Pith. sign in

REVIEW 2 major objections 6 minor 66 references

Interleaving distance on Reeb cosheaves of proximity graphs yields object-level confidence regions for the unknown Reeb graph of a filtered space from finite samples.

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

2026-07-10 07:08 UTC pith:GUMIVSHF

load-bearing objection Solid object-level Reeb confidence theory with a clean PL-vs-Mapper split; experiments are honest about their weaker calibration. the 2 major comments →

arxiv 2607.08458 v1 pith:GUMIVSHF submitted 2026-07-09 math.ST math.ATstat.TH

Building confidence regions for Reeb graphs using the interleaving distance

Matteo Pegoraro , Alberto Conforti , Mathieu Carri\`ere This is my paper
classification math.ST math.ATstat.TH MSC 62R4055N3162G15
keywords Reeb graphsinterleaving distanceconfidence regionsMapper graphsconstructible cosheavesextended persistencepositive reachPL proximity graphs
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.

The paper asks what can be said about the Reeb graph of an unknown filtered space when one only has a finite sample. It builds a proximity graph on the sample, extends the observed filter values piecewise-linearly, and treats the Reeb cosheaf of that finite filtered graph as the primary estimator. Stability theorems bound the interleaving distance between this estimator and the true Reeb cosheaf by a scale controlled by the sample covering radius (intrinsic metric always; Euclidean metric under positive reach). Probabilistic control of that covering radius—via (a,b)-standard assumptions or Hausdorff-scale subsampling—then produces explicit interleaving balls that cover the unknown Reeb graph with prescribed probability. Mapper graphs appear only afterward as cover-based coarsenings whose extra error is the cover resolution, so statistical claims stay at the PL level while visualization stays readable. A comparison theorem shows that the extended-persistence pseudometric is at most twice the interleaving distance (sharp constant 1 for the H0 components), so the same regions also control ordinary persistence summaries.

Core claim

Under covering-radius control of a finite sample, the Reeb cosheaf of the PL proximity graph on that sample is close in interleaving distance to the true Reeb cosheaf of the filtered space; the resulting interleaving balls are therefore valid confidence regions for the unknown Reeb graph, and they dominate the usual extended-persistence bottleneck pseudometric by a factor of two.

What carries the argument

The PL-Reeb estimator R^{S_n,\rho}_\delta (Reeb cosheaf of the geometric realization of the proximity graph with linear filter extension) together with the interleaving distance d_I on constructible cosheaves; stability maps it to the true Reeb cosheaf at scale \mu_\delta (intrinsic) or \omega_f(\eta_\tau(\delta)) (extrinsic, positive reach).

Load-bearing premise

Extrinsic bounds and several confidence conversions require the underlying support to have positive reach and the Euclidean covering radius to stay below that reach; without positive reach the Euclidean-to-intrinsic distortion control fails.

What would settle it

On a compact positive-reach manifold with known Reeb graph, draw large i.i.d. samples, build the PL estimator at the theoretically prescribed scale, and check whether the true Reeb cosheaf lies inside the claimed interleaving ball at the nominal rate; systematic failure of coverage falsifies the confidence claim.

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

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

2 major / 6 minor

Summary. The paper constructs interleaving-distance confidence regions for Reeb graphs from finite samples. The primary estimator is the Reeb cosheaf of a piecewise-linear filtered proximity graph on the sample (intrinsic or Euclidean); Mapper graphs are treated as controlled cover coarsenings used for visualization. The authors prove a comparison d_Δ ≤ 2 d_I between the extended-persistence pseudometric and the interleaving distance (constant 1 for Ord0, Ext0, Rel1), stability of the PL–Reeb estimators under covering-radius control (with positive-reach distortion for the extrinsic case), and transfer of Hausdorff-scale probabilistic bounds into interleaving balls via (a,b)-standard tails or Fasy-type subsampling. Numerical experiments on a torus and an ant mesh illustrate feature selection on the PL object and pushforward to Mapper graphs.

Significance. If the results hold as stated, the paper supplies a genuine object-level alternative to persistence-based Mapper confidence sets: interleaving balls separate non-isomorphic Reeb graphs, and the PL-versus-Mapper split cleanly separates statistical approximation from visualization. The comparison theorem links this geometry to extended persistence with sharp H0 constants, and the stability theorems give explicit, usable radii under standard geometric hypotheses. Strengths include detailed proofs (including the Ext1 Hall-rank argument and density reduction in Appendix C), an explicit comparison with CMO18 and BBS25 in Appendix A, and open disclosure that rate-corrected subsampling radii and unsplit (a,b) plug-ins are not theorem-level certificates (Remarks 5.1, 6.1). The contribution is solid for the TDA–statistics interface and advances beyond bottleneck-only guarantees.

major comments (2)
  1. Theorem 3.1 asserts that the constant 2 is sharp, but the text only gives a heuristic (a loop of persistence 4ε killed by 2ε-smoothing yields diagonal bottleneck cost 2ε). An explicit pair of Reeb graphs with d_Δ / d_I arbitrarily close to 2 (or a proof that no smaller universal constant works) is needed to justify the sharpness claim; otherwise rephrase as “the factor 2 is forced by the two-sided smoothing argument and is not an artifact of the proof technique.”
  2. Section 6 and Remark 6.1: the experiments label features “significant” under unsplit (a,b) plug-in estimation and empirically rate-corrected subsampling radii whose finite-sample coverage is not proved (Remark 5.1, Section B.4). The abstract and conclusions still present these as identifying statistically significant features. For a statistics journal, either (i) restrict the experimental claims to “features outside the calibrated band under the disclosed heuristic calibration,” with no 1−α language, or (ii) add a formal coverage result / split-sample protocol for the procedures actually used. The deterministic transfer from a valid Hausdorff radius to d_I and d_Δ is fine; the load-bearing gap is the data-driven radius itself.
minor comments (6)
  1. Figure 1 caption: “Matching colors identify the corresponding topological features… which must be inferred using the elder rule” is dense; a short pointer to which bars are Ord0/Ext0/Rel1/Ext1 would help non-specialists.
  2. Notation: R is used both for the Reeb graph and for the real line (e.g. covers of R). A consistent distinction (e.g. mathbb{R} vs script R) would reduce ambiguity in Sections 2–4.
  3. Corollary 5.3–5.4: the O((b_n/n)^{1/4}) remainder is inherited from Fasy et al.; stating the sample-size regime in which the authors treat the remainder as negligible in the experiments would help practitioners.
  4. Appendix A comparison with CMO18 is valuable; a one-line summary table of leading constants (PL vs Mapper vs MultiNerve) would make the improvement at the PL level easier to scan.
  5. Typos / style: “Carri` ere”, “Universit` a”, “Cˆ ote” appear with broken accents in the source; fix for the journal version. Also “dI-confidence” vs “d_I-confidence” inconsistently.
  6. Section 4.1: θ_f and μ_δ are well-defined, but a short remark that μ_δ is computable only if ω_f and geodesic interpolants are known (or bounded) would clarify the gap between theory and practice for non-Lipschitz filters.

Circularity Check

0 steps flagged

No significant circularity: stability, d_Δ–d_I comparison, and confidence regions are derived from covering radii and cosheaf interleavings, not from fitted targets or load-bearing self-citation.

full rationale

The load-bearing claims are Theorem 3.1 (d_Δ ≤ 2 d_I with sharp H0 constants), Theorems 4.1–4.2 (PL–Reeb interleaving stability under intrinsic/extrinsic covering-radius control), and Corollaries 5.1–5.4 (Hausdorff tails → interleaving balls). Each is proved from the definitions of constructible cosheaves, ε-interleavings, moduli of continuity, and (for extrinsic bounds) positive-reach distortion; the Ext1 argument uses a standard density reduction via functional distortion (Prop. C.2) that does not import the target inequality. Overlapping-author citations (e.g. CMO18) appear as related-work comparison and experimental baseline, not as uniqueness theorems or ansätze that force the new bounds. Numerical scale choices (unsplit (a,b) plug-in, rate-corrected subsampling) are explicitly labeled empirical calibrations (Remarks 5.1, 6.1) rather than first-principles predictions. Nothing reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 2 invented entities

Theory rests on standard constructible R-space/cosheaf foundations, positive-reach metric distortion, and classical sampling tail bounds. Free parameters appear only in experimental scale selection (pilot C, confidence level, subsample rule, cover length χ=2δ, Isomap k=10). No new physical entities; invented objects are mathematical estimators (PL-Reeb cosheaf, Mapper coarsening of that cosheaf).

free parameters (6)
  • pilot constant C for (a,b) local-mass radius r_n = 2
    Set to C=2 in experiments for the pilot radius r_n = C (log n / n)^{1/(b+2)}; affects â and thus δ_α.
  • confidence level 1-α = 0.85
    Fixed at 0.85 throughout numerical experiments; not derived from data but chosen for display.
  • subsample size rule b_n = floor(n / (log n)^{1.001}) = power 1.001
    Hand-chosen logarithmic power 1.001 in the Fasy-style subsampling construction; controls raw Hausdorff quantile.
  • Mapper cover length χ and overlap = χ=2δ; overlap 0.45
    χ set to 2δ (statistical runs) or hand-tuned values; overlap fraction 0.45; directly adds to Mapper confidence radius.
  • Isomap nearest-neighbor count for intrinsic metric = 10
    Intrinsic distances approximated by 10-NN graph shortest paths; changes graph edges and estimated covering scales.
  • rate-correction intrinsic dimension d = 2 (displayed experiments)
    Used in δ_log and δ_pow corrections; treated as known (d=2 for torus/ant displays).
axioms (6)
  • standard math Constructible R-spaces and Reeb–cosheaf equivalence (de Silva–Munch–Patel) so d_I is a metric up to isomorphism on Reeb graphs.
    Background Section 2.1; foundation for object-level confidence balls.
  • domain assumption Positive reach of the support X implies Euclidean–intrinsic distortion d_X(x,y) ≤ 2τ arcsin(‖x−y‖/(2τ)) for ‖x−y‖<2τ (Boissonnat–Lieutier–Wintraecken).
    Theorem 2.1 / Corollary 2.1; load-bearing for extrinsic PL stability and Euclidean-to-intrinsic scale conversion.
  • domain assumption (a,b)-standard lower mass bounds on metric balls yield exponential Hausdorff covering tails (Chazal et al.).
    Theorem 5.1; used for deterministic-scale confidence regions in Section 5.1.
  • domain assumption Fasy et al. subsampling quantile for Euclidean Hausdorff distance on compact manifolds without boundary, positive reach, noiseless sampling on M.
    Section 5.2 standing hypotheses; restricts the theorem-level subsampling guarantees.
  • domain assumption Filter f continuous with modulus of continuity ω_f w.r.t. intrinsic distance; (X,f) constructible.
    Section 4.1 setting; enters μ_δ and all stability radii.
  • ad hoc to paper Rate-corrected radii δ_log, δ_pow are valid finite-sample calibrations of the raw Fasy radius.
    Remark 5.1 explicitly leaves theoretical analysis to future work; used for main experimental figures.
invented entities (2)
  • PL-Reeb estimator R^{S_n,ρ}_δ (Reeb cosheaf of proximity graph with PL filter extension) independent evidence
    purpose: Primary finite-sample estimator carrying object-level interleaving stability and confidence statements.
    Defined in Section 4.2; standard construction ingredients but packaged as the inference object distinct from Mapper.
  • Mapper coarsening M_U of the PL-Reeb cosheaf with multivalued pushforward Ξ_I independent evidence
    purpose: Visualization layer with explicit resolution error bound, separate from statistical approximation.
    Section 4.4; builds on Bro+21 Mapper transformation but applied to the PL estimator.

pith-pipeline@v1.1.0-grok45 · 59391 in / 4058 out tokens · 40253 ms · 2026-07-10T07:08:17.142504+00:00 · methodology

0 comments
read the original abstract

We develop confidence regions for Reeb graphs from finite samples using the interleaving distance. Given a point cloud equipped with a filter function, we construct a finite proximity graph, extend the filter linearly, and use the Reeb cosheaf of the resulting filtered graph as the primary estimator. Mapper graphs are then treated as controlled cover-based coarsenings of this estimator, separating the statistical approximation problem from the visualization problem. We prove stability bounds for the Reeb estimators obtained both using intrinsic and extrinsic metrics, the latter under positive-reach assumptions, and derive interleaving-distance confidence regions from either \((a,b)\)-standard sampling assumptions or subsampling-based Hausdorff scale estimates. We also compare this object-level metric viewpoint with persistence-based guarantees by showing that the extended-persistence pseudometric is bounded by twice the interleaving distance, with sharp constant \(1\) for the \(H_0\)-related components. Numerical experiments illustrate how statistically significant features can be identified and then projected to Mapper graphs for interpretation.

Figures

Figures reproduced from arXiv: 2607.08458 by Alberto Conforti, Mathieu Carri\`ere, Matteo Pegoraro.

Figure 1
Figure 1. Figure 1: Two non-isomorphic Reeb graphs with identical barcode representations. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Reeb graph of a torus of genus 2: the function used to compute the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A visual representation of the map ι≤t involved in Lemma 3.1. 3.1 Smoothing and persistence modules We first record the precise smoothing facts used below. Let R be a Reeb graph and let δ ≥ 0. Recall that Uδ(R) = D [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Schematic illustrations for the proofs of the stability theorems. [PITH_FULL_IMAGE:figures/full_fig_p046_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Point clouds used in the experiments, coloured by the height filter. The [PITH_FULL_IMAGE:figures/full_fig_p053_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Hand-tuned extrinsic torus. Both signatures display the dominant ex [PITH_FULL_IMAGE:figures/full_fig_p054_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Hand-tuned extrinsic torus. Left: a representative of the dominant Ext [PITH_FULL_IMAGE:figures/full_fig_p054_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Hand-tuned intrinsic ant. The highlighted support in the PL graph rep [PITH_FULL_IMAGE:figures/full_fig_p055_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Hand-tuned ant, relative one-dimensional features on the PL graph and [PITH_FULL_IMAGE:figures/full_fig_p055_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Torus, (a, b)-standard scale selection. The panels compare Mapper and PL confidence bands for the Euclidean (Ex.) and intrinsic (Int.) metrics. The dom￾inant Ext1 feature is separated from the diagonal in all settings, but the separation is much clearer in the PL diagrams than in the Mapper diagrams, and clearer for the intrinsic metric than for the Euclidean metric. (a) Int. Mapper. (b) Int. PL. (c) Int.… view at source ↗
Figure 11
Figure 11. Figure 11: Ant, (a, b)-standard scale selection. Compared with [PITH_FULL_IMAGE:figures/full_fig_p057_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Torus, subsampling with the no-log correction. The dominant Ext [PITH_FULL_IMAGE:figures/full_fig_p059_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Ant, subsampling with the no-log rate correction. Both PL graphs [PITH_FULL_IMAGE:figures/full_fig_p059_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Mean total runtime over ten repetitions. The vertical axis is logarithmic [PITH_FULL_IMAGE:figures/full_fig_p066_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Local downward branch-sliding move. One downward branch [PITH_FULL_IMAGE:figures/full_fig_p076_15.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

66 extracted references · 66 canonical work pages

  1. [1]

    arXiv preprint arXiv:2512.08159 , year=

    Reeb Graph of Sample Thickenings , author=. arXiv preprint arXiv:2512.08159 , year=

    work page arXiv

  2. [2]

    2025 , month=

    Conforti, Alberto and Micheletti,Alessandra , title =. 2025 , month=

    work page 2025

  3. [3]

    Acta Mathematica , year =

    Janson, Svante , title =. Acta Mathematica , year =

    work page

  4. [4]

    Probability Theory and Related Fields , volume=

    Random Euclidean coverage from within , author=. Probability Theory and Related Fields , volume=. 2023 , publisher=

    work page 2023

  5. [5]

    Diestel, Reinhard , title =

    work page

  6. [6]

    Journal of Computational Geometry , volume =

    Bauer, Ulrich and Lesnick, Michael , title =. Journal of Computational Geometry , volume =. 2015 , doi =

    work page 2015

  7. [7]

    On the Stability of Interval Decomposable Persistence Modules , journal =

    Bakke Bjerkevik, H. On the Stability of Interval Decomposable Persistence Modules , journal =. 2021 , doi =

    work page 2021

  8. [8]

    Computing in Geometry and Topology , volume =

    Alharbi, Rehab and Chambers, Erin Wolf and Munch, Elizabeth , title =. Computing in Geometry and Topology , volume =. 2024 , url =

    work page 2024

  9. [9]

    Computing in Geometry and Topology , volume =

    Realizable Piecewise Linear Paths of Persistence Diagrams with Reeb Graphs , author =. Computing in Geometry and Topology , volume =. 2024 , doi =

    work page 2024

  10. [10]

    Proceedings of the 30th Annual Symposium on Computational Geometry , series =

    Ulrich Bauer and Xiaoyin Ge and Yusu Wang , title =. Proceedings of the 30th Annual Symposium on Computational Geometry , series =. 2014 , doi =

    work page 2014

  11. [11]

    31st International Symposium on Computational Geometry (SoCG 2015) , series =

    Ulrich Bauer and Elizabeth Munch and Yusu Wang , title =. 31st International Symposium on Computational Geometry (SoCG 2015) , series =. 2015 , doi =

    work page 2015

  12. [12]

    36th International Symposium on Computational Geometry (SoCG 2020) , pages =

    Kim, Jisu and Shin, Jaehyeok and Chazal, Fr. 36th International Symposium on Computational Geometry (SoCG 2020) , pages =. 2020 , doi =

    work page 2020

  13. [13]

    2021 , month = sep, type =

    Yim, Ka Man , title =. 2021 , month = sep, type =

    work page 2021

  14. [14]

    Transactions of the American Mathematical Society , volume =

    Herbert Federer , title =. Transactions of the American Mathematical Society , volume =

    work page

  15. [15]

    and Carrière, Mathieu and Peterson, Alexis and Liao, Will and Tjärnberg, Andreas and Fung, Jun Hou and Mahbubani, Krishnaa T

    Kandror, Elena K. and Carrière, Mathieu and Peterson, Alexis and Liao, Will and Tjärnberg, Andreas and Fung, Jun Hou and Mahbubani, Krishnaa T. and Loper, Jackson and Pangburn, William and Xu, Yuchen and Saeb-Parsy, Kourosh and Maniatis, Tom and Rizvi, Abbas H. , urldate =. Enhancer Dynamics and Cellular Architecture in the Human Spinal Cord , issn =. doi...

    work page doi:10.1016/j.neuron.2025.12.035 2025

  16. [16]

    Boissonnat, Jean-Daniel and Lieutier, André and Wintraecken, Mathijs , date =. The

    work page

  17. [17]

    and Carlsson, Gunnar , date =

    Nicolau, Monica and Levine, Arnold J. and Carlsson, Gunnar , date =. Topology Based Data Analysis Identifies a Subgroup of Breast Cancers with a Unique Mutational Profile and Excellent Survival , volume =. doi:10.1073/pnas.1102826108 , pages =

    work page doi:10.1073/pnas.1102826108

  18. [18]

    and Cámara, Pablo G

    Rizvi, Abbas H. and Cámara, Pablo G. and Kandror, Elena K. and Roberts, Thomas J. and Schieren, Ira and Maniatis, Tom and Rabadán, Raúl , urldate =. Single-Cell Topological. doi:10.1038/nbt.3854 , pages =

    work page doi:10.1038/nbt.3854

  19. [19]

    Convergence between Categorical Representations of Reeb Space and Mapper , volume =

    Munch, Elizabeth and Wang, Bei , date =. Convergence between Categorical Representations of Reeb Space and Mapper , volume =. 32nd International Symposium on Computational Geometry (. doi:10.4230/LIPIcs.SoCG.2016.53 , series =

    work page doi:10.4230/lipics.socg.2016.53 2016

  20. [20]

    Topological Data Analysis of Single-Cell Hi-C Contact Maps , booktitle =

    Carri. Topological Data Analysis of Single-Cell Hi-C Contact Maps , booktitle =. 2020 , publisher =

    work page 2020

  21. [21]

    Two-Tier Mapper, an unbiased topology-based clustering method for enhanced global gene expression analysis , journal =

    Jeitziner, Rachel and Carri. Two-Tier Mapper, an unbiased topology-based clustering method for enhanced global gene expression analysis , journal =. 2019 , doi =

    work page 2019

  22. [22]

    Differentiable Mapper for Topological Optimization of Data Representation , booktitle =

    Oulhaj, Ziyad and Carri. Differentiable Mapper for Topological Optimization of Data Representation , booktitle =. 2024 , editor =

    work page 2024

  23. [23]

    , title =

    Morozov, Dmitriy and Beketayev, Kenes and Weber, Gunther H. , title =. Discrete & Computational Geometry , year =

    work page

  24. [24]

    Journal of Computational Geometry , year =

    Farahbakhsh Touli, Elena and Wang, Yusu , title =. Journal of Computational Geometry , year =

    work page

  25. [25]

    and Fox, Kyle and Nath, Abhinandan and Sidiropoulos, Anastasios and Wang, Yusu , title =

    Agarwal, Pankaj K. and Fox, Kyle and Nath, Abhinandan and Sidiropoulos, Anastasios and Wang, Yusu , title =. ACM Transactions on Algorithms , year =

    work page

  26. [26]

    La Matematica , year =

    Gasparovic, Ellen and Munch, Elizabeth and Oudot, Steve and Turner, Katharine and Wang, Bei , title =. La Matematica , year =

    work page

  27. [27]

    Journal of Applied and Computational Topology , year =

    Curry, Justin and Gasparovic, Ellen and Munch, Elizabeth and Oudot, Steve and Wang, Bei , title =. Journal of Applied and Computational Topology , year =

    work page

  28. [28]

    2008 , note =

    Reeb graphs for shape analysis and applications , journal =. 2008 , note =. doi:https://doi.org/10.1016/j.tcs.2007.10.018 , author =

    work page doi:10.1016/j.tcs.2007.10.018 2008

  29. [29]

    Saunders Mac Lane , title =

    work page

  30. [30]

    Categorified Reeb Graphs , volume=

    de Silva, Vin and Munch, Elizabeth and Patel, Amit , year=. Categorified Reeb Graphs , volume=. Discrete & Computational Geometry , publisher=. doi:10.1007/s00454-016-9763-9 , number=

    work page doi:10.1007/s00454-016-9763-9

  31. [31]

    Journal of Applied and Computational Topology , volume =

    Adam Brown and Omer Bobrowski and Elizabeth Munch and Bei Wang , title =. Journal of Applied and Computational Topology , volume =. 2021 , doi =

    work page 2021

  32. [32]

    , title =

    Funk, J. , title =. Cahiers de Topologie et G\'eom\'etrie Diff\'erentielle Cat\'egoriques , pages =. 1995 , mrnumber =

    work page 1995

  33. [33]

    Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry , pages =

    Chazal, Fr\'. Proximity of persistence modules and their diagrams , year =. Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry , pages =. doi:10.1145/1542362.1542407 , abstract =

    work page doi:10.1145/1542362.1542407

  34. [34]

    , journal =

    Bubenik, Peter and Scott, Jonathan A. , journal =. Categorification of Persistent Homology , volume =

    work page

  35. [35]

    32nd International Symposium on Computational Geometry (SoCG 2016) , pages =

    Carri\`. 32nd International Symposium on Computational Geometry (SoCG 2016) , pages =. 2016 , volume =. doi:10.4230/LIPIcs.SoCG.2016.25 , annote =

    work page doi:10.4230/lipics.socg.2016.25 2016

  36. [36]

    Structure and Stability of the 1-Dimensional Mapper , journal =

    Carri. Structure and Stability of the 1-Dimensional Mapper , journal =

    work page

  37. [37]

    2016 , DOI =

    Chazal, Fr. 2016 , DOI =

    work page 2016

  38. [38]

    Extending Persistence Using Poincar

    Cohen-Steiner, David and Edelsbrunner, Herbert and Harer, John , doi =. Extending Persistence Using Poincar. Foundations of Computational Mathematics , number =. 2009 , bdsk-url-1 =

    work page 2009

  39. [39]

    Proceedings of the Thirtieth Annual Symposium on Computational Geometry , pages =

    Bauer, Ulrich and Ge, Xiaoyin and Wang, Yusu , title =. Proceedings of the Thirtieth Annual Symposium on Computational Geometry , pages =. 2014 , isbn =. doi:10.1145/2582112.2582169 , abstract =

    work page doi:10.1145/2582112.2582169 2014

  40. [40]

    Zigzag Persistence , url =

    Carlsson, Gunnar and de Silva, Vin , date-added =. Zigzag Persistence , url =. Foundations of Computational Mathematics , number =. 2010 , bdsk-url-1 =. doi:10.1007/s10208-010-9066-0 , id =

    work page doi:10.1007/s10208-010-9066-0 2010

  41. [41]

    Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry , pages =

    Carlsson, Gunnar and de Silva, Vin and Morozov, Dmitriy , title =. Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry , pages =. 2009 , isbn =. doi:10.1145/1542362.1542408 , abstract =

    work page doi:10.1145/1542362.1542408 2009

  42. [42]

    33rd International Symposium on Computational Geometry (SoCG 2017) , pages =

    Carri\`. 33rd International Symposium on Computational Geometry (SoCG 2017) , pages =. 2017 , volume =. doi:10.4230/LIPIcs.SoCG.2017.25 , annote =

    work page doi:10.4230/lipics.socg.2017.25 2017

  43. [43]

    Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition , journal =

    Singh, Gurjeet and Mémoli, Facundo and Carlsson, Gunnar , year =. Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition , journal =

    work page

  44. [44]

    Topological Methods for Exploring Low-density States in Biomolecular Folding Pathways , volume =

    Yao, Yuan and Sun, Jian and Huang, Xuhui and Bowman, Gregory and Singh, Gurjeet and Lesnick, Michael and Guibas, Leonidas and Pande, Vijay and Carlsson, Gunnar , year =. Topological Methods for Exploring Low-density States in Biomolecular Folding Pathways , volume =. The Journal of chemical physics , doi =

    work page

  45. [45]

    Journal of allergy and clinical immunology , volume=

    Innate and adaptive T cells in asthmatic patients: relationship to severity and disease mechanisms , author=. Journal of allergy and clinical immunology , volume=. 2015 , publisher=

    work page 2015

  46. [46]

    Extracting insights from the shape of complex data using topology , volume =

    Lum, Pek and Singh, G and Lehman, A and Ishkanov, T and Vejdemo-Johansson, Mikael and Alagappan, Meenatchi and Carlsson, J and Carlsson, Gunnar , year =. Extracting insights from the shape of complex data using topology , volume =. Scientific reports , doi =

    work page

  47. [47]

    Statistical Analysis and Parameter Selection for Mapper , journal =

    Mathieu Carri. Statistical Analysis and Parameter Selection for Mapper , journal =. 2018 , volume =

    work page 2018

  48. [48]

    Convergence Rates for Persistence Diagram Estimation in Topological Data Analysis , journal =

    Fr. Convergence Rates for Persistence Diagram Estimation in Topological Data Analysis , journal =. 2015 , volume =

    work page 2015

  49. [49]

    Confidence Sets for Persistence Diagrams , volume =

    Fasy, Brittany and Lecci, Fabrizio and Rinaldo, Alessandro and Wasserman, Larry and Balakrishnan, Sivaraman and Singh, Aarti , year =. Confidence Sets for Persistence Diagrams , volume =

    work page

  50. [50]

    Geometric Inference for Probability Measures , volume =

    Chazal, Frédéric and Cohen-Steiner, David and Mérigot, Quentin , year =. Geometric Inference for Probability Measures , volume =. Foundations of Computational Mathematics , doi =

    work page

  51. [51]

    ACM Trans

    Chen, Xiaobai and Golovinskiy, Aleksey and Funkhouser, Thomas , title =. ACM Trans. Graph. , month = jul, articleno =. 2009 , issue_date =. doi:10.1145/1531326.1531379 , abstract =

    work page doi:10.1145/1531326.1531379 2009

  52. [52]

    Roweis and Lawrence K

    Sam T. Roweis and Lawrence K. Saul , title =. Science , volume =. 2000 , doi =. https://www.science.org/doi/pdf/10.1126/science.290.5500.2323 , abstract =

    work page doi:10.1126/science.290.5500.2323 2000

  53. [53]

    34th International Symposium on Computational Geometry (SoCG 2018) , pages =

    Boissonnat, Jean-Daniel and Lieutier, Andr\'. 34th International Symposium on Computational Geometry (SoCG 2018) , pages =. 2018 , volume =. doi:10.4230/LIPIcs.SoCG.2018.10 , annote =

    work page doi:10.4230/lipics.socg.2018.10 2018

  54. [54]

    Discrete & Computational Geometry , volume=

    Stability and minimax optimality of tangential Delaunay complexes for manifold reconstruction , author=. Discrete & Computational Geometry , volume=. 2018 , publisher=

    work page 2018

  55. [55]

    and Hamann, Bernd , booktitle =

    Beketayev, Kenes and Yeliussizov, Damir and Morozov, Dmitriy and Weber, Gunther H. and Hamann, Bernd , booktitle =. Measuring the Distance Between Merge Trees , year =

    work page

  56. [56]

    Finding the Homology of Submanifolds with High Confidence from Random Samples , volume =

    Niyogi, Partha and Smale, Stephen and Weinberger, Shmuel , year =. Finding the Homology of Submanifolds with High Confidence from Random Samples , volume =. Discrete & Computational Geometry , doi =

    work page

  57. [57]

    Computational Topology: An Introduction , isbn =

    Edelsbrunner, Herbert and Harer, John , year =. Computational Topology: An Introduction , isbn =

    work page

  58. [58]

    A graph-matching formulation of the interleaving distance between merge trees , journal =

    Matteo Pegoraro , keywords =. A graph-matching formulation of the interleaving distance between merge trees , journal =. 2025 , issn =. doi:10.3934/math.2025586 , url =

    work page doi:10.3934/math.2025586 2025

  59. [59]

    Bounding the interleaving distance for mapper graphs with a loss function , url =

    Chambers, Erin Wolf and Munch, Elizabeth and Percival, Sarah and Wang, Bei , date =. Bounding the interleaving distance for mapper graphs with a loss function , url =. Journal of Applied and Computational Topology , number =. 2025 , bdsk-url-1 =. doi:10.1007/s41468-025-00215-x , id =

    work page doi:10.1007/s41468-025-00215-x 2025

  60. [60]

    1998 , publisher=

    Categories for the Working Mathematician , author=. 1998 , publisher=

    work page 1998

  61. [61]

    On standardness: estimation of the standardness constant and decidability aspects , url =

    Alejandro Cholaquidis and Leonardo Moreno and Beatriz Pateiro-L. On standardness: estimation of the standardness constant and decidability aspects , url =. Journal of Nonparametric Statistics , number =. 2026 , bdsk-url-1 =. doi:10.1080/10485252.2025.2591341 , eprint =

    work page doi:10.1080/10485252.2025.2591341 2026

  62. [62]

    2010 , eprint=

    Intrinsic dimension estimation of data by principal component analysis , author=. 2010 , eprint=

    work page 2010

  63. [63]

    Tenenbaum and Vin de Silva and John C

    Joshua B. Tenenbaum and Vin de Silva and John C. Langford , journal =. A Global Geometric Framework for Nonlinear Dimensionality Reduction , volume =

    work page

  64. [64]

    Cover complex , publisher =

    Mathieu Carri. Cover complex , publisher =. GUDHI User and Reference Manual , url =

    work page

  65. [65]

    and Varoquaux, G

    Pedregosa, F. and Varoquaux, G. and Gramfort, A. and Michel, V. and Thirion, B. and Grisel, O. and Blondel, M. and Prettenhofer, P. and Weiss, R. and Dubourg, V. and Vanderplas, J. and Passos, A. and Cournapeau, D. and Brucher, M. and Perrot, M. and Duchesnay, E. , journal=. Scikit-learn: Machine Learning in

    work page

  66. [66]

    Hagberg and Daniel A

    Aric A. Hagberg and Daniel A. Schult and Pieter J. Swart , title =. Proceedings of the 7th Python in Science Conference , pages =

    work page