pith. sign in

arxiv: 2606.21688 · v1 · pith:OO453W23new · submitted 2026-06-19 · 🧮 math.AT · cs.CG

A Three Axis Evaluation Framework for Mapper Algorithms

Annesha Sen , Shivam Singh , S. P. Tiwari This is my paper

Pith reviewed 2026-06-26 12:28 UTC · model grok-4.3

classification 🧮 math.AT cs.CG
keywords Mapper algorithmtopological data analysisstabilitycluster qualitytopological shape preservationevaluation frameworksynthetic datasets
0
0 comments X

The pith

Mapper variants trade off stability, cluster quality, and topological shape preservation, with none optimal on all three.

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

The paper proposes evaluating Mapper algorithms along three axes instead of relying on visual inspection alone. It applies the axes of stability, cluster quality, and topological shape preservation to Mapper and its variants on synthetic datasets plus the UCI Digits data. The tests reveal that gains on one axis frequently reduce performance on the others. This matters because Mapper summaries of high-dimensional data depend heavily on lens functions, covers, and clustering choices, and conflicting criteria make reliable selection difficult without a structured approach.

Core claim

The three axes of evaluation are often in tension, and no single Mapper variant performs optimally across stability, cluster quality, and topological shape preservation on the tested data.

What carries the argument

Three-axis evaluation framework measuring stability of outputs, quality of clusters, and preservation of topological shape.

If this is right

  • Selection of Mapper variants should be guided by which axis matters most for the intended use case.
  • No universal Mapper configuration exists that balances all three criteria simultaneously.
  • Open challenges remain in creating Mapper methods that reduce the observed tensions between axes.

Where Pith is reading between the lines

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

  • Application-specific tuning of Mapper parameters may be necessary rather than seeking one best variant.
  • The same multi-axis tension could appear in evaluations of related tools such as Reeb graphs or other TDA summaries.
  • Adding metrics like runtime cost or noise robustness might further expose trade-offs not captured by the current three axes.

Load-bearing premise

The three axes of stability, cluster quality, and topological shape preservation are complementary and together sufficient to judge Mapper outputs.

What would settle it

Discovery of one Mapper variant that scores highly on all three axes across the synthetic and UCI Digits datasets would falsify the tension claim.

Figures

Figures reproduced from arXiv: 2606.21688 by Annesha Sen, Shivam Singh, S. P. Tiwari.

Figure 1
Figure 1. Figure 1: Swiss Roll with Hole and Noisy Circle Datasets. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Threshold sensitivity analysis: (left) fraction of bootstrap trials whose observed [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Geometric Stability Heatmap (Ball Mapper). The heatmap displays the β1 count for Ball Mapper on the Swiss Roll across varying noise levels (σ) and radius parameters (ε). The consistently high Betti numbers (ranging from 339 to 482) indicate that the nerve captures the geometry of the cover (the dense ε-net lattice) rather than the underlying manifold topol￾ogy (β1 = 2). This phenomenon, which we term the L… view at source ↗
Figure 4
Figure 4. Figure 4: Visual depiction of instability scenarios. (Top Row) Conventional Mapper on Swiss Roll: For a resolution of Res = 8 and 15, the graph captures the topology correctly (β1 = 1). For a resolution of 25, there is a topological explosion (β1 = 11). (Bottom Row) Ball Mapper on Noisy Circle: With ε = 0.1, a dense lattice forms (β1 = 63). At ε = 0.2, the lattice collapses to correct topology (β1 = 1). At ε = 0.35,… view at source ↗
Figure 5
Figure 5. Figure 5: Bootstrap variability of β1 (Swiss Roll). Ball Mapper (CV = 8.8%) is the most stable, as the lattice artifact is robust to resampling. Conventional Mapper (CV = 14.9%) shows low instability. F-mapper (CV = 37.1%) exhibits higher relative variance; this occurs because fuzzy memberships tend to suppress cycles [9, 10] (mean β1 = 3.6, range 1–6), making occasional detections statistically volatile relative to… view at source ↗
Figure 6
Figure 6. Figure 6: Cluster quality across datasets. Silhouette Coefficient (SC) for Swiss Roll and Noisy Circle, Normalized Mutual Information (NMI) for UCI Digits. Low SC values on the Swiss Roll reflect manifold geometry and the effect of Voronoi hardening; the Ensemble is slightly more conservative in these runs. Numerical values are reported in [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Parameter sensitivity on Swiss Roll. Heatmaps show Silhouette scores as functions of resolution n and overlap p. Conventional and F-mapper methods degrade due to fragmentation at high resolution; the Ensemble reduces volatility by selecting consensus maps. 3. Effect of Lens Function Lens selection is often one of the most important user choices [13,65]. We tested three lenses on the Swiss Roll ( [PITH_FUL… view at source ↗
Figure 8
Figure 8. Figure 8: Effect of lens function on Swiss Roll. All lenses resulted in low or negative silhouette values, confirming the manifold’s continuous nature and the limits of Voronoi hardening. Noisy Circle (Figure 9a). For the Noisy Circle, we observe a reasonably stable regime. In our runs, the Conventional Mapper achieved a peak silhouette near ε = 0.1 (Conventional ≈ 0.326), then plateaued around ≈ 0.238 for larger ra… view at source ↗
Figure 9
Figure 9. Figure 9: Effect of DBSCAN ε on cluster quality. (a) The Noisy Circle shows a stable plateau with a clear regime of good silhouette. (b) The Swiss Roll exhibits uniformly low silhouette values across parameters; small positive values at larger ε do not indicate preserved topology. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: How resolution drives topological explosion. The blue curve (Conventional Mapper) shows spurious loops multiplying as resolution increases. F-mapper (green) reduces this problem substantially. Ball Mapper (orange dashed line) maintains a constant but very high Betti number (β1 ≈ 455), because it is capturing the geometric lattice covering the surface rather than adapting to resolution changes in the skele… view at source ↗
Figure 11
Figure 11. Figure 11: Comparing graph structures at N = 28. On the left, Conventional Mapper creates a mesh-like structure (E/V ≈ 1.2). In the middle, F-Mapper distills things down to a skeletal chain. On the right, Ball Mapper wraps the manifold surface in a dense lattice (E/V ≈ 8.1, with β1 ≈ 482). The Connectivity-Cycle Tradeoff. There is another interesting pattern here: these methods behave in opposite ways when it comes … view at source ↗
Figure 12
Figure 12. Figure 12: Visual Intuition for Theorem 3.4 (Interleaving). The blue Ball Mapper graph is geometrically contained within the red Vietoris-Rips complex, consistent with theoretical interleaving bounds [20]. 5.4 Open Problems in Topological and Shape Preservation Although substantial progress has been made in understanding the preservation properties of Mapper and its variants, several important open problems remain. … view at source ↗
Figure 13
Figure 13. Figure 13: Conventional Mapper Instability Heatmap. This heatmap is obtained for the Swiss Roll dataset using the same parameter settings as in [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: F-mapper on the Swiss Roll Dataset. This result is obtained using the parameter settings as in with n = 10 and τ = 0.3. 6.3 Relation of Preservation with Stability • If we look at Ball Mapper’s Bootstrap Stability in [PITH_FULL_IMAGE:figures/full_fig_p030_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Conventional Mapper Visualization of the Swiss Roll Dataset. The graph is obtained using the First coordinate projection lens function f(x,y) = x, with (n, p) = (12,0.7) and DBSCAN as the pullback clustering algorithm. Ensemble Mapper (Algorithm 2) proposed by Kang and Lim [37]. Unlike simple averaging strategies, this method uses a Selection-and-Consensus approach: 1. Grid Search: For a target resolution… view at source ↗
Figure 16
Figure 16. Figure 16: Integrative Stability Analysis. (a) The single-run Mapper (grey) on the Swiss Roll exhibits a topological explosion (β0 rises to 9) at high resolution. The Ensemble (blue) prevents this explosion, maintaining a constant β0 = 1, although the β1 count is higher due to the "thick" meta-graph structure. (b) On UCI Digits, the Ensemble handles fragmentation caused by sparsity better than the single-run Mapper,… view at source ↗
read the original abstract

Mapper is a well-known tool in topological data analysis, which visualizes and summarizes high-dimensional data. However, its output is sensitive to choices of lens functions, cover parameters, and clustering strategies, making evaluation challenging. Most works that have attempted to evaluate the Mapper algorithm have done so visually. In this paper, we review a roadmap for assessing Mapper algorithms along three complementary axes: stability, cluster quality, and topological shape preservation. We analyze Mapper and its variants on synthetic datasets and the UCI Digits dataset. These modes include topological explosion at high resolutions. Our findings indicate that these axes of evaluation are often in tension and that no single Mapper variant performs optimally across all three. This review provides practical guidelines for choosing Mapper variants and identifies open challenges toward a principled Mapper analysis.

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 / 2 minor

Summary. The manuscript proposes a three-axis evaluation framework for Mapper algorithms in topological data analysis—stability, cluster quality, and topological shape preservation—and applies it to Mapper variants on synthetic datasets and the UCI Digits dataset. The central claim is that these axes are frequently in tension and that no single variant optimizes all three simultaneously, yielding practical guidelines and identifying open challenges.

Significance. If the experimental demonstration of axis tensions holds under the reported conditions, the work supplies a concrete, multi-criteria alternative to purely visual Mapper assessment. This could inform variant selection in applied TDA pipelines and surface trade-offs that single-metric studies miss.

major comments (2)
  1. [§4] §4 (Experiments): the reported tension between axes is demonstrated only for the chosen synthetic constructions and UCI Digits; it is unclear whether the same conflicts appear under different lens functions or cover parameter regimes not tested in the high-resolution explosion cases.
  2. [§3.2] §3.2 (Cluster quality axis): the metric relies on a fixed external clustering reference; the claim that this axis is independent of the stability axis would be strengthened by an ablation showing that the tension persists when the reference clustering is replaced by an internal validity index.
minor comments (2)
  1. Notation for the three axes is introduced without a consolidated table; a summary table listing each axis, its quantitative proxy, and the datasets on which it is evaluated would improve readability.
  2. Figure captions for the high-resolution Mapper outputs do not state the exact resolution parameter values used in the 'topological explosion' examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: [§4] §4 (Experiments): the reported tension between axes is demonstrated only for the chosen synthetic constructions and UCI Digits; it is unclear whether the same conflicts appear under different lens functions or cover parameter regimes not tested in the high-resolution explosion cases.

    Authors: Our experiments in §4 focus on synthetic constructions and the UCI Digits dataset precisely because they exhibit the high-resolution topological explosion behavior central to the framework. The tensions are demonstrated consistently across these cases. We agree that the scope is limited and will add an expanded discussion subsection in §4 on the rationale for the chosen lens functions and cover parameters, along with explicit caveats about generalizability to untested regimes. revision: partial

  2. Referee: [§3.2] §3.2 (Cluster quality axis): the metric relies on a fixed external clustering reference; the claim that this axis is independent of the stability axis would be strengthened by an ablation showing that the tension persists when the reference clustering is replaced by an internal validity index.

    Authors: The external reference was selected to provide a ground-truth benchmark for cluster quality on datasets with known structure. We accept the suggestion and will add an ablation in §3.2 (or a new subsection) that replaces the external reference with an internal index such as the silhouette score, confirming that the reported tensions with the stability axis persist under this change. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents an empirical evaluation framework for Mapper algorithms across three axes (stability, cluster quality, topological shape preservation) using synthetic and UCI Digits datasets. No derivation chain, first-principles predictions, fitted parameters renamed as outputs, or self-citation load-bearing premises are present; the central claim of tension between axes rests on direct experimental comparison rather than any reduction to inputs by construction. This is a standard evaluation review with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only input provides no equations, parameters, or explicit assumptions; ledger entries cannot be extracted.

pith-pipeline@v0.9.1-grok · 5656 in / 1101 out tokens · 16179 ms · 2026-06-26T12:28:23.868122+00:00 · methodology

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

75 extracted references · 3 linked inside Pith

  1. [1]

    Adams H., et al., Persistence images: A stable vector representation of persistent homology, Journal of Machine Learning Research 18.8 (2017): 1–35

    2017

  2. [2]

    Alvarado E., et al., G-mapper: Learning a cover in the mapper construction, SIAM Journal on Mathematics of Data Science 7.2 (2025): 572–596

    2025

  3. [3]

    J., et al., Genomics data analysis via spectral shape and topology, PLOS ONE 18.4 (2023): e0284820

    Amézquita E. J., et al., Genomics data analysis via spectral shape and topology, PLOS ONE 18.4 (2023): e0284820

    2023

  4. [4]

    Belchí F., et al., A numerical measure of the instability of mapper-type algorithms, Journal of Machine Learning Research 21 (2020): 1–45

    2020

  5. [5]

    C., Pattern Recognition with Fuzzy Objective Function Algorithms, Springer Science & Business Media, 1981

    Bezdek J. C., Pattern Recognition with Fuzzy Objective Function Algorithms, Springer Science & Business Media, 1981. 35

    1981

  6. [6]

    Biasotti S., et al., Reeb graphs for shape analysis and applications, Theoretical Computer Science 392.1–3 (2008): 5–22

    2008

  7. [7]

    Bodnar C., Cangea C., and Liò P., Deep Graph Mapper: Seeing graphs through the neural lens, Frontiers in Big Data 4 (2021): 680535

    2021

  8. [8]

    Brown A., et al., Probabilistic convergence and stability of random mapper graphs, Journal of Applied and Computational Topology 5.1 (2021): 99–140

    2021

  9. [9]

    Bui Q.-T., et al., F-mapper: A fuzzy mapper clustering algorithm, Knowledge-Based Systems 189 (2020): 105107

    2020

  10. [10]

    Bui Q.-T., et al., SFCM: A fuzzy clustering algorithm for extracting the shape informa- tion of data, IEEE Transactions on Fuzzy Systems 29.1 (2020): 75–89

    2020

  11. [11]

    Bungula W., and Darcy I., Bi-filtration and stability of TDA mapper for point cloud data, arXiv preprint arXiv:2409.17360 (2024)

    arXiv 2024

  12. [12]

    F., A fuzzy-logic mapper for audiovisual media, Computer Music Journal 30.1 (2006): 67–82

    Cadiz R. F., A fuzzy-logic mapper for audiovisual media, Computer Music Journal 30.1 (2006): 67–82

    2006

  13. [13]

    Carrière M., Michel B., and Oudot S., Statistical analysis and parameter selection for mapper, Journal of Machine Learning Research 19 (2018): 1–39

    2018

  14. [14]

    Carrière M., and Oudot S., Structure and stability of the one-dimensional mapper, Foundations of Computational Mathematics 18.6 (2018): 1333–1396

    2018

  15. [15]

    M., Carlsson G., and Rabadán R., Topology of viral evolution, Proceedings of the National Academy of Sciences 110.46 (2013): 18566–18571

    Chan J. M., Carlsson G., and Rabadán R., Topology of viral evolution, Proceedings of the National Academy of Sciences 110.46 (2013): 18566–18571

    2013

  16. [16]

    Chen Y ., and V oli´c I., Topological data analysis model for the spread of the coronavirus, PLOS ONE 16.8 (2021): e0255584

    2021

  17. [17]

    Cuerno M., et al., Topological data analysis in air traffic management: The shape of big flight data sets, PLOS ONE 20.2 (2025): e0318108

    2025

  18. [18]

    L., and Bouldin D

    Davies D. L., and Bouldin D. W., A cluster separation measure, IEEE Transactions on Pattern Analysis and Machine Intelligence 1.2 (1979): 224–227

    1979

  19. [19]

    Dey T. K., Mémoli F., and Wang Y ., Multiscale Mapper: Topological summarization via codomain covers, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM (2016): 1935–1951

    2016

  20. [20]

    Dłotko P., Ball Mapper: A shape summary for topological data analysis, arXiv preprint arXiv:1901.07410 (2019)

    Pith/arXiv arXiv 1901

  21. [21]

    Dłotko P., Qiu W., and Rudkin S., Topological Data Analysis Ball Mapper for finance, arXiv preprint arXiv:2206.03622 (2022)

    arXiv 2022

  22. [22]

    N., Topological signatures of altered brain network centrality in ADHD: A TDA Mapper study, NeurIPS 2025 Workshop on Symmetry and Geometry in Neural Representations (2025)

    Duman A. N., Topological signatures of altered brain network centrality in ADHD: A TDA Mapper study, NeurIPS 2025 Workshop on Symmetry and Geometry in Neural Representations (2025). 36

    2025

  23. [23]

    C., A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters, Journal of Cybernetics 3.3 (1973): 32–57

    Dunn J. C., A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters, Journal of Cybernetics 3.3 (1973): 32–57

    1973

  24. [24]

    Ester M., et al., A density-based algorithm for discovering clusters in large spatial databases with noise, Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD) (1996): 226–231

    1996

  25. [25]

    Fitzpatrick P., et al., Ensemble learning for mapper parameter optimization, Proceedings of the 2023 IEEE 35th International Conference on Tools with Artificial Intelligence (ICTAI), IEEE (2023): 924–931

    2023

  26. [26]

    Geniesse C., Chowdhury S., and Saggar M., NeuMapper: A scalable computational framework for multiscale exploration of the brain’s dynamical organization, Network Neuroscience 6.2 (2022): 467–498

    2022

  27. [27]

    Gidea M., and Katz Y ., Topological data analysis of financial time series: Landscapes of crashes, Physica A: Statistical Mechanics and Its Applications 491 (2018): 820–834

    2018

  28. [28]

    Guzmán-Sáenz A., et al., Signal enrichment with strain-level resolution in metagenomes using topological data analysis, BMC Genomics 20.Suppl 2 (2019): 194

    2019

  29. [29]

    Ha¸ segan D., et al., Deconstructing the Mapper algorithm to extract richer topological and temporal features from functional neuroimaging data, Network Neuroscience 8.4 (2024): 1355–1382

    2024

  30. [30]

    Hatcher A., Algebraic Topology, Cambridge University Press, 2005

    2005

  31. [31]

    Hiraoka Y ., et al., Hierarchical structures of amorphous solids characterized by persistent homology, Proceedings of the National Academy of Sciences 113.26 (2016): 7035– 7040

    2016

  32. [32]

    Hofer C., et al., Deep learning with topological signatures, Advances in Neural Infor- mation Processing Systems (NeurIPS) 30 (2017): 1634–1644

    2017

  33. [33]

    D., Kwitt R., and Niethammer M., Learning representations of persistence barcodes, Journal of Machine Learning Research 20 (2019): 1–45

    Hofer C. D., Kwitt R., and Niethammer M., Learning representations of persistence barcodes, Journal of Machine Learning Research 20 (2019): 1–45

    2019

  34. [34]

    Hubert L., and Arabie P., Comparing partitions, Journal of Classification 2.1 (1985): 193–218

    1985

  35. [35]

    Indah D., et al., Topological data analysis for driver behavior classification driven by vehicle trajectory data, Machine Learning with Applications 16 (2025): 100719

    2025

  36. [36]

    Jeitziner R., et al., Two-tier Mapper: A user-independent clustering method for global gene expression analysis based on topology, arXiv preprint arXiv:1801.01841 (2018)

    Pith/arXiv arXiv 2018

  37. [37]

    J., and Lim Y ., Ensemble Mapper, Stat 10.1 (2021): e405

    Kang S. J., and Lim Y ., Ensemble Mapper, Stat 10.1 (2021): e405

    2021

  38. [38]

    Kappe C., Böttinger M., and Leitte H., Topology-based feature analysis of scalar field ensembles: An application to climate (change) analysis, Computers & Graphics 104 (2022): 59–71. 37

    2022

  39. [39]

    Kramár M., et al., Quantifying force networks in particulate systems, Physica D: Nonlinear Phenomena 283 (2014): 37–53

    2014

  40. [40]

    Kyeong S., et al., A new approach to investigate the association between brain func- tional connectivity and disease characteristics of attention-deficit/hyperactivity disorder: Topological neuroimaging data analysis, PLOS ONE 10.9 (2015): e0137296

    2015

  41. [41]

    G., Topological data analysis and machine learning, Advances in Physics: X 8.1 (2023): 2202331

    Leykam D., and Angelakis D. G., Topological data analysis and machine learning, Advances in Physics: X 8.1 (2023): 2202331

    2023

  42. [42]

    Y ., et al., Extracting insights from the shape of complex data using topology, Scientific Reports 3.1 (2013): 1236

    Lum P. Y ., et al., Extracting insights from the shape of complex data using topology, Scientific Reports 3.1 (2013): 1236

    2013

  43. [43]

    Lymberopoulos E., et al., Topological data analysis highlights novel geographical signatures of the human gut microbiome, Frontiers in Artificial Intelligence 4 (2021): 680564

    2021

  44. [44]

    N., Ugoala B

    Madukpe V . N., Ugoala B. C., and Zulkepli N. F. S., A comprehensive review of the mapper algorithm, a topological data analysis technique, and its applications across various fields (2007–2025), International Journal of Data Science and Analytics (2026): 1–56

    2007

  45. [45]

    Manogaran G., Lopez D., and Chilamkurti N., In-Mapper combiner based MapReduce algorithm for processing of big climate data, Future Generation Computer Systems 86 (2018): 433–445

    2018

  46. [46]

    W., Morse Theory, Princeton University Press, 1963

    Milnor J. W., Morse Theory, Princeton University Press, 1963

    1963

  47. [47]

    S., Topological Data Analysis and Applications to Influenza, M.S

    Morrison K. S., Topological Data Analysis and Applications to Influenza, M.S. thesis, Miami University, 2020

    2020

  48. [48]

    Munch E., A user’s guide to topological data analysis, Journal of Learning Analytics 4.2 (2017): 47–61

    2017

  49. [49]

    Munch E., and Wang B., Convergence between categorical representations of Reeb space and mapper, arXiv preprint arXiv:1512.04108 (2015)

    Pith/arXiv arXiv 2015

  50. [50]

    Muszynski G., et al., Topological data analysis and machine learning for recognizing atmospheric river patterns in large climate datasets, Geoscientific Model Development 12.2 (2019): 613–628

    2019

  51. [51]

    Naitzat G., Zhitnikov A., and Lim L.-H., Topology of deep neural networks, Journal of Machine Learning Research 21 (2020): 1–40

    2020

  52. [52]

    Nicolau M., Levine A. J., and Carlsson G., Topology-based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival, Proceedings of the National Academy of Sciences 108.17 (2011): 7265–7270

    2011

  53. [53]

    Oulhaj Z., et al., Deep Mapper: Efficient Visualization of Plausible Conformational Pathways, arXiv preprint arXiv:2402.19177 (2024). 38

    arXiv 2024

  54. [54]

    Parida L., et al., Host trait prediction of metagenomic data for topology-based visual- ization, International Conference on Distributed Computing and Internet Technology, Springer International Publishing (2015): 7–17

    2015

  55. [55]

    J., Topological Data Analysis for Genomics and Evolu- tion: Topology in Biology, Cambridge University Press, 2019

    Rabadán R., and Blumberg A. J., Topological Data Analysis for Genomics and Evolu- tion: Topology in Biology, Cambridge University Press, 2019

    2019

  56. [56]

    Rieck B., et al., Neural persistence: A complexity measure for deep neural networks using algebraic topology, arXiv preprint arXiv:1812.09764 (2018)

    arXiv 2018

  57. [57]

    Riihimäki H., et al., A topological data analysis based classification method for multiple measurements, BMC Bioinformatics 21.1 (2020): 336

    2020

  58. [58]

    H., et al., Single-cell topological RNA-seq analysis reveals insights into cellular differentiation and development, Nature Biotechnology 35.6 (2017): 551–560

    Rizvi A. H., et al., Single-cell topological RNA-seq analysis reveals insights into cellular differentiation and development, Nature Biotechnology 35.6 (2017): 551–560

    2017

  59. [59]

    J., Silhouettes: A graphical aid to the interpretation and validation of cluster analysis, Journal of Computational and Applied Mathematics 20 (1987): 53–65

    Rousseeuw P. J., Silhouettes: A graphical aid to the interpretation and validation of cluster analysis, Journal of Computational and Applied Mathematics 20 (1987): 53–65

    1987

  60. [60]

    Routh A., and Johnson J. E., Discovery of functional genomic motifs in viruses with ViReMa–a Virus Recombination Mapper–for analysis of next-generation sequencing data, Nucleic Acids Research 42.2 (2014): e11

    2014

  61. [61]

    Rudkin S., An Introduction to Topological Data Analysis Ball Mapper in R, arXiv preprint arXiv:2504.14081 (2025)

    arXiv 2025

  62. [62]

    Rudkin S., and Qiu W., ballmapper: Applying Topological Data Analysis Ball Mapper in Stata, arXiv preprint arXiv:2601.00508 (2026)

    arXiv 2026

  63. [63]

    Saggar M., et al., Towards a new approach to reveal dynamical organization of the brain using topological data analysis, Nature Communications 9.1 (2018): 1399

    2018

  64. [64]

    A., Topological data analysis to model the shape of immune responses during co-infections, Communications in Nonlinear Science and Numerical Simulation 85 (2020): 105228

    Sasaki K., Bruder D., and Hernandez-Vargas E. A., Topological data analysis to model the shape of immune responses during co-infections, Communications in Nonlinear Science and Numerical Simulation 85 (2020): 105228

    2020

  65. [65]

    Singh G., Mémoli F., and Carlsson G. E., Topological methods for the analysis of high dimensional data sets and 3D object recognition, Proceedings of the Eurographics Symposium on Point-Based Graphics (PBG) (2007): 91–100

    2007

  66. [66]

    Skaf Y ., and Laubenbacher R., Topological data analysis in biomedicine: A review, Journal of Biomedical Informatics 130 (2022): 104082

    2022

  67. [67]

    Sotcheff S., et al., ViReMa: A virus recombination mapper of next-generation sequenc- ing data characterizes diverse recombinant viral nucleic acids, GigaScience 12 (2023): giad009

    2023

  68. [68]

    Strehl A., and Ghosh J., Cluster ensembles—a knowledge reuse framework for combin- ing multiple partitions, Journal of Machine Learning Research 3 (2002): 583–617

    2002

  69. [69]

    Tao Y ., and Ge S., A Mapper algorithm with implicit intervals and its optimization, Journal of Computational Biology (2025). 39

    2025

  70. [70]

    Tao Y ., and Ge S., A distribution-guided Mapper algorithm, BMC Bioinformatics 26.1 (2025): 73

    2025

  71. [71]

    Wee J., and Jiang J., A review of topological data analysis and topological deep learning in molecular sciences, Journal of Chemical Information and Modeling 65.23 (2025): 12691–12706

    2025

  72. [72]

    thesis, The Chinese University of Hong Kong (Hong Kong), 2020

    Wei Y ., An Integrative Framework Based on Topological Data Analysis for Population- scale Microbiome Stratification, Association and Comparative Studies, Ph.D. thesis, The Chinese University of Hong Kong (Hong Kong), 2020

    2020

  73. [73]

    Wei G., Topological data analysis and topological deep learning beyond persistent homology, 2026 Joint Mathematics Meetings (JMM 2026), AMS (to appear)

    2026

  74. [74]

    Zhang M., Chowdhury S., and Saggar M., Temporal Mapper: Transition networks in simulated and real neural dynamics, Network Neuroscience 7.2 (2023): 431–460

    2023

  75. [75]

    Zhou Y ., et al., Comparing Mapper graphs of artificial neuron activations, Proceedings of the 2023 Topological Data Analysis and Visualization (TopoInVis), IEEE (2023): 41–50. 40

    2023