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REVIEW 4 major objections 4 minor 48 references

Unreduced Persistence Diagrams for Topological Machine Learning

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Unreduced persistence diagrams can replace fully reduced ones in machine learning pipelines without hurting task performance.

desk verdict 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. read the letter →

arxiv 2507.07156 v2 pith:PP4LTTZ7 submitted 2025-07-09 stat.ML cs.CGcs.LGmath.AT

Nicole Abreu , Parker B. Edwards , Francis Motta This is my paper · ORCID
classification stat.MLcs.CGcs.LGmath.AT MSC 55N31
keywords persistenthomologyunreducedpersistencediagramstopologicalmachinelearningVietoris-Ripsfiltrationapparentpairsquasi-apparentlow-onesdiagram
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

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.

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 (4)
  1. [Section 5.1.4 and Table 4] 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.
  2. [Section 6 and Table 4] 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.
  3. [Sections 4 and 6] 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.
  4. [Section 4, second zero-AP paragraph; Table 3] 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.
minor comments (4)
  1. [Table 2] 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.
  2. [Section 3.2] The text refers to "Theorem 3.2" when discussing the instability construction, but the construction is Example 3.2, not a theorem.
  3. [Figure 3] 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.
  4. [Section 5.2.2] There is a duplicated word in "useful useful for classification;" this should be corrected.

Circularity Check

1 steps flagged · score 3.0 of 10

One evaluation step is partly circular: brain-artery regression hyperparameters are selected on the same folds later reported as test performance, so Table 4 can re-rank PD types by construction; the theoretical derivations and the classification experiments remain independent.

  1. fitted input called prediction [Section 5.1.4 (Hyperparameter tuning), Table 4, Section 5.2.1, Section 6 (Conclusion)]
    "Hyperparameter tuning was conducted using 5-fold cross validation with hyperopt. Only vectorization hyperparameters were tuned while random forest hyperparameters were not. Final performance assessment also used 5-fold cross validation with the best identified hyperparameters for every experiment. A hyperparameter tuning holdout set was used for the shape and Fashion-MNIST tasks, but this was not done for the brain artery regression as there were too few data entries available."

    For the brain-artery regression, the vectorization hyperparameters were selected by optimizing 5-fold cross-validation performance on the same folds that are subsequently labeled 'testing performance' in Table 4. The reported R^2 values are therefore selection-optimized in-sample scores rather than independent out-of-sample predictions, and differences between PD types in that table can be created or re-ranked by the tuning procedure itself. This matters because Section 5.2.1 and the Conclusion use this regression task as part of the load-bearing evidence that unreduced PDs 'can serve as effective substitutes' for FR diagrams; the abstract's headline claim that unreduced-diagram models 'can perform on par and even outperform' fully reduced ones partly rests on this contaminated comparison.

full rationale

The theoretical portions of the paper are self-contained and are not circular: Definition 2.1 defines FR, AP, qAP-down, qAP-up, and L1 diagrams directly from the boundary matrix and the beta function; Proposition 2.2, Proposition 3.1, Example 3.2, Proposition 3.3, and Proposition 4.1 are proven from these definitions with stated assumptions that do not include the ML-performance conclusions. The computational benchmarks use external point-cloud datasets and compare against the existing Ripser implementation, with code released publicly, and the memory and wall-time claims are stated as benchmark results rather than derived from the conclusions. The only self-citations, [27] and [37], appear in the introductory list of TDA application areas and are not load-bearing. The one genuinely circular element is the brain-artery regression evaluation: Section 5.1.4 states that no hyperparameter-tuning holdout was used for that task, so the same 5-fold cross-validation folds were used both to choose vectorization hyperparameters and to produce the final R^2 values in Table 4; this biases and can re-rank the reported comparisons. That limitation is explicitly acknowledged by the authors, but it still means the regression-based substitution evidence is partly forced by the evaluation protocol rather than by an independent prediction. Section 6 also concedes that L1 diagrams take 'significantly longer times to vectorize,' so the end-to-end computational advantage is not yet established; this is a scope caveat, not a circularity. Finally, the Conclusion's statement that 'L1 diagrams performed as well or better than FR diagrams in all cases' is contradicted by Table 4's Adcock-Carlsson row, where L1 (0.1844) is below FR (0.3042); this is an overstatement or internal inconsistency, not a circular step. Overall, the central theoretical derivation and the shape and Fashion-MNIST comparisons, which did use holdout tuning, are independent, so the paper is not substantially circular; the regression-based comparison should be treated as partly circular until corrected with nested cross-validation.

Assumptions & free parameters 4 free parameters · 5 assumptions · 4 invented entities

All four constructions are mathematical definitions, so they need no external evidence to exist; the independent_evidence flag here records that each was put to falsifiable tests (stability experiments and held-out ML benchmarks). The paper adds no unmeasured physical entities. The free parameters are pipeline hyperparameters and experimental design choices, not constants in a derivation, and the axioms are standard algebraic facts plus domain assumptions about representativeness of the benchmarks.

free parameters (4)
  • Persistence image hyperparameters (resolution, bandwidth, weight exponent) = varied per task, selected by hyperopt 5-fold CV
    Each PD's vectorization hyperparameters were tuned independently per homology degree (Section 5.1.6); for the brain regression task no separate holdout existed during tuning (Section 5.1.4), so these choices affect the reported R2 comparisons.
  • Adcock-Carlsson coordinate powers = selected by hyperopt
    Raising birth, death, or persistence coordinates to tuned positive powers is part of the AC vectorization (Section 5.1.6); these exponents are fitted values.
  • Number of principal components (13) = 13
    The feature-importance analysis projects 800-dimensional vectorizations onto the first 13 PCs, chosen as the minimum number explaining over 90% of FR training variance (Section 5.1.5); this is a hand-chosen number for the explanatory analysis, not for the headline parity claim.
  • Stability experiment noise levels = 1e-6, 1e-5, 1e-4, 1e-3, 1e-2, 0.1, plus torus geometry
    Selected by hand for the experimental stability study (Section 3.2); they govern which instability events are observed but do not feed the main ML claim.
assumptions (5)
  • standard math Standard boundary matrix reduction facts: low(M_j) only decreases under column additions; beta(M_j)=z implies the reduced column R_j is negative; low(R_j) lies in [beta(M_j), low(M_j)] (Mendoza-Smith [34, Thm. 5]).
    Invoked in Section 2 to justify the qAP and AP definitions and the containment relations of Proposition 2.2.
  • standard math Stability of fully reduced PDs under the bottleneck distance with Lipschitz constant 1, and d_GH <= d_H for geometric complexes.
    Used in Section 3 as the benchmark against which unreduced stability is judged, and to interpret the bottleneck/Hausdorff ratio experiment.
  • domain assumption The three ML tasks (shape classification, Fashion-MNIST, brain artery regression) are representative of PH-ML pipelines, and random forests on vectorized diagrams are a representative downstream model.
    Section 5 extrapolates from these tasks to a general claim about unreduced PDs; the authors themselves flag this as a limitation in Section 6.
  • domain assumption For VR complexes, f(sigma) = max filtration value of the 1-dimensional faces, so non-H0 unreduced diagrams contain only ephemeral pairs.
    Section 2 uses this to justify focusing the computational implementation on qAP-up arrow diagrams for VR complexes, and it shapes which diagrams enter the ML comparisons.
  • domain assumption The modified qAP-up arrow construction that skips certain zero-AP columns still yields a useful diagram.
    Section 4 states the implementation loses some information by skipping columns that Ripser also skips; the ML results for qAP-up arrow with VR complexes use this lossy variant.
invented entities (4)
  • Apparent pair PD (AP) independent evidence
    purpose: Multiset of pairs (low(M_j), j) where beta(M_j) = low(M_j), capturing columns that are already fully reduced without additional work.
    Defined in Definition 2.1; it is cheap to compute but empty in most homology degrees for flag-type filtrations (Section 2), which limits its usefulness for VR complexes.
  • Lower quasi-apparent PD (qAP down arrow) independent evidence
    purpose: Pairs (low(M_j), j) when beta(M_j) is not -1, corresponding to the lowest possible death row index.
    Defined in Definition 2.1; showed stability spikes in Figure 3 and was used in the ML experiments, so it has a falsifiable track record.
  • Upper quasi-apparent PD (qAP up arrow) independent evidence
    purpose: Pairs (beta(M_j), j) when beta(M_j) is not -1, corresponding to the highest possible death row index; the basis of the parallel algorithm.
    Defined in Definition 2.1 and computed by Algorithm 1; this is the construction behind the benchmarked modified Ripser implementation and the memory/time claims.
  • Low-ones PD (L1) independent evidence
    purpose: All pairs (low(M_j), j) with low(M_j) not -1, the naive and largest diagram.
    Defined in Definition 2.1; globally stable by Proposition 3.1 and the best-performing or tied-best type across the paper's ML experiments, with the caveat that vectorization is slower due to the larger number of points (Section 6).

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Cite this review

Pith. "Pith review of Unreduced Persistence Diagrams for Topological Machine Learning." pith.science (2026). https://pith.science/paper/PP4LTTZ7

@misc{pith2026250707156,
  author       = {Pith},
  title        = {Pith review of: Unreduced Persistence Diagrams for Topological Machine Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PP4LTTZ7}},
  note         = {Machine review of arXiv:2507.07156}
}
read the original abstract

Supervised machine learning pipelines trained on features derived from persistent homology have been experimentally observed to ignore much of the information contained in a persistence diagram. Computing persistence diagrams is often the most computationally demanding step in such a pipeline, however. To explore this dynamic, we introduce several methods to generate topological feature vectors from unreduced boundary matrices and investigate their theoretical and computational properties. We compared the performance of pipelines trained on vectorizations of unreduced PDs to vectorizations of fully-reduced PDs across several data and task types. Our results indicate that models trained on PDs built from unreduced diagrams can perform on par and even outperform those trained on fully-reduced diagrams on some tasks. We also benchmarked the computational performance of an algorithm for computing unreduced diagrams, which was implemented as a heavily modified version of Ripser. These computations are parallelizable and required an order of magnitude less memory on average compared to computing full persistence diagrams. Our results suggest that machine learning pipelines which incorporate topology-based features may benefit in terms of computational cost and performance by utilizing information contained in unreduced boundary matrices.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.