x(Delta) - v_x'(Delta)| = | avg_t avg_t' (w_x(t,t+Delta) - w_x'(t',t'+Delta)) | does not hold; introducing a second independent averaging variable t' changes the quantity. The intended bound can be obtained directly by writing v_x(Delta) - v_x'(Delta) = avg_t (w_x(t,t+Delta) - w_x'(t,t+Delta)) and then applying the previously established sup-norm bound on w_x - w_x'. This is a fixable error, but as it stands the Appendix does not give a correct derivation of Method 3's stability.","section":"Appendix A.3, Eq. (4)"},{"comment":"The threshold parameters (epsilon, delta, d, tau) are tuned on the same benchmark sections for which errors are reported; Table 2 lists one set of selected parameters per method, and the evaluation is in-sample. No train/test split, cross-validation, or parameter-sensitivity analysis is provided. The reported accuracies are therefore estimates of in-sample fit rather than evidence of generalization to new series or to other parameter settings. Please report a parameter sensitivity analysis or split the data into tuning and evaluation portions.","section":"Section 4.2 and Table 2"}],"minor_comments":[{"comment":"The runtime complexity statement reads 'O(nα(n)' with a missing closing parenthesis; it should be O(n α(n)), and the reference to the inverse Ackermann function would benefit from a citation to a precise theorem.","section":"Section 2.1"},{"comment":"The sentence 'The strength of this approach lies in it's real-world applicability' contains a typo: 'it's' should be 'its'.","section":"Section 3, introductory paragraph"},{"comment":"The caption begins 'Distribtion of the absolute error'; this should be 'Distribution'.","section":"Figure 3 caption"},{"comment":"The delay parameter is denoted tau in Section 3.2 but tau is also used for cycle length in Section 1.2; Table 2 lists 'tau = 500' for Method 2. Please use a distinct symbol, e.g., Delta, to avoid ambiguity.","section":"Section 3.2 and Table 2"},{"comment":"The statement 'if x is epsilon-delta-approximately recurring then there is a recurring x-hat with ||x - x-hat||_inf < epsilon' is asserted without proof. A short construction or a reference would make the hierarchy of definitions clearer.","section":"Section 1.2, Definition 5"},{"comment":"No quantitative comparison against classical baselines (e.g., FFT or autocorrelation on the evenly sampled sections) is provided. A small baseline comparison would help calibrate the practical gain of the topological methods.","section":"Section 4.2"},{"comment":"Definition 1 defines periodicity by x(t+tau)=x(t) for all t in [0,T-tau], but the paper also allows time series with finitely many discontinuities. These two notions are not compatible in general; please clarify how the periodic definition is interpreted for discontinuous series.","section":"Section 1.2, Definition 1"}],"recommendation":"major_revision","confidential_remarks":"The paper has genuine assets: a concrete industrial benchmark with annotated cycles, a public code repository, and a clean taxonomy of cyclic behaviors. The main risk is overclaiming stability: the proofs in Appendix A concern only the persistence diagram, while the actual output of the algorithms is a thresholded selection that is not shown to be stable. The empirical section compounds this by censoring failure cases. Both problems are addressable in revision, and the manuscript does not contain any indication of an integrity problem. If the authors add margin assumptions or weaken the stability claim, and report failure rates, the paper could become a solid contribution suitable for a computational geometry / applied topology venue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: the paper is worth reading and worth sending out for review, but the abstract overstates what is proven. The contribution is real: they define three levels of cyclic behavior (periodic, repetitive, recurring), build three surrogate functions, use sublevel persistence, and produce a new annotated industrial dataset. The individual stability bounds in Appendix A are correct as far as they go—they show Lipschitz maps from series to persistence diagrams. The benchmark is a genuine asset, with code and data released.\n\nThe soft spot is exactly where the stress-test note points. The algorithms do not output a persistence diagram. They threshold it: select points with birth < eps and persistence > delta, then map back to time-domain recurrence times. That selection map is not continuous in bottleneck distance under the stated assumptions. A point near the threshold can pop in or out under an arbitrarily small L-infinity perturbation. So the 'provably stable' claim in the abstract and the 'stability guarantees of all three methods' in Section 5 are not established. The reader's worry is on target. I checked whether Appendix A state any margin assumption or separation condition; it doesn't. So the gap is real.\n\nThe evaluation has a related, smaller issue: Table 2 and Figure 3 omit sections where the method found the wrong number of cycles. They are transparent about that, but it means the reported MAE/MARE are conditional on a successful count. That's a reasonable way to report localization error, but it shouldn't be sold as overall accuracy.\n\nNothing here is fatal. The framework and experiments are a solid start, and the problem is genuinely under-served. To make the claims match the proofs, they need to either weaken the stability language to 'the persistence diagram is stable' or prove stability for the thresholded selection under a margin assumption (e.g., a gap between signal and noise persistence). The benchmark alone is enough to justify a referee.\n\nMy recommendation: send it out. It's a legitimate contribution with a clear, fixable overclaim. A good referee will ask for a revised claims section and maybe a more careful evaluation protocol.","headline":"A useful framework and benchmark for recovering recurrence times, but the 'provably stable' claim outruns the proofs: only persistence diagrams are shown stable, not the thresholded selection.","tokens_in":12217,"tokens_out":1464,"would_cite":true,"duration_ms":14384,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N31","62M10","68U05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Persistent homology can recover recurrence times in noisy multi-variate time series, with stability guarantees for periodic, repetitive, and recurring signals.","keywords":["persistent homology","time series analysis","recurrence time estimation","cycle detection","periodicity","delay embedding","industrial automation","injection molding benchmark"],"falsifier":"Run Method 1 on a synthetic recurring series with known cycle boundaries, adding bounded noise whose amplitude is swept from zero up to the assumed $\\varepsilon$, and record whether the number of detected recurrence times changes before the perturbation reaches the bottleneck bound.","tokens_in":11190,"feed_emoji":"🔁","tokens_out":7693,"duration_ms":71056,"temperature":0.7,"pith_summary":"The paper sets out to recover the time points at which a multi-variate time series returns to a previous state, for three increasingly strict senses of repetition: recurring, repetitive, and periodic. It proposes a persistent-homology framework that turns the series into scalar-valued surrogate functions and reads recurrence times from significant local minima of their sublevel-set filtrations. The paper claims all three methods are stable under bounded perturbations of the input and computationally efficient, with $O(n \\log n)$ runtime, and it validates them on real industrial data from an injection molding machine. If the framework holds, cycle boundaries can be recovered from unevenly sampled, noisy sensor data where classical Fourier or autocorrelation methods fail.","feed_headline":"Persistence topology finds cycle boundaries in noisy time series","feed_subtitle":"Recovers recurring, repetitive, and periodic patterns in multi-variate sensor data, even with uneven sampling.","key_machinery":"The mechanism that carries the argument is the sublevel-set persistence diagram of a surrogate scalar function. For Method 1, $v_x(t)=\\|x(t)-x(0)\\|_2$ captures how far the trajectory has strayed from its starting point; for Method 2, a delay embedding $U_x(t)=(x(t),x(t+\\Delta),\\ldots,x(t+(d-1)\\Delta))$ with $v_x(t)=\\|U_x(t)-U_x(0)\\|_p$ separates self-intersecting repetitions; for Method 3, $w_x(t_1,t_2)=\\|x(t_1)-x(t_2)\\|_2$ averaged over diagonals, $v_x(\\Delta)=\\operatorname{avg}_t w_x(t,t+\\Delta)$, isolates constant period. In all three cases, zero-dimensional persistent homology tracks births and deaths of connected components in the sublevel filtration, and significant local minima\\u2014points with birth $<\\varepsilon$ and persistence $>\\delta$\\u2014are mapped back to time to yield recurrence times. The load-bearing identity is the bottleneck stability bound $d_B(D(v_x),D(v_{x'}))\\le c\\|x-x'\\|_\\infty$, which transfers input perturbations to diagram perturbations and underpins the 'provably stable' claim.","core_discovery":"On the paper's own terms, the central discovery is that the problem of estimating recurrence times reduces to a stable question about zero-dimensional persistent homology. Given a time series $x$, each method constructs a nonnegative scalar function $v$ on the time interval\\u2014distance to the starting point (Method 1), distance of a delay embedding to the embedded starting point (Method 2), or diagonal averages of the pairwise distance function (Method 3)\\u2014and then computes the sublevel-set persistence diagram. The recurrence times $T_i$ are recovered by selecting diagram points with birth below $\\varepsilon$ and persistence above $\\delta$ and mapping the corresponding local minima back to the time domain. The stability argument shows that the composed map from input series to persistence diagram is Lipschitz, with bottleneck distance bounded by $2\\|x-x'\\|_\\infty$ for Methods 1 and 3 and by $2\\sqrt{d}\\|x-x'\\|_\\infty$ for the delay-embedding method. The authors conclude that the stability guarantees plus experiments on their benchmark establish practical tools for periodic, repetitive, and recurring behavior.","pith_inferences":["A natural extension is to make the thresholds $\\varepsilon$ and $\\delta$ adaptive rather than fixed, since the paper's stability proofs concern the full persistence diagram and do not directly guarantee that the selected point set\\u2014and hence the recovered times\\u2014is stable.","The evaluation reports errors only for sections where the correct number of cycles was found, so the published MAE/MARE numbers likely understate failures; a metric that counts missed or extra cycles would better expose where each method breaks.","The distance-to-start surrogate of Method 1 could be extended to quasi-periodic or chaotic series by tracking higher-dimensional persistence or by replacing the fixed start point with a sliding reference point.","A direct stress test would be to run Method 1 on synthetic recurring series with known cycle boundaries while sweeping the noise amplitude from zero to $\\varepsilon$ and recording when the detected cycle count changes."],"forward_implications":["Recurrence boundaries can be estimated for unevenly sampled multi-variate series, a regime where FFT-based and autocorrelation-based period detection is undefined.","A single framework covers three levels of cyclic behavior, so practitioners can choose Method 1 for recurring signals, Method 2 for self-intersecting repetitive signals, and Method 3 for periodic signals.","The $O(n \\log n)$ runtime makes the approach feasible for online monitoring and control tasks in industrial settings.","The stability bounds imply that small changes in the input series produce small changes in the persistence diagram, giving a theoretical foundation for reliability under measurement noise.","The published benchmark with annotated cycle boundaries provides a reusable evaluation protocol for future recurrence-estimation methods."],"supporting_citations":[{"why":"Documents the lack of a robust, general method for repetitive behavior in industrial time series, setting up the problem.","marker":"[1]"},{"why":"Establishes that classical frequency-domain, autocorrelation, and wavelet methods require evenly spaced samples, defining the gap the framework fills.","marker":"[2]"},{"why":"Prior approach for estimating period length in video data that this work extends but does not handle repetitive or recurring behavior.","marker":"[7]"},{"why":"Topological method for reparameterized periodic functions that counts periods but does not extract their lengths and is limited to scalar series.","marker":"[9]"},{"why":"Prior cycling-signature method using derivatives that motivates the delay-embedding alternative and provides a noise-sensitivity comparison.","marker":"[10]"},{"why":"Supplies the bottleneck stability theorem for persistence diagrams that all three stability proofs invoke.","marker":"[13]"},{"why":"Provides the algorithmic background for sublevel-set filtrations and the $O(n \\log n)$ runtime claim.","marker":"[14]"}],"fun_headline_variants":["Topology-driven method finds repetition times in series","Persistent homology uncovers cycles in time series","Stable persistence framework spots recurrence","Recurrence detection via sublevel-set topology","Multivariate series: topology identifies repetition"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that fixed thresholds $\\varepsilon$ and $\\delta$ can separate true recurrence points from noise for every series in the perturbation class; the stability proofs cover the whole persistence diagram, not the thresholded selection that actually produces the times.","fun_headline_variants_meta":{"raw":{"variants":["Topology-driven method finds repetition times in series","Persistent homology uncovers cycles in time series","Stable persistence framework spots recurrence","Recurrence detection via sublevel-set topology","Multivariate series: topology identifies repetition"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000538,"raw_usage":{"total_tokens":2549,"prompt_tokens":878,"completion_tokens":1671,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":494,"completion_tokens_details":{"reasoning_tokens":1605}},"tokens_in":494,"tokens_out":1671,"duration_ms":13845,"temperature":1.0,"reasoning_tokens":1605,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:19:00.829422+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Method 1 on a synthetic recurring series with known cycle boundaries, adding bounded noise whose amplitude is swept from zero up to the assumed $\\varepsilon$, and record whether the number of detected recurrence times changes before the perturbation reaches the bottleneck bound.","supporting_citations":[{"cited_title":"Open Challenges in Time Series Anomaly Detection: An Industry Perspective","cited_arxiv_id":"2502.05392","evidence_quote":"Documents the lack of a robust, general method for repetitive behavior in industrial time series, setting up the problem."},{"cited_title":"Transforms, Wavelets","cited_arxiv_id":null,"evidence_quote":"Establishes that classical frequency-domain, autocorrelation, and wavelet methods require evenly spaced samples, defining the gap the framework fills."},{"cited_title":"(Quasi) periodicity quantification in video data, using topology","cited_arxiv_id":null,"evidence_quote":"Prior approach for estimating period length in video data that this work extends but does not handle repetitive or recurring behavior."},{"cited_title":"Topological phase estimation method for reparameterized periodic functions","cited_arxiv_id":null,"evidence_quote":"Topological method for reparameterized periodic functions that counts periods but does not extract their lengths and is limited to scalar series."},{"cited_title":"Stability of persistence diagrams","cited_arxiv_id":null,"evidence_quote":"Supplies the bottleneck stability theorem for persistence diagrams that all three stability proofs invoke."},{"cited_title":"Computational topology: an introduction","cited_arxiv_id":null,"evidence_quote":"Provides the algorithmic background for sublevel-set filtrations and the $O(n \\log n)$ runtime claim."}],"review_version":1}