Pith. sign in

REVIEW 4 major objections 8 minor 69 references

Expected Persistence Diagrams can be vectorized by counting mass in data-dependent Voronoi cells instead of smoothing each feature with a fixed kernel.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-30 11:43 UTC pith:3UEXI67P

load-bearing objection Solid EPD tooling with an honest coarse-vs-smooth trade-off, but the Lipschitz stability theorem does not actually cover the atomic histograms used in the experiments. the 4 major comments →

arxiv 2607.27126 v1 pith:3UEXI67P submitted 2026-07-29 cs.LG

Voronoi Histograms for Adaptive Vectorization of Expected Persistence Diagrams

Kaifeng Zhang , Kai Ming Ting This is my paper
classification cs.LG MSC 55N3168T10
keywords Expected Persistence DiagramVoronoi histogramtopological data analysispersistence diagram vectorizationWasserstein stabilitypoint cloud classificationEPD representation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Persistence diagrams capture the topology of a point cloud, but computing them is expensive. Expected Persistence Diagrams average many diagrams from random subsets, turning topology into a distribution of birth–death features. Most existing vectorizations still smooth each feature with a preset function such as a Gaussian or a landscape and then discretize. This paper proposes Vrep: sample codebooks from the empirical EPD, build Voronoi cells, and record the normalized mass in each cell, optionally with a cell for the diagonal to catch near-diagonal noise. Under normalization and separation conditions the histograms are Lipschitz-stable to small EPD moves and can lower-bound Wasserstein separation when the codebook approximates the measures well. On topology-sensitive classification and dimensionality-reduction tasks the method is competitive with persistence images, silhouettes, and landscapes, and its cost barely grows with the number of sampled diagrams per EPD.

Core claim

A Voronoi histogram of a normalized empirical Expected Persistence Diagram—mass counts inside cells of randomly sampled codebooks, concatenated across codebooks—is a stable finite vectorization that needs no explicit smooth point-transformation model, and under stated separation conditions it can preserve Wasserstein-scale differences at the cell level while matching or beating common smooth EPD summaries on topology-rich data.

What carries the argument

Vrep (Voronoi-based Representation): Φ(μ̄, C) = [μ̄(V(c₁)), …, μ̄(V(c_k))], the histogram of normalized EPD mass over Voronoi cells of a codebook C, concatenated over many sampled codebooks (Vrepd adds a diagonal cell). It carries the argument by replacing smooth functional summaries with adaptive partition-based mass counts, enabling the stability and conditional Wasserstein bounds.

Load-bearing premise

Every EPD is normalized to unit total mass before analysis and representation, so any signal in the absolute number or total mass of topological features is thrown away.

What would settle it

Build synthetic EPD pairs whose Wasserstein gap is pure coarse mass shift across well-separated regions with small codebook approximation error: if Vrep’s ℓ₁ distances fail to track W₁ better than PI/PS/PL there—or if tasks that depend on total feature mass lose accuracy under the paper’s unit-mass normalization—the central trade-off claim does not hold.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • When the useful signal is coarse mass allocation on the birth–death plane, adaptive Voronoi counts can replace fixed-kernel EPD vectorizations.
  • With support subsampling, EPD vectorization cost need not grow with the number of sampled persistence diagrams.
  • A diagonal Voronoi cell can separate near-diagonal noise from persistent features more stably than interior-only codebooks.
  • The same unsupervised vectors can feed dimensionality reduction where labeled end-to-end point-cloud models cannot.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Hard cell assignment will under-resolve tasks driven by fine within-cell shape; hybrid pipelines that keep a smooth local descriptor inside large cells are a natural next test.
  • Re-attaching total mass or unnormalized counts as extra coordinates would check how much the unit-mass step is costing on density-sensitive problems.
  • Learned or quantized codebooks under the same hard-histogram readout could tighten the Wasserstein lower bound when random support samples miss modes.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 8 minor

Summary. The paper proposes Vrep/Vrepd, a vectorization of empirical Expected Persistence Diagrams (EPDs). An empirical EPD (the average measure of n sampled PDs, ℓ1-normalized to unit mass) is represented by concatenating, over t codebooks of size k sampled from EPD supports, the histogram of EPD mass in each Voronoi cell; Vrepd adds a cell centered at the diagonal. The contributions are: (i) the representation itself; (ii) a stability analysis (Lemmas 4.3–4.4, Thm. 4.5: ‖Φ̂(μ̄)−Φ̂(μ̄′)‖₁ ≤ LΔ) and a conditional Wasserstein lower bound (Thm. 4.6) relating histogram distance to W1 separation minus codebook approximation error; (iii) experiments: Random-Forest classification on Protein, CAD (with two noise levels), and five time-series-derived point-cloud datasets, where Vrep/Vrepd report higher mean accuracy than PI/PS/PL on all datasets; ablations over codebook sampling schemes; scale-up tests showing near-constant cost in n; comparisons to PWGK/SWK (kernels win on several datasets at much higher runtime); an unsupervised t-SNE/CH study; and a synthetic W1-correlation study. The paper is unusually candid about non-dominance and limitations, and the normalization premise is explicitly disclosed.

Significance. If the claims hold, this is a useful, simple, and computationally attractive EPD vectorization: near-constant cost in the number of sampled PDs (Fig. 6b) is a genuine practical advantage over PI/PS/PL, and the data-dependent Voronoi partition is a sensible alternative bias to fixed smooth kernels. The empirical package is stronger than typical: codebook-choice ablations (Table 2), scale-up tests, honest non-dominance results against PWGK/SWK (App. F.2), a controlled synthetic W1-correlation study (App. F.7) that explicitly shows where PI is preferable, codebook-construction alternatives (App. F.8), and promised code. The candid limitation section (D.5) and the conditional reading of Thm. 4.6 are commendable. The weak half is the theory: as written, the stability theorem does not cover the atomic object used in every experiment, and Lemma 4.3's proof has concrete gaps. Fixing or rescoping this would make the paper a solid contribution.

major comments (4)
  1. [§4.2, Def. 4.2, Lemma 4.3, Thm. 4.5] Population/empirical mismatch in the stability results. Def. 4.2, Lemma 4.3 and Thm. 4.5 quantify over the population EPD μ̄ = lim_{n→∞}(1/n)Σμ_i, and Lemma 4.3's proof requires μ̄ to admit a Lipschitz density p (via [16]) perturbed by convolution with Gaussian noise. The deployed pipeline (§4.1, App. E.1) instead histograms an atomic empirical EPD (n=50 sampled PDs, support subsampled to |S_μ̄|=50). For an atomic measure, μ ↦ μ(V(c)) is not Lipschitz in W1: an atom of mass ~1/50 sitting ε from a Voronoi edge crosses it under an ε-perturbation, shifting ‖Φ‖₁ by ~2/50 regardless of how small ε is. Thus no deterministic Lipschitz-in-W1 bound covers the regime of Table 1 and Fig. 5, and Fig. 5 — the only stability evidence — is itself run in that uncovered regime and shows averages on one dataset only. Additionally, Lemma 4.3 is stated for arbitrary μ̄′ (via an optimal matching η), but the
  2. [App. C.1 (proof of Lemma 4.3)] Several specific gaps need repair: (a) the final equality Σ_j M·W1·Vol(V(c_j)) = M·W1 implicitly uses Σ_j Vol(V(c_j)) = 1, but the cells partition Ω′, so the sum is Vol(Ω′); the constant should be M·Vol(Ω′) (directionally harmless, but the stated constant is wrong). (b) The text asserts 'α and ϵ is dependent of each other' and then writes r = p∗q, which is valid only under independence — either assume independence or replace the convolution step. (c) 'W1(μ̄,μ̄′) = ≤ ∫q(z)‖z‖dz': only ≤ holds; W1 is an infimum over couplings. (d) Def. 4.2 uses Δ both as a deterministic radius bound (‖r̂_j − r_j‖ < Δ) and as a Gaussian random variable Δ ~ N(0,Σ); Gaussian noise is unbounded, so the hard bound fails almost surely. Each item is individually fixable, but together they require a careful rewrite of the perturbation model and the proof.
  3. [§4.1 normalization; App. D.4/D.5] All theory and the main experiments ℓ1-normalize EPDs to unit mass, discarding total feature count / total persistence mass and making Wasserstein (rather than OTp) applicable. The authors disclose this in App. D.4/D.5, which is appropriate, but no experiment quantifies the cost. Since the motivation (§2) explicitly invokes OTp for unequal-mass EPDs, a cheap ablation would substantially clarify when the premise is safe: e.g., augment Φ̂ with log μ̄(Ω) as an extra feature, or an unnormalized variant, on a dataset where class plausibly correlates with feature count. Without it, the 'Wasserstein-scale variation' framing applies to a geometry whose match to the tasks is asserted, not checked.
  4. [§5, Tables 1–2] Accuracies are means over 10 random splits on small datasets (BirdChicken n=40, Beef n=60, Protein n=99), with no standard deviations, confidence intervals, or paired tests reported, yet the text claims Vrep/Vrepd 'outperform PI, PS and PL over all the datasets.' Several margins look within plausible split noise (e.g., CAD0.01: 0.912 vs PI 0.911; CAD0.01/CAD0.05 vs PI at 0.900–0.911; Earthquakes vs PL). Please report per-split standard deviations and a paired significance test, mark which differences are significant, and soften the blanket claim where they are not. This is load-bearing for the main empirical claim, not a presentational nicety.
minor comments (8)
  1. [App. C.2 (proof of Lemma 4.4)] The proof invokes 'Theorem 5.1 and Remark 5.3 in [4]', but [4] is Beer, 'The Hausdorff metric and convergence in measure'; footnote 7's arXiv link (1103.4125) shows Reem [49] is intended. Also 'd_H(V(c_i), V(c_j)) ≤ Δ/γ' should be V(c′_i), and 'bounder area' → 'bounded area'.
  2. [App. C.5 vs App. D.3 (Thm. D.1)] d_min(C) is defined with ‖·‖₁ in App. C.5 but with ‖·‖₂ in the App. D.3 restatement of Thm. D.1 — align. In the C.5 proof, '‖Φ(ν̄,C)−Φ(ν̄,C)‖' appears three times where (μ̄,ν̄) is meant.
  3. [App. C.4 (proof of Thm. 4.6)] The transported mass in the designed plan is exactly ½‖Φ(μ̄,C)−Φ(ν̄,C)‖₁ (the ℓ1 norm double-counts excess and deficit), so the stated bound holds with factor-2 slack — worth a remark, or tighten. Also the phrase 'optimal partial transport metric W1' in the proof is a misnomer since masses are equal here.
  4. [Thm. 4.5 constant L] L = mt(M_max + kC_0^max) grows linearly with dataset size m; App. D.2 shows mt cancels in the normalized δ, but the raw bound is vacuous at realistic m. A sentence on the practical meaning (and in-principle estimability) of the constants M and C_0, both of which depend on the unknown EPD density, would help readers gauge the bound's content.
  5. [Lemma 4.4 statement] Main text says C_0 is 'determined by μ̄ and C'; the App. C.2 restatement says 'determined by μ̄' — align.
  6. [Citation/formatting] Bracketed citation numbers render without brackets throughout ('As pointed by 69', 'motivated by Lemma 2 in 25'). In App. D.3, 'Theorem 0.5' does not exist — presumably Thm. 4.6 is meant. 'V oronoi' appears with a spurious space throughout. Fig. 5 would benefit from error bars across the 450 EPDs.
  7. [App. D.1 (PI/PS comparison example)] The example assumes codebook size k very large to get ‖Φ(μ,C)−Φ(ν,C)‖₁ = 2W(μ,ν); please comment on how the conclusion degrades at the k values used in practice (k ≤ 20, App. E.1).
  8. [§3 / Def. 4.1] Since Persistence Bag-of-Words [68] is the closest histogram-style prior art, consider discussing it in §3 rather than only footnote 3, with one sentence on hard Voronoi vs GMM soft assignment. Also state near Def. 4.1 what §5 only says later: for a test EPD the concatenation uses codebooks sampled from training EPDs only.

Circularity Check

0 steps flagged

No significant circularity: Vrep is an independently defined histogram; stability/W1 bounds are conditional inequalities, not fits renamed as predictions.

full rationale

The paper defines Φ as normalized EPD mass in Voronoi cells of sampled codebooks (Def. 4.1), then proves Lipschitz stability under stated perturbation/normalization/codebook conditions (Lemmas 4.3–4.4, Thm 4.5) and a conditional W1 lower/upper bound via triangle inequality and a designed transport plan (Thm 4.6, App. C). Those bounds are mathematical consequences of the definitions plus external facts (e.g. Chazal–Divol density existence [16], Hausdorff stability of Voronoi cells [4], Divol–Lacombe quantization motivation [25]); they do not fit a target metric from the classification labels or redefine the claimed quantity as its own input. Empirical tables compare unsupervised vectors plus Random Forest against PI/PS/PL and kernels on external datasets; accuracies are not forced by construction from fitted constants. Modeling choices (l1 normalization, codebook sampling) are assumptions, not circular reductions. Correctness gaps about atomic vs density EPDs are outside circularity. No self-citation uniqueness chain or ansatz-smuggling from the same authors is load-bearing.

Axiom & Free-Parameter Ledger

6 free parameters · 7 axioms · 1 invented entities

The central method claim rests on standard TDA measure geometry (PDs as measures, EPD as expectation/average, Wasserstein/OTp), plus paper-specific modeling choices: unit-mass normalization, random codebook sampling, hard Voronoi assignment, and separation/regularity conditions used in the stability proofs. Free parameters are the usual representation hyperparameters (k, t, subsample size, codebook scheme). No new physical entities are postulated.

free parameters (6)
  • codebook size k = CV over 2–20 (t fixed at 10)
    Number of Voronoi sites per codebook; chosen by CV in {2..20}; controls resolution vs stability (Lemma 4.4).
  • number of codebooks t = 10
    How many random codebooks are concatenated into Vrep; fixed to 10 in main experiments.
  • EPD support subsample size |S_μ̄| = 50
    Support is subsampled (default 50) to make histogram construction O(mtk) rather than depending on full support size.
  • codebook sampling scheme = default in main table; ablation in Table 2
    Default uniform-on-support vs persistence-weighted vs uniform-in-bounding-box; changes which mass regions are resolved; selected/compared empirically.
  • PI/PS/PL and kernel bandwidths / resolutions = PI 10×10; bandwidths in {1e-4..1e-1}; PL kmax in {2,4,6,8}; resolution 100
    Baseline hyperparameters selected by 3-fold CV; affect comparative accuracy claims.
  • number and size of sampled PDs forming each EPD = n=50; dataset-specific subsample fractions
    n=50 sampled PDs; subset fractions differ by dataset (e.g. 2% CAD, 50 points Protein, 50% time series); defines the empirical EPD being vectorized.
axioms (7)
  • domain assumption A persistence diagram is a finite atomic measure on the birth-death half-plane; empirical EPD is the average of sampled PD measures.
    Sec. 2 background; standard in Chazal–Divol and follow-on EPD work.
  • ad hoc to paper After normalizing EPD total mass to 1, Wasserstein distance is the appropriate dissimilarity and equal-mass comparison is valid.
    Sec. 4.1 and App. D.4 adopt normalization to avoid OTp and treat EPDs as distributions; discards total mass information.
  • domain assumption EPD densities exist and are Lipschitz (or C^k) on a bounded filtration window so measure perturbation controls cell-mass change.
    Invoked in Lemma 4.3 proof via Chazal–Divol density results and finite max filtration.
  • standard math Voronoi cells of perturbed codebooks have Hausdorff distance O(Δ) under minimum-separation and domain assumptions (Beer-type geometric stability).
    Lemma 4.4 cites geometric stability of Voronoi diagrams to bound codebook error.
  • domain assumption Lemma 2 of Divol–Lacombe: Voronoi coefficients μ(V(ci)) optimally represent a measure among measures supported on codebook C in OTp.
    Sec. 4 motivation for using cell masses as representation coordinates.
  • ad hoc to paper For the W1 lower bound to be informative, codebook approximation errors must be small relative to W1(μ̄,ν̄) (well-resolved codebooks).
    Explicit caveat in Sec. 4.3 / Thm. 4.6; not a general distance-preservation theorem.
  • domain assumption Random-forest accuracy on selected topology-sensitive datasets is a valid proxy for representation quality.
    Sec. 5 experimental protocol following ATOL-style evaluation choices.
invented entities (1)
  • Vrep / Vrepd (Voronoi-based EPD histogram representation) independent evidence
    purpose: Finite-dimensional vectorization of empirical EPDs via concatenated normalized cell masses of data-dependent Voronoi partitions, optionally with a diagonal cell.
    Primary proposed object; defined in Def. 4.1. Operational and falsifiable via downstream tasks; not a hidden physical mediator.

pith-pipeline@v1.2.0-grok45-kimik3 · 32059 in / 4222 out tokens · 98141 ms · 2026-07-30T11:43:54.164748+00:00 · methodology

0 comments
read the original abstract

Persistence Diagram (PD) is known to capture point cloud topology effectively, but its computation has high time complexity. Expected Persistence Diagram (EPD) has been developed to reduce the time cost by studying the topology of multiple subsets of a point cloud and it serves as a distribution of topological features. Existing EPD vectorizations often rely on predefined point transformations, such as Gaussian or landscape functions. We study an alternative discretization based on Voronoi histograms, which trades smooth functional approximation for adaptive partition-based counting. We propose to use Voronoi Diagram-based histogram as the vectorization of EPD, without imposing an explicit smooth point transformation model. Under stated separation and normalization conditions, we establish stability bounds and characterize when the histogram representation preserves Wasserstein-scale variation. We demonstrate the effectiveness of our proposed representation on real-world datasets which have significant topological features for classification and dimensionality reduction tasks.

Figures

Figures reproduced from arXiv: 2607.27126 by Kaifeng Zhang, Kai Ming Ting.

Figure 1
Figure 1. Figure 1: Rips filtration on a 2D point cloud and the corresponding 1-dimensional PD. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) A torus-shaped point cloud with 20000 points, and (b) its 1-dimensional PD, EPD and EPD Quantization outcomes. The transparent blue area contains noise topological features. Given a finite set {µ1, µ2, ..., µn}, consisting of sampled PDs from P, where n is referred as the number of sampled PDs, the empirical EPD is defined as µ¯ = 1 n Pn i=1 µi . The support of µ¯ is Sµ¯ = ∪ n i=1Di , where Di = {rj = … view at source ↗
Figure 3
Figure 3. Figure 3: Example of EPD (a) and Voronoi Diagram built [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: As an alternative to Vrep, if each codebook C ∈ S has diagonal ck = ∂Ω while the other k −1 points are in the open half plane Ω, we denote it as Vrepd . Vrep is data-dependent because for an EPD µ¯i , Φ(¯ ˆ µi) is influenced by both the codebook set Si from EPD µ¯i and the codebook set ∪j̸=iSj from other EPDs in the dataset. We will elaborate this dependence in the Appendix D.3. There are three possible ch… view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of Vrep Φ(¯ ˆ µi) of EPD µ¯i . Each Si consists of t codebooks, with each codebook C sampled from EPD µ¯i . ci is sampled from a uniform distribution supported on a fixed rectangle above the diagonal ∂Ω. The former two choices are relevant to EPD and the last one is not. For Vrepd, these three choices determine the sampling method for the remaining k −1 points in codebook C, while ck is fixed … view at source ↗
Figure 5
Figure 5. Figure 5: Average change δ in Vrep and Vrepd under different EPD perturbation levels α. We use a 3d dynamical system dataset [26, 43], which describes a discrete food chain model. This dataset con￾tains 9 classes, each class contains 50 point clouds with each point cloud having 2000 points. Each class corre￾sponds to a parameter of the dynamical system. With the dataset of 450 point clouds, for each point cloud Xi ,… view at source ↗
Figure 6
Figure 6. Figure 6: (a) Scaleup test on the dataset size, where the dataset size is 126 at data size ratio =1. [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (a) Distribution ν is obtained via the translation of µ. The arrows indicate the direction of translation. The blue and red arrows are for marking the length of rectangle. (b) Illustration of the bijection map η. (c) Illustration of the point transformation function of PS. (d) Illustration of the point transformation function of PI. ∥Ψ(µ) − Ψ(ν)∥1 ∥Ψ(µ)∥1 = ∥ P x∈Sµ\A f(x) − P y∈Sν \A f(y)∥1 ∥Ψ(µ)∥1 = ∥ P … view at source ↗
Figure 8
Figure 8. Figure 8: Two measures µ, ¯ ν¯ on Ω and codebook C (red points) sampled from ν¯. We provide an upper bound BU for the l1 distance between Φ(¯µ, C) and Φ(¯ν, C). A larger BU would indicate a possible larger l1 distance. Next we focus on the case where C is sampled from ν¯ and its effect on BU . If the codebook is sampled from ν¯, the key of the analysis is to disentangle µ¯ from C in the W1(¯µ, µˆ(C)) term of BU . As… view at source ↗
Figure 9
Figure 9. Figure 9: A point cloud in CAD dataset. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Hyperparameter sensitivity of Vrep (a) and Vrep [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Hyperparameter sensitivity of Vrep (a) and Vrep [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Hyperparameter sensitivity of Vrep (a) and Vrep [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Scaleup test on the dataset size, where the dataset size is 126 at data size ratio =1. The [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Synthetic EPD families used in the Wasserstein correlation experiment. Columns show [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

69 extracted references · 2 linked inside Pith

  1. [1]

    Persistence images: A stable vector representation of persistent homology.Journal of Machine Learning Research, 18, 2017

    Henry Adams, Tegan Emerson, Michael Kirby, Rachel Neville, Chris Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson, Francis Motta, and Lori Ziegelmeier. Persistence images: A stable vector representation of persistent homology.Journal of Machine Learning Research, 18, 2017

    2017

  2. [2]

    Persistent homology for breast tumor classification using mammogram scans.Mathematics, 10(21):4039, 2022

    Aras Asaad, Dashti Ali, Taban Majeed, and Rasber Rashid. Persistent homology for breast tumor classification using mammogram scans.Mathematics, 10(21):4039, 2022

    2022

  3. [3]

    On the stability of persistent entropy and new summary functions for topological data analysis.Pattern Recognition, 107:107509, 2020

    Nieves Atienza, Rocío González-Díaz, and Manuel Soriano-Trigueros. On the stability of persistent entropy and new summary functions for topological data analysis.Pattern Recognition, 107:107509, 2020

    2020

  4. [4]

    The hausdorff metric and convergence in measure.Michigan Mathematical Journal, 20(4):63–64, 1974

    Gerald A Beer. The hausdorff metric and convergence in measure.Michigan Mathematical Journal, 20(4):63–64, 1974

    1974

  5. [5]

    Springer Science & Business Media, 2007

    Ingwer Borg and Patrick JF Groenen.Modern multidimensional scaling: Theory and applications. Springer Science & Business Media, 2007

    2007

  6. [6]

    Random forests.Machine Learning, 45:5–32, 2001

    Leo Breiman. Random forests.Machine Learning, 45:5–32, 2001

    2001

  7. [7]

    P. Bubenik. Statistical topological data analysis using persistence landscapes.Journal of Machine Learning Research, 16:77–102, 01 2015

    2015

  8. [8]

    The persistence landscape and some of its properties.Topological Data Analysis, 2020

    Peter Bubenik. The persistence landscape and some of its properties.Topological Data Analysis, 2020

    2020

  9. [9]

    A persistence landscapes toolbox for topological statistics.Journal of Symbolic Computation, 78:91–114, 2017

    Peter Bubenik and Paweł Dłotko. A persistence landscapes toolbox for topological statistics.Journal of Symbolic Computation, 78:91–114, 2017

    2017

  10. [10]

    A dendrite method for cluster analysis.Communications in Statistics-theory and Methods, 3(1):1–27, 1974

    Tadeusz Cali´nski and Jerzy Harabasz. A dendrite method for cluster analysis.Communications in Statistics-theory and Methods, 3(1):1–27, 1974

    1974

  11. [11]

    Approximating persistent homology for large datasets.arXiv preprint arXiv:2204.09155, 2022

    Yueqi Cao and Anthea Monod. Approximating persistent homology for large datasets.arXiv preprint arXiv:2204.09155, 2022

    arXiv 2022

  12. [12]

    Perslay: A neural network layer for persistence diagrams and new graph topological signatures.International Conference on Artificial Intelligence and Statistics, 2020

    Mathieu Carrière, Frédéric Chazal, Yuichi Ike, Théo Lacombe, Martin Royer, and Yuhei Umeda. Perslay: A neural network layer for persistence diagrams and new graph topological signatures.International Conference on Artificial Intelligence and Statistics, 2020

    2020

  13. [13]

    Sliced wasserstein kernel for persistence diagrams

    Mathieu Carrière, Marco Cuturi, and Steve Oudot. Sliced wasserstein kernel for persistence diagrams. International Conference on Machine Learning, 2017

    2017

  14. [14]

    John-Marc Chandonia, Lindsey Guan, Shiangyi Lin, Changhua Yu, Naomi K Fox, and Steven E Brenner. Scope: improvements to the structural classification of proteins–extended database to facilitate variant interpretation and machine learning.Nucleic Acids Research, 50(D1):D553–D559, 2022

    2022

  15. [15]

    Pointnet: Deep learning on point sets for 3d classification and segmentation.IEEE Conference on Computer Vision and Pattern Recognition, 2017

    R Qi Charles, Hao Su, Mo Kaichun, and Leonidas J Guibas. Pointnet: Deep learning on point sets for 3d classification and segmentation.IEEE Conference on Computer Vision and Pattern Recognition, 2017

    2017

  16. [16]

    The density of expected persistence diagrams and its kernel based estimation

    Frédéric Chazal and Vincent Divol. The density of expected persistence diagrams and its kernel based estimation. InProceedings of International Symposium on Computational Geometry, 2018

    2018

  17. [17]

    Subsampling methods for persistent homology.International Conference on Machine Learning, 2015

    Frédéric Chazal, Brittany Fasy, Fabrizio Lecci, Bertrand Michel, Alessandro Rinaldo, and Larry Wasserman. Subsampling methods for persistent homology.International Conference on Machine Learning, 2015

    2015

  18. [18]

    Stochas- tic convergence of persistence landscapes and silhouettes

    Frédéric Chazal, Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, and Larry Wasserman. Stochas- tic convergence of persistence landscapes and silhouettes. InProceedings of Annual Symposium on Computational Geometry, pages 474–483, 2014

    2014

  19. [19]

    An introduction to topological data analysis: Fundamental and practical aspects for data scientists.Frontiers in Artificial Intelligence, 4:667963, 2021

    Frédéric Chazal and Bertrand Michel. An introduction to topological data analysis: Fundamental and practical aspects for data scientists.Frontiers in Artificial Intelligence, 4:667963, 2021

    2021

  20. [20]

    Persistence paths and signature features in topological data analysis.IEEE Transactions on Pattern Analysis and Machine Intelligence, 42(1):192–202, 2018

    Ilya Chevyrev, Vidit Nanda, and Harald Oberhauser. Persistence paths and signature features in topological data analysis.IEEE Transactions on Pattern Analysis and Machine Intelligence, 42(1):192–202, 2018

    2018

  21. [21]

    An entropy-based persistence barcode.Pattern Recognition, 48(2):391–401, 2015

    Harish Chintakunta, Thanos Gentimis, Rocio Gonzalez-Diaz, Maria-Jose Jimenez, and Hamid Krim. An entropy-based persistence barcode.Pattern Recognition, 48(2):391–401, 2015

    2015

  22. [22]

    Persistence curves: A canonical framework for summarizing persis- tence diagrams.Advances in Computational Mathematics, 48(1):6, 2022

    Yu-Min Chung and Austin Lawson. Persistence curves: A canonical framework for summarizing persis- tence diagrams.Advances in Computational Mathematics, 48(1):6, 2022. 10

    2022

  23. [23]

    The ucr time series archive.IEEE/CAA Journal of Automatica Sinica, 6(6):1293–1305, 2019

    Hoang Anh Dau, Anthony Bagnall, Kaveh Kamgar, Chin-Chia Michael Yeh, Yan Zhu, Shaghayegh Gharghabi, Chotirat Ann Ratanamahatana, and Eamonn Keogh. The ucr time series archive.IEEE/CAA Journal of Automatica Sinica, 6(6):1293–1305, 2019

    2019

  24. [24]

    Comparing persistence diagrams through complex vectors

    Barbara Di Fabio and Massimo Ferri. Comparing persistence diagrams through complex vectors. InImage Analysis and Processing, pages 294–305. Springer, 2015

    2015

  25. [25]

    Estimation and quantization of expected persistence diagrams.Interna- tional Conference On Machine Learning, 2021

    Vincent Divol and Théo Lacombe. Estimation and quantization of expected persistence diagrams.Interna- tional Conference On Machine Learning, 2021

    2021

  26. [26]

    Persistence b-spline grids: stable vector representation of persistence diagrams based on data fitting.Machine Learning, 113(3):1373–1420, 2024

    Zhetong Dong, Hongwei Lin, Chi Zhou, Ben Zhang, and Gengchen Li. Persistence b-spline grids: stable vector representation of persistence diagrams based on data fitting.Machine Learning, 113(3):1373–1420, 2024

    2024

  27. [27]

    Edelsbrunner, D

    H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. In Proceedings of Annual Symposium on Foundations of Computer Science, 2000

    2000

  28. [28]

    The union of balls and its dual shape

    Herbert Edelsbrunner. The union of balls and its dual shape. InProceedings of Annual Symposium on Computational Geometry, pages 218–231, 1993

    1993

  29. [29]

    Representing size functions by complex polynomials.Proc

    Massimo Ferri and Claudia Landi. Representing size functions by complex polynomials.Proc. Math. Met. in Pattern Recognition, 9:16–19, 1999

    1999

  30. [30]

    The optimal partial transport problem.Archive for Rational Mechanics and Analysis, 195(2):533–560, 2010

    Alessio Figalli. The optimal partial transport problem.Archive for Rational Mechanics and Analysis, 195(2):533–560, 2010

    2010

  31. [31]

    Scope: Structural classification of pro- teins—extended, integrating scop and astral data and classification of new structures.Nucleic Acids Research, 42(D1):D304–D309, 2014

    Naomi K Fox, Steven E Brenner, and John-Marc Chandonia. Scope: Structural classification of pro- teins—extended, integrating scop and astral data and classification of new structures.Nucleic Acids Research, 42(D1):D304–D309, 2014

    2014

  32. [32]

    On the vietoris-rips complexes and a cohomology theory for metric spaces

    Jean-Claude Hausmann et al. On the vietoris-rips complexes and a cohomology theory for metric spaces. Annals of Mathematics Studies, 138:175–188, 1995

    1995

  33. [33]

    Graph filtration learning.International Conference on Machine Learning, 2020

    Christoph Hofer, Florian Graf, Bastian Rieck, Marc Niethammer, and Roland Kwitt. Graph filtration learning.International Conference on Machine Learning, 2020

    2020

  34. [34]

    Deep learning with topological signatures.Advances in Neural Information Processing Systems, 2017

    Christoph Hofer, Roland Kwitt, Marc Niethammer, and Andreas Uhl. Deep learning with topological signatures.Advances in Neural Information Processing Systems, 2017

    2017

  35. [35]

    Hofer, Roland Kwitt, and Marc Niethammer

    Christoph D. Hofer, Roland Kwitt, and Marc Niethammer. Learning representations of persistence barcodes. Journal of Machine Learning Research, 20(126):1–45, 2019

    2019

  36. [36]

    Topological graph neural networks.International Conference on Learning Representations, 2021

    Max Horn, Edward De Brouwer, Michael Moor, Yves Moreau, Bastian Rieck, and Karsten Borgwardt. Topological graph neural networks.International Conference on Learning Representations, 2021

    2021

  37. [37]

    Tropical coordinates on the space of persistence barcodes.Foundations of Computational Mathematics, 19(1):101–129, 2019

    Sara Kališnik. Tropical coordinates on the space of persistence barcodes.Foundations of Computational Mathematics, 19(1):101–129, 2019

    2019

  38. [38]

    A large-scale annotated mechanical components benchmark for classification and retrieval tasks with deep neural networks

    Sangpil Kim, Hyung gun Chi, Xiao Hu, Qixing Huang, and Karthik Ramani. A large-scale annotated mechanical components benchmark for classification and retrieval tasks with deep neural networks. European Conference on Computer Vision, 2020

    2020

  39. [39]

    Kernel method for persistence diagrams via kernel embedding and weight factor.Journal of Machine Learning Research, 18(1):6947–6987, 2017

    Genki Kusano, Kenji Fukumizu, and Yasuaki Hiraoka. Kernel method for persistence diagrams via kernel embedding and weight factor.Journal of Machine Learning Research, 18(1):6947–6987, 2017

    2017

  40. [40]

    Persistence weighted gaussian kernel for topological data analysis.International Conference on Machine Learning, 2016

    Genki Kusano, Yasuaki Hiraoka, and Kenji Fukumizu. Persistence weighted gaussian kernel for topological data analysis.International Conference on Machine Learning, 2016

    2016

  41. [41]

    Persistence fisher kernel: A riemannian manifold kernel for persistence diagrams.Advances in Neural Information Processing Systems, 2018

    Tam Le and Makoto Yamada. Persistence fisher kernel: A riemannian manifold kernel for persistence diagrams.Advances in Neural Information Processing Systems, 2018

    2018

  42. [42]

    Quantifying similarity of pore-geometry in nanoporous materials.Nature Communications, 8(1):1–8, 2017

    Yongjin Lee, Senja D Barthel, Paweł Dłotko, S Mohamad Moosavi, Kathryn Hess, and Berend Smit. Quantifying similarity of pore-geometry in nanoporous materials.Nature Communications, 8(1):1–8, 2017

    2017

  43. [43]

    On the dynamics of discrete food chains: Low-and high-frequency behavior and optimality of chaos.Journal of Mathematical Biology, 45(5):396–418, 2002

    Torsten Lindström. On the dynamics of discrete food chains: Low-and high-frequency behavior and optimality of chaos.Journal of Mathematical Biology, 45(5):396–418, 2002

    2002

  44. [44]

    Dowker complex based machine learning (dcml) models for protein-ligand binding affinity prediction.PLOS Computational Biology, 18(4):1 – 17, 2022

    Xiang Liu, Huitao Feng, Jie Wu, and Kelin Xia. Dowker complex based machine learning (dcml) models for protein-ligand binding affinity prediction.PLOS Computational Biology, 18(4):1 – 17, 2022. 11

    2022

  45. [45]

    Weighted persistent homology for biomolecular data analysis.Scientific Reports, 10(1):1–15, 2020

    Zhenyu Meng, D Vijay Anand, Yunpeng Lu, Jie Wu, and Kelin Xia. Weighted persistent homology for biomolecular data analysis.Scientific Reports, 10(1):1–15, 2020

    2020

  46. [46]

    Adaptive topological feature via persistent homology: filtration learning for point clouds.Advances in Neural Information Processing Systems, 2024

    Naoki Nishikawa, Yuichi Ike, and Kenji Yamanishi. Adaptive topological feature via persistent homology: filtration learning for point clouds.Advances in Neural Information Processing Systems, 2024

    2024

  47. [47]

    Approximating continuous functions on persistence diagrams using template functions.Foundations of Computational Mathematics, 23(4):1215– 1272, 2023

    Jose A Perea, Elizabeth Munch, and Firas A Khasawneh. Approximating continuous functions on persistence diagrams using template functions.Foundations of Computational Mathematics, 23(4):1215– 1272, 2023

    2023

  48. [48]

    Luis Polanco and Jose A. Perea. Adaptive template systems: Data-driven feature selection for learning with persistence diagrams.International Conference On Machine Learning And Applications, 2019

    2019

  49. [49]

    The geometric stability of voronoi diagrams with respect to small changes of the sites

    Daniel Reem. The geometric stability of voronoi diagrams with respect to small changes of the sites. In Proceedings of Annual Symposium on Computational Geometry, pages 254–263, 2011

    2011

  50. [50]

    A stable multi-scale kernel for topological machine learning.IEEE Conference on Computer Vision and Pattern Recognition, 2015

    Jan Reininghaus, Stefan Huber, Ulrich Bauer, and Roland Kwitt. A stable multi-scale kernel for topological machine learning.IEEE Conference on Computer Vision and Pattern Recognition, 2015

    2015

  51. [51]

    Gaussian mixture models.Encyclopedia of biometrics, 741(659-663), 2009

    Douglas A Reynolds et al. Gaussian mixture models.Encyclopedia of biometrics, 741(659-663), 2009

    2009

  52. [52]

    Atol: measure vectoriza- tion for automatic topologically-oriented learning

    Martin Royer, Frédéric Chazal, Clément Levrard, Yuhei Umeda, and Yuichi Ike. Atol: measure vectoriza- tion for automatic topologically-oriented learning. InInternational Conference on Artificial Intelligence and Statistics, pages 1000–1008. PMLR, 2021

    2021

  53. [53]

    Characterisation of the idiotypic immune network through persistent entropy.European Conference on Complex Systems, 2016

    Matteo Rucco, Filippo Castiglione, Emanuela Merelli, and Marco Pettini. Characterisation of the idiotypic immune network through persistent entropy.European Conference on Complex Systems, 2016

    2016

  54. [54]

    Simplicial complexes and complex systems

    Vsevolod Salnikov, Daniele Cassese, and Renaud Lambiotte. Simplicial complexes and complex systems. European Journal of Physics, 40(1):014001, 2018

    2018

  55. [55]

    On time-series topological data analysis: New data and opportunities.IEEE Conference on Computer Vision and Pattern Recognition workshops, 2016

    Lee M Seversky, Shelby Davis, and Matthew Berger. On time-series topological data analysis: New data and opportunities.IEEE Conference on Computer Vision and Pattern Recognition workshops, 2016

    2016

  56. [56]

    Support vector machine.Machine learning models and algorithms for big data classification: thinking with examples for effective learning, pages 207–235, 2016

    Shan Suthaharan and Shan Suthaharan. Support vector machine.Machine learning models and algorithms for big data classification: thinking with examples for effective learning, pages 207–235, 2016

    2016

  57. [57]

    Representation of molecular structures with persistent homology for machine learning applications in chemistry.Nature Communications, 11(1):1–9, 2020

    Jacob Townsend, Cassie Putman Micucci, John H Hymel, Vasileios Maroulas, and Konstantinos D V ogiatzis. Representation of molecular structures with persistent homology for machine learning applications in chemistry.Nature Communications, 11(1):1–9, 2020

    2020

  58. [58]

    Visualizing data using t-sne.Journal of Machine Learning Research, 9(11), 2008

    Laurens Van der Maaten and Geoffrey Hinton. Visualizing data using t-sne.Journal of Machine Learning Research, 9(11), 2008

    2008

  59. [59]

    Attention is all you need.Advances in Neural Information Processing Systems, 2017

    A Vaswani. Attention is all you need.Advances in Neural Information Processing Systems, 2017

    2017

  60. [60]

    The wasserstein distances.Optimal transport: old and new, pages 93–111, 2009

    Cédric Villani. The wasserstein distances.Optimal transport: old and new, pages 93–111, 2009

    2009

  61. [61]

    Multiset transformer: Advancing representation learning in persistence diagrams.arXiv preprint arXiv:2411.14662, 2024

    Minghua Wang, Ziyun Huang, and Jinhui Xu. Multiset transformer: Advancing representation learning in persistence diagrams.arXiv preprint arXiv:2411.14662, 2024

    Pith/arXiv arXiv 2024

  62. [62]

    Topological data analysis.Annual Review of Statistics and Its Application, 5:501–532, 2018

    Larry Wasserman. Topological data analysis.Annual Review of Statistics and Its Application, 5:501–532, 2018

    2018

  63. [63]

    On the estimation of persistence intensity functions and linear representations of persistence diagrams.International Conference on Artificial Intelligence and Statistics, 2024

    Weichen Wu, Jisu Kim, and Alessandro Rinaldo. On the estimation of persistence intensity functions and linear representations of persistence diagrams.International Conference on Artificial Intelligence and Statistics, 2024

    2024

  64. [64]

    Multiscale persistent functions for biomolecular structure characteri- zation.Bulletin of mathematical biology, 80:1–31, 2018

    Kelin Xia, Zhiming Li, and Lin Mu. Multiscale persistent functions for biomolecular structure characteri- zation.Bulletin of mathematical biology, 80:1–31, 2018

    2018

  65. [65]

    Multidimensional persistence in biomolecular data.Journal of Computational Chemistry, 36(20):1502–1520, 2015

    Kelin Xia and Guo-Wei Wei. Multidimensional persistence in biomolecular data.Journal of Computational Chemistry, 36(20):1502–1520, 2015

    2015

  66. [66]

    Deep sets.Advances in Neural Information Processing Systems, 2017

    Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R Salakhutdinov, and Alexander J Smola. Deep sets.Advances in Neural Information Processing Systems, 2017

    2017

  67. [67]

    Gefl: extended filtration learning for graph classifica- tion.Learning on Graphs Conference, 2022

    Simon Zhang, Soham Mukherjee, and Tamal K Dey. Gefl: extended filtration learning for graph classifica- tion.Learning on Graphs Conference, 2022. 12

    2022

  68. [68]

    Persistence bag-of-words for topological data analysis.International Joint Conference on Artificial Intelligence, 2019

    Bartosz Zieli´nski, Michał Lipi´nski, Mateusz Juda, Matthias Zeppelzauer, and Paweł Dłotko. Persistence bag-of-words for topological data analysis.International Joint Conference on Artificial Intelligence, 2019

    2019

  69. [69]

    Computing persistent homology

    Afra Zomorodian and Gunnar Carlsson. Computing persistent homology. InProceedings of Annual Symposium on Computational Geometry, page 347–356, 2004. A Vectorization methods of PD Here we introduce unsupervised and supervised vectorization methods for PD. • For unsupervised methods, the simplest category is the basic statistical quantities of PD, including...

    Pith/arXiv arXiv 2004