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arxiv: 2602.22784 · v2 · pith:QUN65XO4new · submitted 2026-02-26 · 🧮 math.AT

It's All About Covers: Persistent Homology of Cover Refinements

Ant\'onio Leit\~ao This is my paper

Pith reviewed 2026-05-15 19:27 UTC · model grok-4.3

classification 🧮 math.AT
keywords persistent homologycover refinementsVietoris-Ripsinterleaving distancenerve functorfiltrationtopological data analysissimplicial complexes
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The pith

Reframing persistent homology around cover refinements produces smaller filtrations with unconditional log-3 interleaving to Vietoris-Rips.

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

The paper recasts persistent homology filtrations in terms of relations between covers of a point cloud instead of building simplicial complexes directly. Using functors such as the Nerve and Co-Nerve that preserve contiguity of refinement maps, the interleaving properties transfer from the cover level to the resulting persistence modules. This construction yields a robust approximation to the Vietoris-Rips filtration that remains log-3 interleaved for every metric space. The filtrations stay near-linear in the number of data points, removing the exponential growth that limits standard methods. The change makes high-degree homology groups computable on large data sets without extra assumptions on the metric.

Core claim

The paper establishes that filtrations constructed from sequences of cover refinements, when passed through a contiguity-preserving functor like the Nerve or Co-Nerve, produce persistence modules that are log-3 interleaved with the Vietoris-Rips persistence module for every metric space. The size of these filtrations grows near-linearly with the number of data points, in contrast to the exponential growth possible in standard constructions.

What carries the argument

Cover refinement relations and contiguity-preserving functors (exemplified by the Nerve and Co-Nerve) that transfer interleaving bounds from the cover level to the simplicial level.

If this is right

  • The new filtration approximates the Vietoris-Rips filtration with a multiplicative log-3 factor in the filtration values for all metric spaces.
  • Near-linear scaling in the number of data points makes the method feasible for large data sets.
  • High-degree homology can be computed efficiently without the exponential blowup typical of simplicial filtrations.
  • The framework applies unconditionally, requiring no special properties of the metric space.

Where Pith is reading between the lines

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

  • This perspective could extend to other topological invariants by choosing different functors on the covers.
  • Optimal choice of cover refinements might be data-driven to minimize the size while preserving the interleaving.
  • The method suggests that many existing approximations in topological data analysis can be unified under a cover-based view.
  • Practical implementations could integrate this with existing software by replacing the complex construction step.

Load-bearing premise

The chosen refinements of covers must be such that the refinement maps remain contiguous when passed through the functors, allowing the interleaving to propagate without metric-specific adjustments.

What would settle it

Run the construction on a point cloud in a high-dimensional or irregular metric space and measure the actual interleaving distance between the cover-based persistence module and the Vietoris-Rips module; if it exceeds log 3, the guarantee fails.

Figures

Figures reproduced from arXiv: 2602.22784 by Ant\'onio Leit\~ao.

Figure 1
Figure 1. Figure 1: Illustration of a cover (a) and the resulting simplicial complexes obtained from [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Quotienting a cover commutes with building its Co-Nerve. Given a cover and a [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: How approximations propagate from covers to simplicial complexes. The per [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Runtime comparison between Vietoris-Rips filtration and our method. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Example of two contiguous refinement maps [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The ball cover Furthermore Nerve and the Co-Nerve of the ball cover are the same simplicial complex. This self-duality is a consequence of two properties: the index set coincides with the space (I = X), and the distance is symmetric. Proposition 3.14. CoNrv(B(r)) = Nrv(B(r)) ∀r ≥ 0 Proof. Since the index set of B(r) is X itself, both complexes have vertex set X. For a finite 18 [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 7
Figure 7. Figure 7: The maximal clique cover. To help visualize the clique structure we added grey [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A persistent clique-partition P. In this case it is a maximal clique partition (each partition is a maximal clique). We added all the edges whenever d(x, y) ≤ r. 4.2 Quotients of covers via pushforwards of relations The objective is to reduce the size of simplicial complexes in a controlled manner such that we guarantee an interleaving at the homology level. We do this by considering the usual simplicial c… view at source ↗
Figure 9
Figure 9. Figure 9: Example of an instance of the maximal clique cover [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of dendrograms for the line example( [PITH_FULL_IMAGE:figures/full_fig_p035_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: (a) Filtration size reduction factor (y-axis) and subsequent computation speedup (marker size) as a function of the number of simplices in the Vietoris-Rips filtration per data point (x−axis). (b) The Empirical Order of Growth of the filtration size (S) relative to the number of points (N). The y−value represents the empirical exponent. 7 Conclusion The central message of this work is that covers, not sim… view at source ↗
read the original abstract

The computational cost of persistent homology is often dominated by the growth of the underlying simplicial filtrations. Many different filtrations exist, each with its own assumptions and trade-offs, but all face some form of this growth which can be exponential in the worst case, as for the Vietoris-Rips. We recast this problem at the level of covers, developing a framework in which filtrations and persistence modules can be constructed, analyzed, and compared through simple relations between covers rather than at the level of simplicial complexes. The guarantees propagate through any functor that preserves the contiguity of refinement maps, we give the example of two such functors: the Nerve and the Co-Nerve. Working at this level is drastically simpler, with stronger, more general consequences. We explore this perspective and show how it can be used to construct a robust approximation of the Vietoris-Rips filtration that is orders of magnitude smaller, while maintaining a log 3-interleaving unconditionally for any metric space. The resulting filtration restores near-linear scaling in the number of data points and enable us to efficiently capture homology at high degrees.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a framework for persistent homology at the level of covers and their refinements rather than simplicial complexes. It asserts that interleaving distances and other persistence guarantees propagate through any contiguity-preserving functor, with the Nerve and Co-Nerve functors given as examples. The central construction is a cover-refinement filtration that approximates the Vietoris-Rips filtration by a log-3 interleaving that holds unconditionally for arbitrary metric spaces, yielding a complex whose size scales near-linearly in the number of data points and enabling efficient computation of high-degree homology.

Significance. If the unconditional log-3 interleaving and propagation claims are established, the work would offer a practical route to scalable persistent homology, especially for high-dimensional features, while simplifying analysis via cover relations. The framework's generality across functors and its avoidance of fitted parameters are positive features that could influence computational topology.

major comments (2)
  1. [Abstract and functor-propagation section] Abstract and the section introducing functor propagation: the claim that 'guarantees propagate through any functor that preserves the contiguity of refinement maps' is stated without an explicit derivation or verification that the specific refinements (e.g., ball covers) induce the log-3 factor; the abstract gives the Nerve/Co-Nerve examples but supplies no step-by-step reasoning or counter-example checks for the interleaving distance.
  2. [Cover-refinement filtration construction] Section constructing the cover-refinement filtration: the log-3 interleaving with Vietoris-Rips is asserted to hold unconditionally for any metric space, yet the argument relies on the chosen refinements satisfying contiguity preservation; this step must be shown to be independent of metric properties such as the triangle inequality, since failure in discrete or ultrametric spaces would undermine the unconditional claim.
minor comments (1)
  1. [Notation and definitions] Notation for cover relations and refinement maps could be illustrated with one or two concrete low-dimensional examples to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight areas where the presentation of the functor-propagation result and the unconditional interleaving claim can be strengthened with additional derivations and verifications. We address each point below and will incorporate the suggested clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract and functor-propagation section] Abstract and the section introducing functor propagation: the claim that 'guarantees propagate through any functor that preserves the contiguity of refinement maps' is stated without an explicit derivation or verification that the specific refinements (e.g., ball covers) induce the log-3 factor; the abstract gives the Nerve/Co-Nerve examples but supplies no step-by-step reasoning or counter-example checks for the interleaving distance.

    Authors: We agree that an explicit derivation of the propagation property is needed. In the revision we will insert a short subsection that proves the general statement: if F is any functor preserving contiguity of refinement maps, then the interleaving distance between the persistence modules induced by two cover filtrations is at most the interleaving distance between the cover filtrations themselves. We will then specialize the argument to the ball-cover refinements used in the paper, walking through the three successive refinements that produce the factor of log 3 and verifying the bound for both the Nerve and Co-Nerve functors. A brief counter-example showing that contiguity preservation is necessary will also be added. revision: yes

  2. Referee: [Cover-refinement filtration construction] Section constructing the cover-refinement filtration: the log-3 interleaving with Vietoris-Rips is asserted to hold unconditionally for any metric space, yet the argument relies on the chosen refinements satisfying contiguity preservation; this step must be shown to be independent of metric properties such as the triangle inequality, since failure in discrete or ultrametric spaces would undermine the unconditional claim.

    Authors: The contiguity of the chosen ball-cover refinements follows directly from the definition of open balls in an arbitrary metric space and does not invoke the triangle inequality beyond the metric axioms themselves. In the revision we will add an explicit paragraph confirming that the same three-step refinement argument applies verbatim to discrete metric spaces and to ultrametric spaces, where the triangle inequality is replaced by the stronger ultrametric inequality; the log-3 bound is recovered unchanged. This establishes that the interleaving is independent of any further metric properties. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper constructs filtrations via cover refinements and propagates interleaving guarantees through standard functors (Nerve, Co-Nerve) that preserve contiguity of refinement maps. The log-3 interleaving with Vietoris-Rips is presented as holding unconditionally for arbitrary metric spaces via these combinatorial relations, without reducing to fitted parameters, self-referential definitions, or load-bearing self-citations. No equations equate the claimed distance to inputs by construction, and the framework relies on external topological facts rather than renaming known results or smuggling ansatzes. The derivation remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard algebraic topology (nerve theorem, contiguity of maps) plus the new definition of cover refinements; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Refinement maps between covers induce contiguous simplicial maps after applying the Nerve or Co-Nerve functor.
    Invoked to propagate interleaving guarantees from the cover level to the persistence module level.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. An Algebraic Introduction to Persistence

    math.AT 2026-04 unverdicted novelty 2.0

    The paper surveys algebraic properties of poset representations and their stability under the interleaving distance in persistence theory.

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages · cited by 1 Pith paper · 5 internal anchors

  1. [1]

    Dory: Computation of persistence diagrams up to dimension two for Vietoris–Rips filtrations of large data sets

    Manu Aggarwal and Vipul Periwal. “Dory: Computation of persistence diagrams up to dimension two for Vietoris–Rips filtrations of large data sets”. In:Journal of Com- putational Science79 (2024), p. 102290.ISSN: 1877-7503.DOI:https://doi.org/10. 1016 / j . jocs . 2024 . 102290.URL:https : / / www . sciencedirect . com / science / article/pii/S1877750324000838

    work page 2024

  2. [2]

    Strong homotopy types, nerves and collapses

    Jonathan Ariel Barmak and Elias Gabriel Minian. “Strong homotopy types, nerves and collapses”. In:Discrete & Computational Geometry47.2 (2012), pp. 301–328

    work page 2012

  3. [3]

    Ripser: efficient Computation of Vietoris–Rips Persistence Barcodes

    Ulrich Bauer. “Ripser: efficient computation of Vietoris-Rips persistence barcodes”. In:J. Appl. Comput. Topol.5.3 (2021), pp. 391–423.ISSN: 2367-1726.DOI:10 . 1007 / s41468-021-00071-5.URL:https://doi.org/10.1007/s41468-021-00071-5

    work page doi:10.1007/s41468-021-00071-5 2021

  4. [4]

    Phat–persistent homology algorithms toolbox

    Ulrich Bauer et al. “Phat–persistent homology algorithms toolbox”. In:Journal of symbolic computation78 (2017), pp. 76–90

    work page 2017

  5. [5]

    The Vietoris Mapping Theorem for Bicompact Spaces

    Edward G. Begle. “The Vietoris Mapping Theorem for Bicompact Spaces”. In:Annals of Mathematics51.3 (1950), pp. 534–543.ISSN: 0003486X, 19398980.URL:http://www. jstor.org/stable/1969366(visited on 07/21/2025)

    work page arXiv 1950

  6. [6]

    Topological methods

    Anders Björner. “Topological methods”. In:Handbook of combinatorics2 (1995), pp. 1819– 1872

    work page 1995

  7. [7]

    Blumberg and Michael Lesnick.Universality of the Homotopy Interleaving Distance

    Andrew J. Blumberg and Michael Lesnick.Universality of the Homotopy Interleaving Distance. 2022. arXiv:1705.01690 [math.AT].URL:https://arxiv.org/abs/1705. 01690

    work page arXiv 2022

  8. [8]

    Computing Persistent Homology of Flag Complexes via Strong Collapses

    Jean-Daniel Boissonnat and Siddharth Pritam. “Computing Persistent Homology of Flag Complexes via Strong Collapses”. In:SoCG. 2019

    work page 2019

  9. [9]

    Metrics for generalized persistence modules

    Peter Bubenik, Vin De Silva, and Jonathan Scott. “Metrics for generalized persistence modules”. In:Foundations of Computational Mathematics15.6 (2015), pp. 1501–1531

    work page 2015

  10. [10]

    Sparse higher order Čech filtrations

    Mickaël Buchet, Bianca B Dornelas, and Michael Kerber. “Sparse higher order Čech filtrations”. In:Journal of the ACM71.4 (2024), pp. 1–23

    work page 2024

  11. [11]

    Efficient and robust persistent homology for measures

    Mickaël Buchet et al. “Efficient and robust persistent homology for measures”. In: Computational Geometry58 (2016), pp. 70–96

    work page 2016

  12. [12]

    Robert Cardona et al.The Universalℓ p-Metric on Merge Trees. 2022. arXiv:2112.12165 [cs.CG].URL:https://arxiv.org/abs/2112.12165

    work page arXiv 2022

  13. [13]

    Topology and data

    Gunnar Carlsson. “Topology and data”. In:Bulletin of the American Mathematical So- ciety46.2 (2009), pp. 255–308. 41

    work page 2009

  14. [14]

    Gunnar Carlsson and Facundo Memoli.Classifying Clustering Schemes. 2010. arXiv: 1011.5270 [stat.ML].URL:https://arxiv.org/abs/1011.5270

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  15. [15]

    Persistent Clustering and a Theorem of J. Kleinberg

    Gunnar Carlsson and Facundo Memoli.Persistent Clustering and a Theorem of J. Klein- berg. 2008. arXiv:0808.2241 [stat.ML].URL:https://arxiv.org/abs/0808.2241

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  16. [16]

    Characterization, Stability and Convergence of Hierarchical Clustering Methods

    Gunnar Carlsson and Facundo Mémoli. “Characterization, Stability and Convergence of Hierarchical Clustering Methods”. In:Journal of Machine Learning Research11.47 (2010), pp. 1425–1470.URL:http://jmlr.org/papers/v11/carlsson10a.html

    work page 2010

  17. [17]

    Hierarchical clustering of asymmetric networks

    Gunnar Carlsson et al. “Hierarchical clustering of asymmetric networks”. In:Ad- vances in Data Analysis and Classification12.1 (2018), pp. 65–105

    work page 2018

  18. [18]

    A Geometric Perspective on Sparse Filtrations

    Nicholas J. Cavanna, Mahmoodreza Jahanseir, and Donald R. Sheehy.A Geometric Perspective on Sparse Filtrations. 2015. arXiv:1506 . 03797 [cs.CG].URL:http : / / arxiv.org/pdf/1506.03797v1

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  19. [19]

    An introduction to topological data analysis: fundamental and practical aspects for data scientists

    Frédéric Chazal and Bertrand Michel. “An introduction to topological data analysis: fundamental and practical aspects for data scientists”. In:Frontiers in artificial intel- ligence4 (2021), p. 667963

    work page 2021

  20. [20]

    Gromov-Hausdorff stable signatures for shapes using persis- tence

    Frédéric Chazal et al. “Gromov-Hausdorff stable signatures for shapes using persis- tence”. In:Computer Graphics Forum. Vol. 28. 5. Wiley Online Library. 2009, pp. 1393– 1403

    work page 2009

  21. [21]

    Proximity of persistence modules and their diagrams

    Frédéric Chazal et al. “Proximity of persistence modules and their diagrams”. In:Pro- ceedings of the twenty-fifth annual symposium on Computational geometry. 2009, pp. 237– 246

    work page 2009

  22. [22]

    A functorial Dowker theorem and persis- tent homology of asymmetric networks

    Samir Chowdhury and Facundo Mémoli. “A functorial Dowker theorem and persis- tent homology of asymmetric networks”. In:Journal of Applied and Computational Topology2.1 (2018), pp. 115–175

    work page 2018

  23. [23]

    Theory of interleavings on categories with a flow

    Vin De Silva, Elizabeth Munch, and Anastasios Stefanou. “Theory of interleavings on categories with a flow”. In:arXiv preprint arXiv:1706.04095(2017)

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  24. [24]

    Multiscale mapper: Topological summarization via codomain covers

    Tamal K Dey, Facundo Mémoli, and Yusu Wang. “Multiscale mapper: Topological summarization via codomain covers”. In:Proceedings of the twenty-seventh annual acm- siam symposium on discrete algorithms. SIAM. 2016, pp. 997–1013

    work page 2016

  25. [25]

    Topological Analysis of Nerves, Reeb Spaces, Mappers, and Multiscale Mappers

    Tamal K Dey, Facundo Mémoli, and Yusu Wang. “Topological analysis of nerves, reeb spaces, mappers, and multiscale mappers”. In:arXiv preprint arXiv:1703.07387(2017)

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  26. [26]

    Simba: An efficient tool for approximating rips-filtration persistence via sim plicial ba tch collapse

    Tamal K Dey, Dayu Shi, and Yusu Wang. “Simba: An efficient tool for approximating rips-filtration persistence via sim plicial ba tch collapse”. In:Journal of Experimental Algorithmics (JEA)24 (2019), pp. 1–16

    work page 2019

  27. [27]

    Computing Topological Persistence for Simplicial Maps

    Tamal K. Dey, Fengtao Fan, and Yusu Wang. “Computing Topological Persistence for Simplicial Maps”. In:Proceedings of the Thirtieth Annual Symposium on Computa- tional Geometry. SOCG’14. Kyoto, Japan: Association for Computing Machinery, 2014, pp. 345–354.ISBN: 9781450325943.DOI:10 . 1145 / 2582112 . 2582165.URL:https : //doi.org/10.1145/2582112.2582165. 42

    work page doi:10.1145/2582112.2582165 2014

  28. [28]

    Simplification of complexes for persistent ho- mology computations

    Paweł Dłotko and Hubert Wagner. “Simplification of complexes for persistent ho- mology computations”. In:Homology, Homotopy and Applications16 (Apr. 2013).DOI: 10.4310/HHA.2014.v16.n1.a3

    work page doi:10.4310/hha.2014.v16.n1.a3 2013

  29. [29]

    Barbara Giunti, J ¯anis Lazovskis, and Bastian Rieck.DONUT: Database of Original & Non-Theoretical Uses of Topology.https://donut.topology.rocks. 2022

    work page 2022

  30. [30]

    Homology groups of relations

    Clifford H Dowker. “Homology groups of relations”. In:Annals of mathematics56.1 (1952), pp. 84–95

    work page 1952

  31. [31]

    Amer- ican Mathematical Soc., 2010

    Herbert Edelsbrunner and John Harer.Computational topology: an introduction. Amer- ican Mathematical Soc., 2010

    work page 2010

  32. [32]

    On the shape of a set of points in the plane

    Herbert Edelsbrunner, David Kirkpatrick, and Raimund Seidel. “On the shape of a set of points in the plane”. In:IEEE Transactions on information theory29.4 (2003), pp. 551–559

    work page 2003

  33. [33]

    Čech Theory: Its Past, Present, and Fu- ture

    David A. Edwards and Harold M. Hastings. “Čech Theory: Its Past, Present, and Fu- ture”. In:Rocky Mountain Journal of Mathematics10.3 (1980), pp. 429–468

    work page 1980

  34. [34]

    General theory of natural equivalences

    Samuel Eilenberg and Saunders MacLane. “General theory of natural equivalences”. In:Transactions of the american mathematical society58.2 (1945), pp. 231–294

    work page 1945

  35. [35]

    Swap, shift and trim to edge collapse a filtration

    Marc Glisse and Siddharth Pritam. “Swap, shift and trim to edge collapse a filtration”. In:arXiv preprint arXiv:2203.07022(2022)

    work page arXiv 2022

  36. [36]

    Florian Graf et al.The Flood Complex: Large-Scale Persistent Homology on Millions of Points. 2026. arXiv:2509 . 22432 [cs.LG].URL:https : / / arxiv . org / abs / 2509 . 22432

    work page 2026

  37. [37]

    Hatcher and Cambridge University Press.Algebraic Topology

    A. Hatcher and Cambridge University Press.Algebraic Topology. Algebraic Topology. Cambridge University Press, 2002.ISBN: 9780521795401

    work page 2002

  38. [38]

    Prentice-Hall, Inc., 1988

    Anil K Jain and Richard C Dubes.Algorithms for clustering data. Prentice-Hall, Inc., 1988

    work page 1988

  39. [39]

    Jardine and R

    N. Jardine and R. Sibson.Mathematical Taxonomy. Wiley series in probability and mathematical statistics. Wiley, 1971.ISBN: 9780471440505.URL:https : / / books . google.it/books?id=ka4KAQAAIAAJ

    work page 1971

  40. [40]

    Barcodes of towers and a streaming algo- rithm for persistent homology

    Michael Kerber and Hannah Schreiber. “Barcodes of towers and a streaming algo- rithm for persistent homology”. In:Discrete & computational geometry61.4 (2019), pp. 852–879

    work page 2019

  41. [41]

    Approximate Čech complex in low and high dimensions

    Michael Kerber and Raghvendra Sharathkumar. “Approximate Čech complex in low and high dimensions”. In:International Symposium on Algorithms and Computation. Springer. 2013, pp. 666–676

    work page 2013

  42. [42]

    Musashi Ayrton Koyama, Vanessa Robins, and Katharine Turner.The distilled Vietoris Rips filtration for persistent homology and a new memory efficient algorithm. 2024. arXiv: 2412.07805 [math.AT].URL:https://arxiv.org/abs/2412.07805

    work page arXiv 2024

  43. [43]

    The theory of the interleaving distance on multidimensional per- sistence modules

    Michael Lesnick. “The theory of the interleaving distance on multidimensional per- sistence modules”. In:Foundations of Computational Mathematics15.3 (2015), pp. 613– 650. 43

    work page 2015

  44. [44]

    Enumerating All Maximal Clique-Partitions of an Undirected Graph

    Mircea Marin et al. “Enumerating All Maximal Clique-Partitions of an Undirected Graph”. In:Electronic Proceedings in Theoretical Computer Science389 (Sept. 2023), pp. 65– 79.ISSN: 2075-2180.DOI:10.4204/eptcs.389.6.URL:http://dx.doi.org/10.4204/ EPTCS.389.6

    work page doi:10.4204/eptcs.389.6.url:http://dx.doi.org/10.4204/ 2023

  45. [45]

    Modern hierarchical, agglomerative clustering algorithms

    Daniel Müllner. “Modern hierarchical, agglomerative clustering algorithms”. In:arXiv preprint arXiv:1109 .2378(2011)

    work page 2011

  46. [46]

    CRC press, 2018

    James R Munkres.Elements of algebraic topology. CRC press, 2018

    work page 2018

  47. [47]

    A roadmap for the computation of persistent homology

    Nina Otter et al. “A roadmap for the computation of persistent homology”. In:EPJ Data Science6.1 (2017), p. 17

    work page 2017

  48. [48]

    Steve Y Oudot.Persistence theory: from quiver representations to data analysis. Vol. 209. American Mathematical Society Providence, 2015

    work page 2015

  49. [49]

    A hierarchical clustering algorithm and an im- provement of the single linkage criterion to deal with noise

    Frédéric Ros and Serge Guillaume. “A hierarchical clustering algorithm and an im- provement of the single linkage criterion to deal with noise”. In:Expert Systems with Applications128 (2019), pp. 96–108

    work page 2019

  50. [50]

    Spatial reconstruction of single-cell gene expression data

    Rahul Satija et al. “Spatial reconstruction of single-cell gene expression data”. In: Nature biotechnology33.5 (2015), pp. 495–502

    work page 2015

  51. [51]

    Linear-size approximations to the Vietoris-Rips filtration

    Donald R Sheehy. “Linear-size approximations to the Vietoris-Rips filtration”. In: Proceedings of the twenty-eighth annual symposium on Computational geometry. 2012, pp. 239–248

    work page 2012

  52. [52]

    Topological estimation using witness com- plexes

    Vin de Silva and Gunnar E. Carlsson. “Topological estimation using witness com- plexes”. In:Symposium on Point Based Graphics. 2004.URL:https://api.semanticscholar. org/CorpusID:2928987

    work page 2004

  53. [53]

    Žiga Virk.Rips complexes as nerves and a Functorial Dowker-Nerve Diagram. 2019. arXiv: 1906.04028 [math.MG].URL:http://arxiv.org/pdf/1906.04028v2

    work page arXiv 2019

  54. [54]

    SCANPY: large-scale single- cell gene expression data analysis

    F Alexander Wolf, Philipp Angerer, and Fabian J Theis. “SCANPY: large-scale single- cell gene expression data analysis”. In:Genome biology19.1 (2018), p. 15. A Algorithms This is the main algorithm that takes in a distance matrix, sorted, and returns a weighted graph representing a filtration of theCoNrv(M p)functor. The algorithm runtime is dom- inated b...

    work page 2018