{"id":"b39a5c99-035b-4218-b60a-c21cb4293e90","arxiv_id":"2412.05900","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A gradient-descent method selects small sets of intervals that approximate full generalized persistence diagram domains, reducing computation time severalfold with comparable classification accuracy.","lead":"This paper sparsifies the domain of generalized persistence diagrams in 2-parameter persistent homology by solving a gradient descent problem over interval sets, and shows the resulting sparse diagrams are much faster to compute while keeping classification accuracy on time-series benchmarks. It is useful for anyone who wants multiparameter persistence descriptors to scale to larger datasets.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The optimized sparse domain J* is independent of the data because Eq. (2) collapses to a geometric distance between interval sets; the choice of the full grid I is unvalidated and controls the method's accuracy.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing concern: the loss depends only on I and J, not on the persistence modules, and the choice of I is arbitrary and untested. My reading confirms this is the point where the central empirical claim is least secure. The theoretical results (sparse erosion distance, closed-form formula, Lipschitz stability, a.e. differentiability) appear internally coherent, and the reported computational speedups are plausible given the complexity reduction. However, the method's empirical value hinges on whether the fixed grid I contains the discriminative intervals for the datasets; if not, neither the full nor the sparse GPDs can be informative, and the optimized J* cannot repair this because it only approximates I. The paper itself notes in the conclusion that dataset-dependent terms could improve performance, which underscores that the current loss is geometric rather than data-adaptive. The concrete test proposed above would directly assess whether this concern affects the reported accuracies. Since the reader's verdict is already CONDITIONAL and this concern does not invalidate the mathematical core, the appropriate verdict remains unchanged.","tokens_in":17829,"tokens_out":9670,"duration_ms":104307,"concrete_test":"Fix one UCR dataset and m=400. Compute J* for three choices of I: (i) the original 1,600-interval grid; (ii) a different grid with the same cardinality, e.g., shifted or randomly sampled within the same filtration ranges; (iii) the original grid augmented with the top-variance intervals over the training modules. Compare downstream classification accuracies. If accuracies vary substantially across choices of I, the method is controlled by the arbitrary choice of I; if the data-dependent augmentation (iii) outperforms, a data-dependent loss would be preferable. Also verify whether the optimized J* is identical across all datasets in Table 1, as the theory implies; if so, the sparsification is dataset-agnostic and the 'automatic' claim should be qualified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central simplification in Proposition 6(iii) makes the loss in Eq. (2) equal to t * \\hat d(I,J), with no dependence on the persistence modules M_i. Consequently, for a fixed I and m, the gradient descent output J* is the same for every dataset. The empirical claim of 'maintaining classification accuracies comparable to full GPDs' therefore rests on two unvalidated assumptions: (a) the hand-chosen full domain I in Section 5.0.0.2, namely 1,600 (2,1)-intervals obtained by evenly sampling 10 values for x,y and 2 values for a,b,c,d, is a sufficiently rich domain for the datasets considered; and (b) Hausdorff approximation of I by m intervals preserves enough discriminative information for the downstream vectorized classifiers. If a dataset's discriminative topology lives on intervals outside I, neither the full GPDs nor the optimized sparse GPDs can detect it, and J* cannot recover it because the optimization only seeks proximity to I, not to any data-dependent signal. This is not an internal inconsistency, but it is the least secure load-bearing step for the paper's empirical headline: the proposed loss is not actually data-driven, and no sensitivity analysis over I or m is provided.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a method for sparsifying the domain of generalized persistence diagrams (GPDs) by solving an optimization problem with gradient descent. The authors define a sparse erosion distance between GPDs relative to (possibly different) sampled interval sets, and prove that for a fixed persistence module this distance equals a purely geometric distance between the interval sets (Proposition 6(iii)). They derive a closed-form formula for this distance when intervals are (1,1)- or (2,1)-intervals (Theorem 9), establish convexity of the search space (Proposition 13), Lipschitz stability (Theorem 14), and almost-everywhere differentiability (Proposition 16) of the resulting loss. They then use the loss to select a sparse set of intervals, and report experiments on UCR time-series classification showing reduced GPD computation times with accuracy comparable to full GPDs. The theoretical results appear coherent and are proved in appendices; the main weakness is that the loss function is independent of the data and labels, so the optimization is a geometric approximation of a fixed full domain, and the experimental evaluation is a single-run comparison without error bars or sensitivity analysis.","tokens_in":18100,"tokens_out":3421,"duration_ms":34413,"significance":"The mathematical contribution is valuable: a metric between GPDs over different interval domains, with a closed form for practically relevant interval shapes, is a useful tool for the multi-parameter persistence community. The proofs are self-contained and the paper explicitly identifies the dependence of the loss on the interval sets only, which is honest about the method's data-adaptive limits. If the empirical claims were supported by sensitivity analyses and repeated trials, the work would constitute a practical step toward scaling GPD-based machine learning. The inclusion of a closed-form formula and the theoretical stability results are strengths; however, the current experimental evidence does not substantiate the headline that the sparsification 'maintains classification accuracies comparable to full GPDs' because the choices of the full domain and the sparsification level are unvalidated and the reported accuracies are from single runs.","major_comments":[{"comment":"The loss function L_{d_E,m}(v_J) = t * d_hat(I,J) depends only on the interval sets I and J, not on the persistence modules M_i or their labels, as shown by Proposition 6(iii). Consequently, for fixed I and m, gradient descent returns the same optimized sparse domain J* for every dataset considered in Section 5. The abstract and Section 1 phrase the method as 'optimizing the selection of intervals, balancing computational efficiency and discriminative accuracy,' which overstates data adaptivity. In reality, the optimization is a geometric approximation of the hand-chosen grid I, and any discriminative information not represented in I is lost regardless of the optimization. The paper should either revise these claims to describe the method as a data-independent domain compression, or add experiments that explicitly demonstrate how the optimized domain depends on the full domain choice and how the accuracy of the downstream classifier varies with that choice.","section":"§5.0.0.2, Eq. (2)"},{"comment":"The fixed full domain I is constructed by evenly sampling 10 values for x and y and 2 values for a, b, c, d within their filtration ranges, yielding 1,600 (2,1)-intervals; the sparsification level is set to m = 400. No criterion is given for these choices, and no sensitivity analysis is reported. Since d_hat(I,J) is a Hausdorff-like distance, the optimized J* can only approximate I; if I omits intervals that are discriminative for a given dataset, neither the full nor the sparse GPDs can detect those features. The empirical headline therefore rests on the unvalidated assumption that this particular grid is a faithful proxy for the informative interval space. Please report accuracy and running time for several grid densities and values of m (e.g., 5, 10, 20 samples for x,y; m = 100, 200, 400) and, ideally, for alternative schemes for constructing I, to demonstrate that the reported performance is not an artifact of this specific choice.","section":"§5.0.0.2"},{"comment":"The accuracy scores in Table 1 are reported as single numbers with no error bars, repeated trials, or cross-validation variance. Random forest classifiers are randomized, and several differences between the optimized and full GPDs are small (e.g., DPC: 0.721 vs 0.743, GP: 0.959 vs 0.946, PC: 0.956 vs 0.990, SC: 0.487 vs 0.503). Without an estimate of variability, the claim that optimized GPDs 'maintain classification accuracies comparable to those achieved using full GPDs' is not statistically substantiated. Please report means and standard deviations over at least 10 independent training runs or over cross-validation folds, and state the number of repeats used to obtain the reported values.","section":"Table 1"}],"minor_comments":[{"comment":"In the discussion of running times, 'strinkingly fast' should read 'strikingly fast'.","section":"Section 5"},{"comment":"The phrase 'Our code is fully available at sparse GPDs' is not a working link; the manuscript should include a URL or a note that code will be made public upon publication.","section":"Section 5.0.0.2 and code availability"},{"comment":"The time-delay embedding is described as transforming each time series into a point cloud in R3 via triples (f(t_i), f(t_{i+1}), f(t_{i+2})); please state the delay and embedding dimension explicitly and justify the choice, as these parameters affect the resulting persistence modules.","section":"Section 5.0.0.1"},{"comment":"The notation d_{W,p} is used without defining the Wasserstein distance between diagrams; please add a definition or a reference.","section":"Appendix A, Remark 17"},{"comment":"The entry for the UCR archive is formatted as 'In CoRR. arXiv:1810.07758'; please provide the journal or conference name and year consistently with the other references.","section":"Reference [14]"}],"recommendation":"major_revision","confidential_remarks":"The theoretical core of the paper is solid and the closed-form formula is a genuine contribution. The main gap is that the empirical evaluation, which is the paper's headline, does not yet support the claims of data-adaptive sparsification and comparable accuracy: the loss is data-independent, the full domain is unvalidated, and the results have no error bars. These issues are fixable within the scope of the manuscript by adding sensitivity analyses and repeated-trial experiments, and by rephrasing the claims to match what the method actually does. I would be comfortable with acceptance after such a revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe thing to know: the paper's optimization is data-independent. The loss reduces to \\hat d(I,J), a purely geometric distance between the full interval set I and the sparse set J. Proposition 6(iii) is explicit about this. It makes the gradient descent tractable, but it also means J* is the same for every dataset. The empirical claim of 'comparable classification accuracy' therefore rests on the choice of I: a hand-chosen grid of 1,600 (2,1)-intervals. If that grid misses discriminative intervals, neither the full nor the sparse GPDs can detect them. The stress-test note is right on this point.\n\nThe theoretical contribution is genuinely new: a sparse erosion distance between GPDs over possibly different interval sets, a closed-form formula for \\hat d(I,J) for (1,1)- and (2,1)-intervals, Lipschitz stability, and a.e. differentiability. The proofs in the appendices look coherent, and the complexity reduction from O(n^3) to O(m^3) is real. The experiments show 5–15x speedups with comparable accuracies over a spread of UCR datasets, which is useful evidence of practical value.\n\nSoft spots, in proportion. First, the lack of sensitivity analysis over I and m. The grid size, the sampling density, and the sparsification size m=400 are chosen without justification, and there is no study of how results degrade as m shrinks or as the grid thins. Second, the accuracy numbers are single-run, with no error bars; for a claim of 'maintaining comparable accuracy', that's a real gap. Third, the code link is a placeholder in the text ('sparse GPDs'), which should be fixed before publication. None of these are fatal to the theory, but they temper the empirical conclusion.\n\nThis paper is for the TDA/ML community, specifically anyone computing GPDs for classification and looking for a principled way to shrink the domain. It deserves a serious referee: the math is original and appears correct, and the method is likely to be used. My recommendation is to send it to review, and to ask for error bars, a sensitivity analysis over I and m, and a working code repository before acceptance.","headline":"The loss is a purely geometric distance between interval sets, making the sparse domain independent of the data, so the empirical claim rests on the unvalidated choice of the full grid; the theory itself is novel and sound.","tokens_in":18620,"tokens_out":3896,"would_cite":true,"duration_ms":38372,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N31"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper introduces a gradient-descent method that sparsifies the domain of generalized persistence diagrams, cutting computation time 5-15x while keeping classification accuracy.","keywords":["multi-parameter persistent homology","generalized persistence diagram","generalized rank invariant","sparse erosion distance","gradient descent","domain sparsification","time series classification"],"falsifier":"Take a classification dataset, optimize the sparse domain from the paper's fixed grid $I$, then repeat the optimization from a denser or shifted grid $I'$ and compare the classification accuracy of the resulting sparse GPDs; a material accuracy change would show that the preservation-of-information claim depends on the arbitrary choice of $I$ rather than on the optimization. A sharper test is to add a hand-built interval known to separate two classes to $I$, re-optimize, and check whether the optimized sparse domain actually exploits it.","tokens_in":17617,"feed_emoji":"⚡","tokens_out":7945,"duration_ms":72274,"temperature":0.7,"pith_summary":"Generalized persistence diagrams (GPDs) extend persistence barcodes to multi-parameter filtrations by recording multiplicities over intervals, but the full diagram is computationally prohibitive because the interval set is enormous. The paper's goal is to show that a small, optimized subset of intervals can replace the full domain without much loss of discriminative power. To do this, it introduces a sparse erosion distance between GPDs defined on different interval sets and proves a key simplification: for one persistence module the distance equals a purely geometric distance between the interval domains. The resulting closed-form loss can be minimized by gradient descent, and experiments on time-series classification show the sparse GPDs compute 5 to 15 times faster than full GPDs while keeping classification accuracy comparable or better.","feed_headline":"Learned sparse domains compute persistence diagrams 5-15x faster","feed_subtitle":"A gradient descent method picks which intervals to keep, preserving accuracy while cutting GPD computation time.","key_machinery":"The central object is the sparse erosion distance $\\hat d_E$, an adaptation of the erosion distance that compares GPDs defined on possibly different interval sets via $\\epsilon$-correspondences between intervals. Its key simplification is Proposition 6(iii): for the same persistence module the distance collapses to $\\hat d(I,J)$, a pure domain distance. The paper then embeds $(1,1)$- and $(2,1)$-intervals of $\\mathbb{R}^2$ into $\\mathbb{R}^6$ as vectors $(x,y,a,b,c,d)$, proves that $\\hat d(I,J)$ equals the maximum over rows and columns of their minima of a matrix of containment costs, and defines the loss $L(v_J)=t\\,\\hat d(I,J)$. This combination makes the loss independent of the underlying persistence modules, convex in the domain variables, Lipschitz stable, and almost everywhere differentiable, which is what lets gradient descent run on a precomputed quantity.","core_discovery":"The central claim is that the domain of a generalized persistence diagram can be optimized rather than hand-picked. For a fixed finitely presentable persistence module $M$ and interval sets $I,J$, the sparse erosion distance satisfies $\\hat d_E((\\mathrm{dgm}^I_M),(\\mathrm{dgm}^J_M)) = \\hat d(I,J)$, where the right-hand side is a Hausdorff-type distance between the interval sets alone, independent of the module (Proposition 6(iii)). For intervals with one or two minimal points and one maximal point, $\\hat d(I,J)$ has a closed-form formula built from pairwise interval-containment costs $\\epsilon_{rs}$ (Theorem 9). The resulting loss $v_J \\mapsto t\\,\\hat d(I,J)$ is convex in the Euclidean embedding of $J$, Lipschitz stable, and differentiable almost everywhere, so standard gradient descent finds a locally optimal sparse domain $J^*$. Experiments on UCR time-series datasets report 5 to 15 times faster GPD computation with accuracy comparable to or better than the full GPD baseline.","pith_inferences":["The domain-only loss means the optimizer cannot discover discriminative intervals that are absent from the initial full grid $I$, so the choice of $I$ is a hidden design parameter.","The same closed-form distance framework could be extended to interval families with more minimal or maximal points, at higher computational cost and with harder embedding constraints.","One could test whether optimized domains transfer across classifiers or datasets, since the loss is computed entirely from interval geometry and not from labels or module data.","The sparse erosion distance might also serve as a training objective beyond classification, for instance in clustering or generative settings where persistence modules must be compared efficiently."],"forward_implications":["Computing GPDs over the optimized sparse domain $J^*$ is reported 5 to 15 times faster than over the full domain $I$ on the tested UCR datasets.","The optimized sparse GPDs maintain classification accuracies comparable to or better than the full GPDs, and usually outperform an initial random sparse domain.","Because the loss depends only on the interval domains, the optimized domain can be reused across persistence modules, sidestepping recomputation of GPDs during optimization.","The Lipschitz stability and almost-everywhere differentiability of the loss guarantee well-behaved gradient-descent iterations, reflected in the observed smooth loss curves.","The sparsification procedure is complementary to other interval-based vectorizations such as signed barcodes and GRIL, which can consume the optimized domains."],"supporting_citations":[{"why":"Defines the GPD via Möbius inversion of the generalized rank invariant and establishes the erosion distance and stability results that the sparse erosion distance adapts.","marker":"[12]"},{"why":"Introduces the generalized rank invariant and GPD over posets, supplying the rank function used throughout the paper.","marker":"[19]"},{"why":"Introduces generalized persistence diagrams and the erosion distance, the conceptual basis for comparing GPDs.","marker":"[30]"},{"why":"Describes zigzag-persistence computation of the GRI and GPD, which sets the computational complexity that sparsification targets.","marker":"[15]"},{"why":"Gives the zigzag persistence algorithm in matrix-multiplication time used in the complexity estimate for GPD computation.","marker":"[28]"},{"why":"Provides signed barcodes as measures and vectorization techniques used as a comparison and complementary pipeline.","marker":"[27]"},{"why":"Introduces GRIL, a worm-based GRI vectorization that the sparsification can be applied to before vectorization.","marker":"[34]"},{"why":"Supplies the UCR time-series datasets used in the classification experiments.","marker":"[14]"}],"fun_headline_variants":["Gradient descent sparsifies GPD domains for 5-15x speedup","Optimized interval subsets speed up GPD computation 5-15x","Learned sparse domains cut persistence time by 5-15x","Pick GPD intervals with gradient descent for 5-15x faster compute","Sparse GPD via gradient descent: accurate, 5-15x faster"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the full domain $I$, a hand-chosen grid of 1,600 $(2,1)$-intervals, already contains the intervals needed to discriminate the classes, because the optimized sparse domain can only approximate the information in $I$ and cannot recover intervals missing from it.","fun_headline_variants_meta":{"raw":{"variants":["Gradient descent sparsifies GPD domains for 5-15x speedup","Optimized interval subsets speed up GPD computation 5-15x","Learned sparse domains cut persistence time by 5-15x","Pick GPD intervals with gradient descent for 5-15x faster compute","Sparse GPD via gradient descent: accurate, 5-15x faster"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000868,"raw_usage":{"total_tokens":3799,"prompt_tokens":1023,"completion_tokens":2776,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":639,"completion_tokens_details":{"reasoning_tokens":2675}},"tokens_in":639,"tokens_out":2776,"duration_ms":18306,"temperature":1.0,"reasoning_tokens":2675,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T20:13:25.345130+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a classification dataset, optimize the sparse domain from the paper's fixed grid $I$, then repeat the optimization from a denser or shifted grid $I'$ and compare the classification accuracy of the resulting sparse GPDs; a material accuracy change would show that the preservation-of-information claim depends on the arbitrary choice of $I$ rather than on the optimization. A sharper test is to add a hand-built interval known to separate two classes to $I$, re-optimize, and check whether the optimized sparse domain actually exploits it.","supporting_citations":[{"cited_title":"Generalized persistence diagrams","cited_arxiv_id":null,"evidence_quote":"Introduces generalized persistence diagrams and the erosion distance, the conceptual basis for comparing GPDs."},{"cited_title":"Computing generalized rank invariant for 2-parameter persistence modules via zigzag persistence and its applications","cited_arxiv_id":null,"evidence_quote":"Describes zigzag-persistence computation of the GRI and GPD, which sets the computational complexity that sparsification targets."},{"cited_title":"Zigzag persistent homology in matrix multiplication time","cited_arxiv_id":null,"evidence_quote":"Gives the zigzag persistence algorithm in matrix-multiplication time used in the complexity estimate for GPD computation."},{"cited_title":"Stable vectorization of multiparameter persistent homology using signed barcodes as measures","cited_arxiv_id":null,"evidence_quote":"Provides signed barcodes as measures and vectorization techniques used as a comparison and complementary pipeline."},{"cited_title":"GRIL: A 2-parameter persistence based vectorization for machine learning","cited_arxiv_id":null,"evidence_quote":"Introduces GRIL, a worm-based GRI vectorization that the sparsification can be applied to before vectorization."}],"review_version":1}