Pith. sign in

REVIEW 7 minor 41 references

Lower Bounds for Approximating the Vietoris-Rips Filtration

T0 review · 0 major / 7 minor · reviewed 2026-07-08 · glm-5.2

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

Pith's one-line read VR approximations must blow up without geometry

desk verdict First explicit lower bounds on c-approximation size for VR(−); clean proofs, correct arguments, deserves a serious referee. read the letter →

arxiv 2607.06524 v1 pith:MWULPXBH submitted 2026-07-07 math.AT cs.CGmath.CO

Kenneth McCabe This is my paper · ORCID
classification math.ATcs.CGmath.CO MSC 55N3168U05
keywords Vietoris-RipsfiltrationapproximationlowerboundshomotopyinterleavingstopologicaldataanalysisflagcomplexesgirthTurangraphs
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

The paper proves that for any fixed approximation factor c, there exist infinite families of finite metric spaces where any c-approximation to the Vietoris-Rips filtration has at least superlinear size. For c in [1, sqrt(2)), the bound is exponential. The key mechanism is Lemma 3.1, which lower-bounds the size of any c-approximation by the rank of the structure map H_i(G)_r -> H_i(G)_{c^2 r}. The author then constructs metric spaces (from Turan graphs, generalized polygons, and Lazebnik-Ustimenko-Woldar graphs) where this rank is large, and extends the results to the intrinsic Cech filtration and to any bifiltration containing VR(-) as a slice.

What carries the argument

homotopy interleavings

What would settle it

Construct a finitely presented c-approximation to VR(X) for the specified metric spaces that achieves linear or sub-superlinear size, which would contradict the rank lower bound of Lemma 3.1.

Watch

Extended reading notes

Core claim

The central observation is that any finitely presented c-approximation F to a filtration G must have size at least the rank of the structure map H_i(G)_r -> H_i(G)_{c^2 r}. By constructing metric spaces from graphs with high girth and large first Betti number (Turan graphs T(3n,n) for exponential bounds, generalized polygon incidence graphs and LUW graphs CD(k,p) for superlinear bounds), the author shows this rank can be made exponentially or superlinearly large, establishing that linear-size approximations are impossible for arbitrary metric spaces at any fixed approximation factor.

Load-bearing premise

The superlinear lower bounds (Theorems 3.6 and 3.9) rely on Lemma 3.5, which uses a result from Adamaszek (2013) stating that if the 1-skeleton of VR(X)_1 has girth at least 3j+1, then the inclusion VR(X)_1 -> VR(X)_j is a homotopy equivalence. If that girth-to-stability result had hidden conditions, the superlinear bounds would fail. The exponential bound (Theorem 3.2) does not depend on this lemma.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

0 major / 7 minor

Summary. The paper proves lower bounds on the size of approximations to the Vietoris-Rips filtration VR(−) for arbitrary finite metric spaces, in the framework of homotopy interleavings. The central tool is Lemma 3.1, which lower-bounds the size of any finitely presented c-approximation by the rank of a structure map H_i(G)_r → H_i(G)_{c²r}. Two main results are established: (1) for c ∈ [1, √2), exponential lower bounds via Turán graphs T(3n, n) (Theorem 3.2), and (2) for any fixed c ≥ 1, superlinear lower bounds via incidence graphs of generalized polygons (Theorem 3.6, for c < √6) and Lazebnik–Ustimenko–Woldar graphs (Theorem 3.9, for all c ≥ 1). Both results extend to the intrinsic Čech filtration (Corollaries 3.10, 3.11) and to any bifiltration containing VR(−) as a 1-parameter slice (Corollaries 3.13, 3.14).

Significance. This paper makes a substantial contribution by providing the first explicit lower bounds on the size of c-approximations to VR(−) for arbitrary metric spaces. The exponential bound (Theorem 3.2) cleanly complements Sheehy's linear-size (1+ε)-approximations for bounded doubling dimension, showing that the geometric assumption is necessary. The superlinear bound (Theorem 3.9) is particularly notable: it shows that no fixed approximation factor yields linear-size approximations for arbitrary metric spaces, closing a natural question in the area. The extensions to the intrinsic Čech filtration and to bifiltrations (function-Rips, degree-Rips, subdivision-Rips) broaden the impact considerably. The proofs are clean and rely on standard, well-established tools (Künneth for joins, Alexander duality, Adamaszek's girth-to-stability result). The reliance on [1, Prop 2.2] for Lemma 3.5 is well-grounded: the girth conditions are verified correctly in all cases, and the exponential bound (Theorem 3.2) is independent of this lemma. The Turán graph construction for the exponential bound is well-motivated by the Beers–Botnan extremal result.

minor comments (7)
  1. §3.1, proof of Theorem 3.2: The claim that VR(X_n)_r = Cl(G_n) for r ∈ [1,2) should specify that this holds because G_n is the 1-skeleton at scale 1 and the metric is twice the shortest path metric, so no new edges appear until scale 2. This is implicit but making it explicit would aid the reader.
  2. §3.1, proof of Theorem 3.6: The table listing the three cases uses q^4, q^6, q^12 for dim H_1(G_q), but the text states |X_q| = Θ(q^3), Θ(q^5), Θ(q^11). The exponents for ϵ(c) follow from these, but a brief sentence explaining the computation (e.g., 'since |X_q| = Θ(q^n) and dim H_1 = Θ(q^{2n-2}), we get ϵ = (2n-2)/n - 1 = (n-2)/n') would make the derivation more transparent.
  3. §3.1, equation (1): The formula d(c) := k(c) − ⌊(k(c)+2)/4⌋ + 1 could use a brief explanation of its origin, or a forward reference to where it appears in [30], as its role is not immediately clear.
  4. §3.2, proof of Corollary 3.10: The ball computation B(x,2) = {x} ∪ (X_n ∖ P_j) is correct but the subsequent argument that ⋂_{x∈σ} B(x,2) ≠ ∅ iff |σ ∩ P_j| ≤ 1 for some j could be stated more carefully; the current phrasing conflates 'there exists j' with 'for a specific j.'
  5. §2.4: The definition of c-interleaving via the category I_c is standard but slightly non-standard in that it uses [0,∞) × {0,1} rather than the more common R × {0,1}. A remark noting that this is equivalent for filtrations indexed by [0,∞) would help readers familiar with the [8] formulation.
  6. Figure 2: The graph G_3 is labeled but the caption could note that this is T(9,3), connecting it to the Turán graph terminology used in Remark 3.3.
  7. References [25] and [31]: Both are dated 2026, which appears to be a forward-dating issue; the authors should verify these are correct.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for a careful reading and for the positive assessment of the paper's contributions. The referee's report recommends minor revision but does not raise any specific major or minor comments requiring changes to the manuscript. We are grateful for the referee's thorough summary of the results and the accurate characterization of the paper's significance. We have reviewed the manuscript in light of the referee's remarks and confirm that the girth conditions, the reliance on [1, Proposition 2.2], and the independence of Theorem 3.2 from Lemma 3.5 are all correctly verified as the referee notes. No revisions to the mathematical content are needed. We will conduct a final proofreading pass to address any typographical issues before the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; derivation is self-contained with appropriate external citations

full rationale

The paper's derivation chain is clean and self-contained. Lemma 3.1 (the central tool) is proved in full via a straightforward diagram chase through the interleaving category I_c; the proof does not depend on any unverified self-citation. The size definition is attributed to [32] (Lesnick-McCabe) but is a standard notion recovering simplex counts for filtrations, and the paper re-derives what it needs. Theorem 3.2 (exponential bound) uses only Lemma 3.1 plus the observation that VR(X_n)_1 = VR(X_n)_{c²} = A_n when c² ∈ [1,2) and distances are even integers—a genuine mathematical fact, not a definitional identity. Lemma 3.5, the other load-bearing ingredient for the superlinear bounds (Theorems 3.6 and 3.9), cites [1, Proposition 2.2] by Adamaszek (Israel J. Math, 2013), who is not an author of this paper; this is an external, peer-reviewed result. The graph-theoretic ingredients (Turán graphs from [5], generalized polygons from [35], LUW graphs from [30]) are all externally sourced. The Čech extensions (Corollaries 3.10–3.11) use standard Alexander duality and the classical VR/Čech interleaving. The bifiltration extension (Lemma 3.12) is a direct restriction argument proved in full. No step reduces to its inputs by construction, and no self-citation is load-bearing in a circular sense.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No new entities, particles, forces, or postulated objects are introduced. The paper works entirely with standard mathematical objects (graphs, simplicial complexes, persistence modules) and externally defined graph families.

assumptions (7)
  • standard math Künneth formula for reduced homology of simplicial joins (Lemma 2.1, cited from [36])
    Used in Theorem 3.2 to compute H̃_{n-1}(D_3^{*n}) ≅ k^{2^n} by induction.
  • standard math Alexander duality for simplicial complexes (Lemma 2.2, cited from [7])
    Used in Corollary 3.10 to relate H̃_i(I(X_n)_2) to H̃_{|X_n|-i-3}(A_n).
  • domain assumption Girth ≥ 3j+1 implies VR(X)_1 ↪ VR(X)_j is a homotopy equivalence (Lemma 3.5, cited from [1, Prop 2.2])
    Load-bearing for Theorems 3.6 and 3.9; ensures the structure map H_1(VR(X))_1 → H_1(VR(X))_{c²} is an isomorphism.
  • domain assumption Existence and properties of generalized polygons (Proposition 3.4, cited from [35])
    Provides the graph families with required girth and vertex counts for Theorem 3.6.
  • domain assumption Properties of LUW graphs CD(k,p): connected, bipartite, p-regular, girth ≥ k+5, |V| ≤ 2p^{d(c)} (Lemma 3.8, cited from [30, Theorem 3.2])
    Provides the graph family with superlinear edge density and sufficient girth for Theorem 3.9.
  • standard math Feit–Higman theorem: finite generalized n-gons exist only for n ∈ {2,3,4,6,8} (Theorem 3.7, cited from [26])
    Explains why generalized polygons cannot handle c ≥ √6, motivating the switch to LUW graphs.
  • standard math Existence and uniqueness of minimal presentations for persistence modules over products of totally ordered sets (Section 2.2, cited from [33])
    Underpins the definition of size via Betti numbers of chain complexes.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Lower Bounds for Approximating the Vietoris-Rips Filtration." pith.science (2026). https://pith.science/paper/MWULPXBH

@misc{pith2026260706524,
  author       = {Pith},
  title        = {Pith review of: Lower Bounds for Approximating the Vietoris-Rips Filtration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MWULPXBH}},
  note         = {Machine review of arXiv:2607.06524}
}
abstract

The Vietoris-Rips filtration $\mathcal{VR}(-)$ is a standard tool for analyzing the shape of data within topological data analysis. Beginning with seminal work of Sheehy, a substantial amount of research has centered on constructing linear-size sparse approximations to $\mathcal{VR}(-)$ and related filtrations for metric spaces of bounded doubling dimension. We show that this geometric assumption is necessary in a precise sense. Working in the framework of homotopy interleavings, we show that for any fixed $c \in [1, \sqrt{2})$, there exists a family of finite metric spaces for which any finitely presented $c$-approximation to $\mathcal{VR}(-)$ has exponential size. We also show that for any fixed $c \geq 1$, there exists a family of finite metric spaces for which any finitely presented $c$-approximation to $\mathcal{VR}(-)$ has superlinear size, yielding an obstruction to linear-size approximations for any fixed approximation factor. Both results extend to the intrinsic \v{C}ech filtration and to any bifiltration containing $\mathcal{VR}(-)$ as a $1$-parameter slice, including the function-Rips, degree-Rips, and subdivision-Rips bifiltrations.

Figures

Figures reproduced from arXiv: 2607.06524 by the authors.

Figure 1
Figure 1. A simplicial complex (left) and its Alexander dual (right). say M is finitely presented if F0 and F1 can be chosen to be finitely generated. A presentation is minimal if the ranks of F0 and F1 are as small as possible; for P = T1 × · · · × Tn with each Ti totally ordered, a minimal presentation exists and is unique up to isomorphism [33]. We write β0(M) and β1(M) for the ranks of F0 and F1 in a minimal presentation,… view at source ↗
Figure 2
Figure 2. The graph G3. 3.1. Lower Bounds for the Vietoris-Rips Filtration. We are now ready to present our lower bounds for approximations to the Vietoris-Rips filtration. Theorem 3.2. For every c ∈ [1, √ 2), there exists an infinite family of finite metric spaces {Xn} such that any finitely presented c-approximation to VR(Xn) has size at least 2 |Xn|/3 . Proof. For n ≥ 2, let Gn be the complete n-partite graph with n parts … view at source ↗
Figure 3
Figure 3. The incidence graph of the generalized quadrangle of order (2, 2). The blue vertices correspond to points, while the red vertices correspond to lines. By [1, Proposition 2.2], the assumption g(H) ≥ 3j + 1 implies that the inclusion Cl(H) ,→ Cl(Hj ) is a homotopy equivalence. □ We now use Lemma 3.5 to give superlinear lower bounds on approximations to the Vietoris-Rips filtration. The metric spaces we will use for th… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

  1. [1]

    Clique complexes and graph powers.Israel Journal of Mathematics, 196:295–319, 2013

    Micha l Adamaszek. Clique complexes and graph powers.Israel Journal of Mathematics, 196:295–319, 2013. doi:10.1007/s11856-012-0166-1

    work page doi:10.1007/s11856-012-0166-1 2013

  2. [2]

    Extremal problems related to Betti numbers of flag complexes.Discrete Applied Mathematics, 173:8–15, 2014

    Micha l Adamaszek. Extremal problems related to Betti numbers of flag complexes.Discrete Applied Mathematics, 173:8–15, 2014. doi:10.1016/j.dam.2014.04.006

    work page doi:10.1016/j.dam.2014.04.006 2014

  3. [3]

    Lower bounds on the homology of Vietoris– Rips complexes of hypercube graphs.Bulletin of the Malaysian Mathematical Sciences Society, 47(3):72, 2024

    Henry Adams and ˇZiga Virk. Lower bounds on the homology of Vietoris– Rips complexes of hypercube graphs.Bulletin of the Malaysian Mathematical Sciences Society, 47(3):72, 2024. doi:10.1007/s40840-024-01663-x

    work page doi:10.1007/s40840-024-01663-x 2024

  4. [4]

    A sparse multicover bifiltration of linear size

    ´Angel Javier Alonso. A sparse multicover bifiltration of linear size. In41st International Symposium on Computational Geometry (SoCG 2025), volume 332 ofLeibniz International Proceedings in Informatics (LIPIcs), pages 6:1– 6:18, 2025. doi:10.4230/LIPIcs.SoCG.2025.6

    work page doi:10.4230/lipics.socg.2025.6 2025

  5. [5]

    Extremal Betti numbers and persis- tence in flag complexes

    Lies Beers and Magnus Bakke Botnan. Extremal Betti numbers and persis- tence in flag complexes. In41st International Symposium on Computational Geometry (SoCG 2025), volume 332 ofLeibniz International Proceedings in In- formatics (LIPIcs), pages 14:1–14:18, 2025. doi:10.4230/LIPIcs.SoCG.2025.14

    work page doi:10.4230/lipics.socg.2025.14 2025

  6. [6]

    Extremal Betti Numbers and Persistence in Flag Complexes

    Lies Beers and Magnus Bakke Botnan. Extremal Betti numbers and persistence in flag complexes, 2025. arXiv preprint, doi:10.48550/arXiv.2502.21294

    work page Pith review arXiv doi:10.48550/arxiv.2502.21294 2025

  7. [7]

    Note: Combinatorial Alexander duality— a short and elementary proof.Discrete & Computational Geometry, 42(4): 586–593, 2009

    Anders Bj¨ orner and Martin Tancer. Note: Combinatorial Alexander duality— a short and elementary proof.Discrete & Computational Geometry, 42(4): 586–593, 2009. doi:10.1007/s00454-008-9102-x

    work page doi:10.1007/s00454-008-9102-x 2009

  8. [8]

    Blumberg and Michael Lesnick

    Andrew J. Blumberg and Michael Lesnick. Universality of the homotopy in- terleaving distance.Transactions of the American Mathematical Society, 376 (12):8269–8307, 2023. doi:10.1090/tran/8738

    work page doi:10.1090/tran/8738 2023

Show all 41 references
  1. [9]

    Blumberg and Michael Lesnick

    Andrew J. Blumberg and Michael Lesnick. Stability of 2-parameter persistent homology.Foundations of Computational Mathematics, 24(2):385–427, 2024. doi:10.1007/s10208-022-09576-6

    2024 doi

  2. [10]

    An introduction to multipa- rameter persistence

    Magnus Bakke Botnan and Michael Lesnick. An introduction to multipa- rameter persistence. In Aslak Bakke Buan, Henning Krause, and Øyvind Solberg, editors,Representations of Algebras and Related Structures, pages 77–150. EMS Press, 2023. doi:10.4171/ecr/19/4

    2023 doi

  3. [11]

    Approximating persistent homol- ogy in Euclidean space through collapses.Applicable Algebra in Engineering, 14 KENNETH MCCABE Communication and Computing, 26(1–2):73–101, 2015

    Magnus Bakke Botnan and Gard Spreemann. Approximating persistent homol- ogy in Euclidean space through collapses.Applicable Algebra in Engineering, 14 KENNETH MCCABE Communication and Computing, 26(1–2):73–101, 2015. doi:10.1007/s00200- 014-0247-y

    2015 doi

  4. [12]

    Sparse Dowker nerves.Journal of Applied and Computational Topology, 3(1–2):1–28, 2019

    Morten Brun and Nello Blaser. Sparse Dowker nerves.Journal of Applied and Computational Topology, 3(1–2):1–28, 2019. doi:10.1007/s41468-019-00028-9

    2019 doi

  5. [13]

    Oudot, and Donald R

    Micka¨ el Buchet, Fr´ ed´ eric Chazal, Steve Y. Oudot, and Donald R. Sheehy. Effi- cient and robust persistent homology for measures.Computational Geometry, 58:70–96, 2016. doi:10.1016/j.comgeo.2016.07.001

    2016 doi

  6. [14]

    Dornelas, and Michael Kerber

    Micka¨ el Buchet, Bianca B. Dornelas, and Michael Kerber. Sparse higher order ˇ cech filtrations. In39th International Symposium on Computational Geometry (SoCG 2023), volume 258 ofLeibniz International Proceedings in Informatics (LIPIcs), pages 20:1–20:17, 2023. doi:10.4230/L...

    2023 doi

  7. [15]

    The theory of multidimen- sional persistence.Discrete & Computational Geometry, 42(1):71–93, 2009

    Gunnar Carlsson and Afra Zomorodian. The theory of multidimen- sional persistence.Discrete & Computational Geometry, 42(1):71–93, 2009. doi:10.1007/s00454-009-9176-0

    2009 doi

  8. [16]

    Cavanna, Mahmoodreza Jahanseir, and Donald R

    Nicholas J. Cavanna, Mahmoodreza Jahanseir, and Donald R. Sheehy. A geo- metric perspective on sparse filtrations. InProceedings of the 27th Cana- dian Conference on Computational Geometry, pages 116–121, 2015. URL https://cccg.ca/proceedings/2015/01.pdf

    2015

  9. [17]

    Persistence stability for geo- metric complexes.Geometriae Dedicata, 173(1), 2014

    Fr´ ed´ eric Chazal, Vin de Silva, and Steve Oudot. Persistence stability for geo- metric complexes.Geometriae Dedicata, 173(1), 2014. doi:10.1007/s10711- 013-9937-z

    2014 doi

  10. [18]

    Improved topo- logical approximations by digitization

    Aruni Choudhary, Michael Kerber, and Sharath Raghvendra. Improved topo- logical approximations by digitization. InProceedings of the Thirtieth An- nual ACM-SIAM Symposium on Discrete Algorithms, pages 2675–2688, 2019. doi:10.1137/1.9781611975482.166

    2019 doi

  11. [19]

    Polynomial-sized topological approximations using the permutahedron.Discrete & Computa- tional Geometry, 61(1):42–80, 2019

    Aruni Choudhary, Michael Kerber, and Sharath Raghvendra. Polynomial-sized topological approximations using the permutahedron.Discrete & Computa- tional Geometry, 61(1):42–80, 2019. doi:10.1007/s00454-017-9951-2

    2019 doi

  12. [20]

    Improved approximate Rips filtrations with shifted integer lattices and cubical com- plexes.Journal of Applied and Computational Topology, 5(3):425–458, 2021

    Aruni Choudhary, Michael Kerber, and Sharath Raghvendra. Improved approximate Rips filtrations with shifted integer lattices and cubical com- plexes.Journal of Applied and Computational Topology, 5(3):425–458, 2021. doi:10.1007/s41468-021-00072-4

    2021 doi

  13. [21]

    Computing the multicover bifiltration.Discrete & Computational Geometry, 70(2):376– 405, 2023

    Ren´ e Corbet, Michael Kerber, Michael Lesnick, and Georg Osang. Computing the multicover bifiltration.Discrete & Computational Geometry, 70(2):376– 405, 2023. doi:10.1007/s00454-022-00476-8

    2023 doi

  14. [22]

    Dey, Fengtao Fan, and Yusu Wang

    Tamal K. Dey, Fengtao Fan, and Yusu Wang. Computing topological persis- tence for simplicial maps. InProceedings of the 30th Annual Symposium on Computational Geometry, pages 345–354, 2014. doi:10.1145/2582112.2582165

    2014 doi

  15. [23]

    Dey, Dayu Shi, and Yusu Wang

    Tamal K. Dey, Dayu Shi, and Yusu Wang. SimBa: An efficient tool for approx- imating Rips-filtration persistence via simplicial batch-collapse.ACM Journal of Experimental Algorithmics, 24:1–16, 2019. doi:10.1145/3284360

    2019 doi

  16. [24]

    The multi-cover persistence of Eu- clidean balls.Discrete & Computational Geometry, 65(4):1296–1313, 2021

    Herbert Edelsbrunner and Georg Osang. The multi-cover persistence of Eu- clidean balls.Discrete & Computational Geometry, 65(4):1296–1313, 2021. doi:10.1007/s00454-021-00281-9

    2021 doi

  17. [25]

    Maximum persis- tent Betti numbers of ˇ cech complexes.Journal of Applied and Computational Topology, 10(1):5, 2026

    Herbert Edelsbrunner, Matthew Kahle, and Shu Kanazawa. Maximum persis- tent Betti numbers of ˇ cech complexes.Journal of Applied and Computational Topology, 10(1):5, 2026. doi:10.1007/s41468-026-00233-3. LOWER BOUNDS FOR APPROXIMATING THE VIETORIS-RIPS FILTRATION 15

    2026 doi

  18. [26]

    The nonexistence of certain generalized poly- gons.Journal of Algebra, 1(2):114–131, 1964

    Walter Feit and Graham Higman. The nonexistence of certain generalized poly- gons.Journal of Algebra, 1(2):114–131, 1964. doi:10.1016/0021-8693(64)90028- 6

    1964 doi

  19. [27]

    Extremal Betti numbers of Vietoris–Rips complexes.Discrete & Computational Geometry, 46(1):132–155, 2011

    Michael Goff. Extremal Betti numbers of Vietoris–Rips complexes.Discrete & Computational Geometry, 46(1):132–155, 2011. doi:10.1007/s00454-010-9274- z

    2011 doi

  20. [28]

    Density sensitive bifiltered Dowker com- plexes via total weight, 2024

    Niklas Hellmer and Jan Spali´ nski. Density sensitive bifiltered Dowker com- plexes via total weight, 2024. arXiv preprint, doi:10.48550/arXiv.2405.15592

    2024 doi arXiv

  21. [29]

    Barcodes of towers and a streaming algorithm for persistent homology.Discrete & Computational Geometry, 61 (4):852–879, 2019

    Michael Kerber and Hannah Schreiber. Barcodes of towers and a streaming algorithm for persistent homology.Discrete & Computational Geometry, 61 (4):852–879, 2019. doi:10.1007/s00454-018-0030-0

    2019 doi

  22. [30]

    Ustimenko, and Andrew J

    Felix Lazebnik, Vladimir A. Ustimenko, and Andrew J. Woldar. A new series of dense graphs of high girth.Bulletin of the American Mathematical Society, 32(1):73–79, 1995. doi:10.1090/S0273-0979-1995-00569-0

    1995 doi

  23. [31]

    It’s all about covers: Persistent homology of cover refinements,

    Ant´ onio Leit˜ ao. It’s all about covers: Persistent homology of cover refinements,

  24. [32]

    arXiv preprint, doi:10.48550/arXiv.2602.22784

    doi arXiv

  25. [33]

    Nerve models of subdivision bifiltra- tions, 2024

    Michael Lesnick and Kenneth McCabe. Nerve models of subdivision bifiltra- tions, 2024. arXiv preprint, doi:10.48550/arXiv.2406.07679

    2024 doi arXiv

  26. [34]

    Sparse approximation of the subdivision-Rips bifiltration for doubling metrics, 2024

    Michael Lesnick and Kenneth McCabe. Sparse approximation of the subdivision-Rips bifiltration for doubling metrics, 2024. arXiv preprint, doi:10.48550/arXiv.2408.16716

    2024 doi arXiv

  27. [35]

    Interactive visualization of 2-D persis- tence modules, 2015

    Michael Lesnick and Matthew Wright. Interactive visualization of 2-D persis- tence modules, 2015. arXiv preprint, doi:10.48550/arXiv.1512.00180

    2015 doi arXiv

  28. [36]

    Springer Basel, Basel, 1998

    Hendrik Van Maldeghem.Generalized Polygons. Springer Basel, Basel, 1998. doi:10.1007/978-3-0348-0271-0

    1998 doi

  29. [37]

    John W. Milnor. Construction of universal bundles, ii.Annals of Mathematics, 63(3):430–436, 1956. doi:10.2307/1970012

    1956 doi

  30. [38]

    Donald R. Sheehy. A multicover nerve for geometric inference. InProceedings of the 24th Canadian Conference on Computational Geometry, pages 309–314,

  31. [39]

    URLhttp://2012.cccg.ca/papers/paper52.pdf

    2012

  32. [40]

    Donald R. Sheehy. Linear-size approximations to the Vietoris–Rips filtration.Discrete & Computational Geometry, 49(4):778–796, 2013. doi:10.1007/s00454-013-9513-1

    2013 doi

  33. [41]

    Donald R. Sheehy. A sparse Delaunay filtration. In37th International Sym- posium on Computational Geometry (SoCG 2021), volume 189 ofLeibniz International Proceedings in Informatics (LIPIcs), pages 58:1–58:16, 2021. doi:10.4230/LIPIcs.SoCG.2021.58. Department of Mathematics, N...

    2021 doi

Pith tools

Reviewed July 8, 2026 · model on record in the stance chip above.