{"id":"0dbd24b4-4c42-4a4b-994f-c26a407b69f4","arxiv_id":"2507.07156","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Skipping boundary matrix reduction and using unreduced persistence diagrams as ML features gives on-par or better task performance with large memory savings.","lead":"This paper asks whether machine learning on topological features can skip the expensive step of fully computing persistence diagrams. It shows that cheap 'unreduced' diagrams built from un-reduced boundary matrices match or beat fully reduced diagrams on three tasks, using far less memory.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Brain-artery regression compares PD types with hyperparameters selected on the same folds used for final evaluation, so Table 4 cannot support the substitution claim until nested CV confirms the result.","rationale":"I read the theory and experiments in good faith. The stability results are internally consistent: Proposition 3.1 gives global stability for L1; Example 3.2 and Proposition 3.3 correctly delimit local stability for AP/qAP; the generic local-Lipschitz claim is caveated. The computational benchmark is honestly scoped: it covers qAP-up for VR complexes, shows strong memory savings on non-thresholded examples, and the paper concedes L1 vectorization cost in Section 6. The ML experiments are the load-bearing part of the substitution claim. Across shape and f-MNIST, holdout-based tuning makes the comparisons credible, and the f-MNIST L1 gain is large. The brain artery regression is the only regression task, and it has selection-on-evaluation-folds (Section 5.1.4). The favorable L1/PI margin is tiny relative to fold variance, and the conclusion's blanket L1 statement is contradicted by the AC row of Table 4. This does not invalidate the paper, because the shape and f-MNIST results still show that unreduced diagrams can match or beat FR on some tasks, but it does mean the broader 'effective substitutes' conclusion needs the nested-CV check before it is adopted. Hence the reader's CONDITIONAL verdict is appropriate and I do not move it.","tokens_in":20507,"tokens_out":11861,"duration_ms":136177,"concrete_test":"Re-run the brain artery regression with nested CV: for each of the five diagram types and both vectorizers, use an outer 5-fold loop for final R^2 and an inner loop (e.g., 4-fold or a single holdout fold) for hyperopt vectorization-hyperparameter selection. Report the corrected mean and SD R^2 for every cell of Table 4. Then check two things: (1) does L1 with persistence images still beat FR in corrected R^2; (2) does at least one unreduced type still meet or exceed FR in either vectorizer. If the corrected L1 advantage disappears and no unreduced type remains on par, the brain regression no longer supports the substitution claim, leaving only the two classification tasks.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing empirical premise is that at least one unreduced PD performs on par or better than FR across several data and task types. The only regression task that supports this is the brain artery experiment, and its protocol (Section 5.1.4) selects vectorization hyperparameters by 5-fold CV on the same folds later used for final evaluation: 'A hyperparameter tuning holdout set was used for the shape and Fashion-MNIST tasks, but this was not done for the brain artery regression.' This selection-on-test-folds biases the Table 4 R^2 values upward and can re-rank diagram types. The headline favorable cell (L1/PI 0.3541 vs FR/PI 0.3445) is smaller than the reported fold standard deviations (0.19 vs 0.12), so the comparison is statistically fragile even before the protocol issue. Moreover, the conclusion that 'L1 diagrams performed as well or better than FR diagrams in all cases' is contradicted by the same table's Adcock-Carlsson row (L1=0.1844 vs FR=0.3042). Because the regression task is the only evidence that the substitution claim extends beyond classification, the central claim is not yet supported without a corrected evaluation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces several \"unreduced\" persistence diagrams—AP, qAP down-arrow, qAP up-arrow, and L1—defined directly from unreduced boundary matrices, and studies their stability, computational cost, and usefulness in supervised ML pipelines. The main theoretical results are Proposition 3.1 (global bottleneck stability of L1), Example 3.2 (instability of AP/qAP constructions), and Proposition 3.3 (local stability of AP/qAP under small filtration perturbations). The computational section benchmarks a modified Ripser implementation for qAP up-arrow diagrams of Vietoris-Rips filtrations, reporting large wall-time and memory savings in several examples. The ML experiments compare FR diagrams against unreduced diagrams on synthetic shape classification, Fashion-MNIST image classification, and brain artery tree regression, with the conclusion that unreduced diagrams can serve as effective substitutes for fully reduced diagrams in PH-ML pipelines.","tokens_in":20682,"tokens_out":4957,"duration_ms":56832,"significance":"If the central claim is established, the paper would make a practically useful contribution: it would show that expensive persistence diagram reduction can be avoided in at least some PH-ML pipelines, with an accompanying open-source implementation and reproducible experiments. The theoretical distinction between global stability of L1 and only local/generic stability of AP/qAP is also a useful clarification, and the explicit instability example is a valuable counterpoint to the usual global stability narrative. The paper ships code, describes its computational experiments in detail, and is transparent about several limitations, which strengthens the presentation. However, the empirical substitution claim is currently supported by only one regression task, and that task has a protocol flaw that can re-rank the reported results; the stated all-cases conclusion is also internally contradicted by one of the paper's own tables. These issues are load-bearing because the ML evidence is what converts the paper from a theoretical/computational exploration into a claimed practical substitute for reduced diagrams.","major_comments":[{"comment":"The brain-artery regression is the only regression task and is therefore the main evidence that the substitution claim extends beyond classification, but its evaluation protocol is not sound for that purpose. Section 5.1.4 states that hyperparameter tuning for this task was done with 5-fold cross-validation on the same folds later used for final performance assessment, because there were too few data entries for a holdout set. This selection-on-evaluation-folds protocol biases the Table 4 R^2 values upward and can re-rank diagram types. Moreover, the headline favorable comparison, L1/PI 0.3541 versus FR/PI 0.3445, is smaller than the reported fold standard deviations (0.19 and 0.12, respectively). Nested cross-validation, a fixed hyperparameter choice, or a separate tuning holdout is required before Table 4 can support the on-par/outperform claim for regression.","section":"Section 5.1.4 and Table 4"},{"comment":"The concluding sentence states that \"L1 diagrams performed as well or better than FR diagrams in all cases.\" This is contradicted by Table 4: in the Adcock-Carlsson row, L1 achieves R^2 = 0.1844 while FR achieves R^2 = 0.3042. The valid weaker statement in Section 5.2.1 is that in every experiment at least one unreduced diagram type performed as well or better than FR. The all-cases L1 claim should be withdrawn or explicitly restricted to the persistence-image vectorization within the brain-artery experiment, where the difference is not statistically stable anyway.","section":"Section 6 and Table 4"},{"comment":"The computational half of the central claim is not yet established at the full-pipeline level. Table 2 reports memory ratios that vary widely with thresholding: the thresholded examples have ratios of roughly 1.28 to 2.63, while the non-thresholded examples average about 13. Section 6 concedes that L1 diagrams take \"significantly longer times to vectorize\" because they contain many more persistence pairs. Since L1 is the unreduced type that performs best in the ML experiments, the advertised savings must be measured end-to-end, including vectorization time and memory, rather than only the qAP up-arrow diagram computation for VR complexes.","section":"Sections 4 and 6"},{"comment":"The qAP up-arrow object used in the Rips shape-classification experiments is not identical to qAP up-arrow(M) in Definition 2.1: the implementation skips columns identified as zero-AP columns of a second type, which the paper says loses some information, and Table 3 labels this a \"slightly modified qAP up-arrow construction.\" This is disclosed, but it means the ML results for qAP up-arrow should be described as applying to a variant rather than to the defined unreduced persistence diagram, and the theoretical containment/stability statements in Section 2 do not directly transfer to that variant.","section":"Section 4, second zero-AP paragraph; Table 3"}],"minor_comments":[{"comment":"In the final row, \"cylco\" is a typo for \"cyclo-;\" also, the table would be easier to read if the average quantities were unambiguously identified as averages over the 63 thread-count trials.","section":"Table 2"},{"comment":"The text refers to \"Theorem 3.2\" when discussing the instability construction, but the construction is Example 3.2, not a theorem.","section":"Section 3.2"},{"comment":"The stability experiment reports only H0 for the AP, qAP down-arrow, and L1 diagrams; since Section 2 notes that other unreduced diagrams for VR complexes have only ephemeral pairs in higher homology degrees, the practical relevance of the empirical stability results for higher-dimensional homology is not demonstrated.","section":"Figure 3"},{"comment":"There is a duplicated word in \"useful useful for classification;\" this should be corrected.","section":"Section 5.2.2"}],"recommendation":"major_revision","confidential_remarks":"The theoretical core is sound and the paper is within scope for stat.ML, so the defects are not fatal. My recommendation is driven by two fixable but load-bearing issues: the brain-artery regression protocol invalidates the only regression-based evidence for the substitution claim, and the conclusion overstates what the results show. A revision that adds a proper nested or holdout evaluation for the regression, corrects the internal contradiction in Section 6, and qualifies the computational claims to the actual measured pipeline would make the paper suitable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe genuinely new thing here is the idea of using the byproducts of an unreduced boundary matrix — apparent pairs, quasi-apparent pairs, and low-ones — as terminal summaries for ML, rather than as acceleration tricks inside a reduction algorithm. The theory is the strongest part: Proposition 3.1 (global stability of L1) is correct and clean, Example 3.2 shows exactly why the quasi-apparent constructions cannot be globally stable, and Proposition 3.3 gives a plausible local stability statement. Those results are original as far as I can tell, and the paper states them with appropriate precision. The computational half is also worth credit: the parallel qAP-up implementation is a real engineering contribution, with memory savings that are genuinely large in the non-thresholded cases (about 1/13th of Ripser's footprint on average), and the code is public.\n\nThe soft spots are empirical, and they are not fatal but they are real. First, the conclusion's claim that “L1 diagrams performed as well or better than FR diagrams in all cases” is contradicted by the paper's own Table 4: with Adcock-Carlsson vectorization on the brain data, L1 gets R^2 = 0.1844 versus FR's 0.3042. The abstract and Section 5.2.1 are more careful (“at least one unreduced PD”, “on some tasks”), so the concluding sentence overreaches. Second, the regression experiment that carries much of the “outperform” claim has a protocol problem: Section 5.1.4 discloses that vectorization hyperparameters for the brain task were tuned on the same five folds used for final evaluation. That can inflate R^2 and re-rank diagram types. The authors are honest about the disclosure, but it means Table 4 should not be used as strong evidence until nested CV confirms it. Third, the only favorable regression cell (L1/PI 0.3541 vs FR 0.3445) is much smaller than the reported fold standard deviations (0.19 vs 0.12), so it is not a robust win. And the computational comparison stops at diagram computation: Section 6 concedes that L1 diagrams take longer to vectorize, so the full-pipeline cost claim remains open.\n\nNone of this sinks the paper. The theoretical and computational contributions stand on their own, and the ML experiments are honestly reported in the per-type tables. But the general substitution claim needs either a corrected regression evaluation or a more circumscribed conclusion. This paper deserves a serious referee — the theory section especially is worth publishing — and the authors are clearly thinking carefully. I would send it to review, asking that the conclusion be reconciled with Table 4 and that the brain-regression protocol be addressed.","headline":"Useful new summaries and a clean stability theorem, but the empirical case for substituting unreduced for fully reduced diagrams is overstated and needs a corrected regression evaluation.","tokens_in":21357,"tokens_out":4943,"would_cite":true,"duration_ms":51018,"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":"Unreduced persistence diagrams can replace fully reduced ones in machine learning pipelines without hurting task performance.","keywords":["persistent homology","unreduced persistence diagrams","topological machine learning","Vietoris-Rips filtration","apparent pairs","quasi-apparent pairs","low-ones diagram"],"falsifier":"An end-to-end benchmark on the same three tasks that times diagram computation plus vectorization, and that tunes vectorization hyperparameters on a true holdout set for the brain-artery regression, would settle the claim: if fully reduced diagrams match or beat low-ones diagrams on the Fashion-MNIST and brain-artery tasks once vectorization time is included, the paper's central claim fails.","tokens_in":20172,"feed_emoji":"📐","tokens_out":7845,"duration_ms":82948,"temperature":0.7,"pith_summary":"This paper argues that supervised machine learning pipelines fed with topological features do not need the fully reduced persistence diagram that standard persistent homology computes. Instead, simple summaries read directly from the unreduced boundary matrix—the low-ones, apparent-pair, and quasi-apparent-pair diagrams—can carry enough signal for classification and regression. Across shape classification, Fashion-MNIST image classification, and brain-artery age regression, at least one unreduced diagram type matched or beat the fully reduced diagram, and the low-ones diagram matched or beat it in every experiment. Computing one such summary, the upper quasi-apparent diagram for Vietoris–Rips filtrations, parallelizes well and used on average an order of magnitude less memory than full reduction in the paper's benchmarks. If the claim holds, topology-based ML pipelines can replace the most expensive step of the standard pipeline with a much cheaper pass over unreduced boundary matrices.","feed_headline":"Unreduced diagrams match full persistence diagrams in ML pipelines","feed_subtitle":"Four cheap diagram constructions match or beat fully reduced ones on shape, image, and brain-age tasks, with far less memory.","key_machinery":"The load-bearing object is the unreduced boundary matrix $M$ of a filtered complex, along with two quantities read from its columns without any column reduction: $\\operatorname{low}(M_j)$, the largest row index containing a 1 in column $j$, and $\\beta(M_j)$, the largest row index $z$ such that the entry $M_{z,j}=1$ and every earlier column has a 0 in row $z$. A column whose $\\operatorname{low}$ equals its $\\beta$ is an apparent pair and is guaranteed to survive reduction; the more permissive quasi-apparent constructions use $\\beta(M_j)$ or $\\operatorname{low}(M_j)$ whenever $\\beta(M_j)\\neq -1$, while the low-ones diagram simply records every column's $\\operatorname{low}$. These quantities let persistence pairs be harvested in one pass, and for upper quasi-apparent diagrams that pass becomes a parallelizable search over the columns of the filtration coboundary matrix.","core_discovery":"The central claim is that for supervised learning purposes, an unreduced persistence diagram can substitute for a fully reduced one. The paper defines four unreduced constructions—the apparent-pair diagram (AP), the lower and upper quasi-apparent diagrams (qAP↓ and qAP↑), and the low-ones diagram (L1)—and reports that in all three experimental settings at least one of them performed as well as or better than the fully reduced diagram on the chosen metric. In the L1 construction, the pair associated to a column is just its lowest nonzero row and the column itself; because every nonzero column contributes a point, L1 diagrams are larger but appear to carry information that full reduction throws away. The paper also reports that an implementation of the qAP↑ diagram for Vietoris–Rips filtrations is parallelizable and required about one thirteenth of the memory of full reduction on the data sets tested, at roughly a twentieth of the wall time when run with 63 threads.","pith_inferences":["We infer that low-ones diagrams may perform well precisely because they do not discard low-persistence pairs: on Fashion-MNIST the most useful L1 principal component concentrated on H0 features, suggesting the extra points act as a denser, more informative persistence image rather than noise.","We infer that the memory savings can be made to survive the full pipeline only if vectorization is streamed straight from the unreduced diagram construction, aggregating each pair's contribution to the vector and discarding the pair—the paper identifies this as a future direction.","We infer that the theoretical instability of AP and qAP diagrams may be an acceptable risk in practice, since instability events in the paper's perturbation experiments were rare; a cheap 'fragility check' that detects whether nearby perturbations change the number of diagram points could screen data sets before deployment.","We infer that if unreduced and fully reduced diagrams encode different signal, an ensemble trained on the concatenation of both may outperform either alone; the paper explicitly leaves this question open."],"forward_implications":["Topology-featured ML pipelines can be run on cheap unreduced summaries: on every task tested, at least one unreduced diagram type matched or beat the fully reduced diagram, so full reduction appears unnecessary for task performance in these settings.","The low-ones diagram is the safest substitute: it matched or beat fully reduced diagrams in every experiment, including a more than 10-percentage-point gain over FR on Fashion-MNIST with both tested vectorizations.","Computing upper quasi-apparent diagrams for Vietoris–Rips filtrations is parallelizable and memory-light: with 63 threads it used on average about 1/20 the wall time and 1/13 the memory of full reduction on non-thresholded examples, and it completed one data set that the reference reduced implementation could not finish.","Stability is theory-limited but not practice-limited: only L1 diagrams are globally stable under the bottleneck distance, yet in perturbation experiments the unstable AP and qAP diagrams produced very large bottleneck/Hausdorff ratios only rarely.","The computational win is not yet end-to-end: low-ones diagrams vectorize more slowly than smaller diagrams, so careful implementation is needed before the savings are realized in a full pipeline."],"supporting_citations":[{"why":"Supplies the Vietoris-Rips persistence algorithm and the notion of apparent pairs that the qAP↑ implementation modifies and benchmarks against.","marker":"[8]"},{"why":"Defines the beta index and quasi-apparent pair construction that the unreduced diagrams are built from.","marker":"[34]"},{"why":"Provides the brain-artery regression data set and the persistence-vectorization baseline the paper reanalyzes.","marker":"[12]"},{"why":"Provides the Fashion-MNIST data set used for the image classification experiment.","marker":"[46]"},{"why":"Establishes the global stability of fully reduced persistence diagrams, the theoretical baseline the unreduced constructions are compared to.","marker":"[22]"},{"why":"Shows how apparent-pair searches can be parallelized, the approach the paper adapts for parallel qAP↑ computation.","marker":"[48]"}],"fun_headline_variants":["Unreduced diagrams rival full persistence for ML","Skip full reduction: unreduced PDs match ML performance","Cheap unreduced diagrams beat full ones in ML tasks","Parallelizable unreduced diagrams cut memory 13x for ML","Unreduced persistence diagrams: same ML power, 1/13 memory"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central claim rests on the assumptions that the three benchmark tasks are fair, representative comparisons and that diagram computation dominates the pipeline: the brain-artery regression tuned vectorization hyperparameters on the same folds used for evaluation, and the reported memory and time savings cover diagram computation only, not the longer vectorization of the much larger low-ones diagrams.","fun_headline_variants_meta":{"raw":{"variants":["Unreduced diagrams rival full persistence for ML","Skip full reduction: unreduced PDs match ML performance","Cheap unreduced diagrams beat full ones in ML tasks","Parallelizable unreduced diagrams cut memory 13x for ML","Unreduced persistence diagrams: same ML power, 1/13 memory"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00015,"raw_usage":{"total_tokens":1188,"prompt_tokens":925,"completion_tokens":263,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":541,"completion_tokens_details":{"reasoning_tokens":179}},"tokens_in":541,"tokens_out":263,"duration_ms":3215,"temperature":1.0,"reasoning_tokens":179,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:50:36.900723+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"An end-to-end benchmark on the same three tasks that times diagram computation plus vectorization, and that tunes vectorization hyperparameters on a true holdout set for the brain-artery regression, would settle the claim: if fully reduced diagrams match or beat low-ones diagrams on the Fashion-MNIST and brain-artery tasks once vectorization time is included, the paper's central claim fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Vietoris-Rips persistence algorithm and the notion of apparent pairs that the qAP↑ implementation modifies and benchmarks against."},{"cited_title":"Mendoza-Smith.Numerical algorithms for the mathematics of information","cited_arxiv_id":null,"evidence_quote":"Defines the beta index and quasi-apparent pair construction that the unreduced diagrams are built from."},{"cited_title":"Bendich, J","cited_arxiv_id":null,"evidence_quote":"Provides the brain-artery regression data set and the persistence-vectorization baseline the paper reanalyzes."},{"cited_title":"Cohen-Steiner, H","cited_arxiv_id":null,"evidence_quote":"Establishes the global stability of fully reduced persistence diagrams, the theoretical baseline the unreduced constructions are compared to."},{"cited_title":"Zhang, M","cited_arxiv_id":null,"evidence_quote":"Shows how apparent-pair searches can be parallelized, the approach the paper adapts for parallel qAP↑ computation."}],"review_version":1}