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REVIEW 3 major objections 5 minor 34 references

Sparsification of the Generalized Persistence Diagrams for Scalability through Gradient Descent

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper introduces a gradient-descent method that sparsifies the domain of generalized persistence diagrams, cutting computation time 5-15x while keeping classification accuracy.

desk verdict 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. read the letter →

arxiv 2412.05900 v2 pith:Z64NWPW5 submitted 2024-12-08 math.AT cs.CG

Mathieu Carrière , Seunghyun Kim , Woojin Kim This is my paper · ORCID
classification math.ATcs.CG MSC 55N31
keywords multi-parameterpersistenthomologygeneralizedpersistencediagramrankinvariantsparseerosiondistancegradientdescentdomainsparsificationtimeseriesclassification
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

3 major / 5 minor

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.

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 (3)
  1. [§5.0.0.2, Eq. (2)] 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.
  2. [§5.0.0.2] 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.
  3. [Table 1] 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.
minor comments (5)
  1. [Section 5] In the discussion of running times, 'strinkingly fast' should read 'strikingly fast'.
  2. [Section 5.0.0.2 and code availability] 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.
  3. [Section 5.0.0.1] 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.
  4. [Appendix A, Remark 17] The notation d_{W,p} is used without defining the Wasserstein distance between diagrams; please add a definition or a reference.
  5. [Reference [14]] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reduction of the loss to \hat d(I,J) is proved from definitions, and the sparse-domain optimization is not a fitted parameter relabeled as a prediction.

full rationale

The paper's central mechanism is the equality \hat d_E((dgmM,I),(dgmM,J)) = \hat d(I,J) (Proposition 6(iii)), which reduces the loss function in Eq. (2) to t·\hat d(I,J). This is not a self-definitional collapse: \hat d_E is defined in Definition 4 as an infimum over correspondences and rank inequalities, and Proposition 6(iii) is proved in Appendix A.2 from the monotonicity of the generalized rank invariant rather than assumed as the loss definition. Theorem 9 then supplies a closed form for \hat d(I,J), and Theorems 14 and Proposition 16 are proved from Lemma 12 and the closed form, not from the paper's empirical conclusions. The fact that J* is independent of the persistence modules is a consequence of the theorem, not a relabeling of inputs as outputs; the paper never claims the loss uses label information, and the final accuracy comparisons are made by an independent classification pipeline (fixed 6-dimensional histogram vectorization, cross-validated hyperparameters, random forests on training labels). The hand-chosen full domain I and sparsification size m are experimental design choices whose sensitivity is not analyzed; that is a robustness limitation, not circular reasoning. Self-citations (e.g., [12], [19]) supply background definitions and stability theorems, but the load-bearing lemmas and propositions of this paper are proved in-appendix from the cited definitions, so the citation chain does not force the main result.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

No new physical or mathematical entities are postulated. The central assumptions are the finite presentability of modules, the faithfulness of the hand-chosen full domain I, and the correctness of the cited erosion stability result.

free parameters (5)
  • n (full domain size) = 1600
    Number of intervals in the full domain I, chosen by hand; directly controls the cost of full GPD computation and the quality of the approximation target.
  • m (sparsification parameter) = 400
    Number of intervals in the sparse domain J, chosen by hand; the paper does not study sensitivity to m.
  • grid sampling density = 10 values for x,y and 2 for a,b,c,d for I; 5 and 2 for J
    Defines the candidate interval sets; arbitrary heuristic with no justification.
  • optimization hyperparameters = lr=0.001, momentum=0.9, 750 epochs, decay=0.99
    Stochastic gradient descent settings chosen by hand; loss decrease plateaus for the first ~300 iterations.
  • Gaussian KDE bandwidth = sigma = 0.1 * diam(X_T)
    Standard heuristic for kernel density estimation over point clouds in preprocessing.
assumptions (3)
  • domain assumption Every persistence module considered is finitely presentable, so its GPD over Int(R^2) exists [12, Theorem C(iii)].
    Invoked implicitly in Section 2 and used throughout; it guarantees the GPDs used in the loss are well-defined.
  • ad hoc to paper The fixed full domain I is a faithful proxy for the full GPD over all intervals, so approximating I preserves discriminative power.
    The entire method optimizes J to match I via \hat d(I,J); if I omits discriminative intervals, the sparse domain cannot recover them. Stated via the grid construction in Section 5.0.0.2.
  • domain assumption Erosion distance stability of d_E (Theorem H in [12]) is correct.
    Used in Corollary 7 to motivate the loss design; not central to the optimization itself but part of the theoretical framing.

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Cite this review

Pith. "Pith review of Sparsification of the Generalized Persistence Diagrams for Scalability through Gradient Descent." pith.science (2026). https://pith.science/paper/Z64NWPW5

@misc{pith2026241205900,
  author       = {Pith},
  title        = {Pith review of: Sparsification of the Generalized Persistence Diagrams for Scalability through Gradient Descent},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z64NWPW5}},
  note         = {Machine review of arXiv:2412.05900}
}
read the original abstract

The generalized persistence diagram (GPD) is a natural extension of the classical persistence barcode to the setting of multi-parameter persistence and beyond. The GPD is defined as an integer-valued function whose domain is the set of intervals in the indexing poset of a persistence module, and is known to be able to capture richer topological information than its single-parameter counterpart. However, computing the GPD is computationally prohibitive due to the sheer size of the interval set. Restricting the GPD to a subset of intervals provides a way to manage this complexity, compromising discriminating power to some extent. However, identifying and computing an effective restriction of the domain that minimizes the loss of discriminating power remains an open challenge. In this work, we introduce a novel method for optimizing the domain of the GPD through gradient descent optimization. To achieve this, we introduce a loss function tailored to optimize the selection of intervals, balancing computational efficiency and discriminative accuracy. The design of the loss function is based on the known erosion stability property of the GPD. We showcase the efficiency of our sparsification method for dataset classification in supervised machine learning. Experimental results demonstrate that our sparsification method significantly reduces the time required for computing the GPDs associated to several datasets, while maintaining classification accuracies comparable to those achieved using full GPDs. Our method thus opens the way for the use of GPD-based methods to applications at an unprecedented scale.

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Reference graph

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