pith. sign in

arxiv: 2606.21671 · v1 · pith:JSMXTRU5new · submitted 2026-06-19 · 🧮 math.AT · cs.CG

GK-Mapper: A Stability Framework for Gustafson-Kessel Fuzzy Mapper Graphs

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

Pith reviewed 2026-06-26 12:31 UTC · model grok-4.3

classification 🧮 math.AT cs.CG
keywords Gustafson-Kessel clusteringFuzzy MapperTopological Data AnalysisGraph stabilityFuzzifier parameterCritical eventsMembership functionsAnisotropic data
0
0 comments X

The pith

Gustafson-Kessel Mapper graphs stay constant between finite critical events set by membership threshold crossings.

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

The paper introduces Gustafson-Kessel Fuzzy Mapper Graphs, which replace spherical covers in prior Mapper variants with ellipsoidal ones drawn from Gustafson-Kessel clustering to better fit anisotropic data. It builds a stability framework proving that membership functions vary smoothly with the fuzzifier, that edges exist under a precise condition on those functions, and that the resulting graph is locally stable to small fuzzifier changes. Graph modifications occur only at discrete critical events when membership values cross thresholds, so the graph remains fixed between such events. When the set of crossings is finite, the graph reaches a freezing threshold and stays unchanged thereafter.

Core claim

The graphs produced by Gustafson-Kessel Fuzzy Mapper and Shape Fuzzy C-Means Mapper are locally stable under small perturbations of the fuzzifier. Membership functions depend smoothly on the fuzzifier. Edges exist precisely when a certain condition on membership values holds. Changes in the graph are completely described by threshold crossings of the membership functions, and the graph is constant between consecutive critical events. When the threshold-crossing set is finite, this produces an eventual freezing threshold beyond which the graph no longer changes.

What carries the argument

The critical-event structure of graph changes, defined by threshold crossings of the membership functions induced by the fuzzifier.

If this is right

  • The graph can be tracked by computing membership functions only up to the last critical event.
  • Small fuzzifier perturbations leave the graph unchanged unless they cross a membership threshold.
  • Ellipsoidal covers allow the method to capture non-spherical cluster shapes in high dimensions.
  • Empirical tests show the graphs remain more stable than those from spherical fuzzy covers on complex data.

Where Pith is reading between the lines

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

  • Parameter tuning for Mapper could shift from visual inspection toward locating the freezing threshold directly from the membership functions.
  • The same threshold-crossing analysis might apply to other fuzzy or soft clustering methods used inside topological summaries.
  • One could monitor the derivative of membership values with respect to the fuzzifier to predict the next critical event without exhaustive search.

Load-bearing premise

The set of threshold crossings of the membership functions is finite for the data under study.

What would settle it

A concrete dataset and range of fuzzifier values in which membership functions cross their edge-existence thresholds at infinitely many distinct points, so that the graph keeps changing without bound.

Figures

Figures reproduced from arXiv: 2606.21671 by Annesha Sen, Shivam Singh, S. P. Tiwari.

Figure 1
Figure 1. Figure 1: Empirical illustrations of Theorems 1 and 2. Subfigure 1a shows the Edgeless [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Empirical illustrations of Theorems 3 and 4. Subfigure 2a shows the conserva [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Unit circle dataset: SFCM (left) and GK-Mapper (right). [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Anisotropic ellipsoidal dataset. 6.2. Anisotropic Ellipsoidal Dataset Three Gaussian clusters (180 points each) in R 2 with aspect ratios 5:1, 8:1, 4:1, deliberately violating spherical-cluster assumptions [23]. Both methods use c = 3, t = 0.20, and m0 = 1.200. GK-Mapper improves tcrit from 0.3723 to 0.4390 and increases r ∗ from 0.0132 to 0.0333. Under the selected perturbation h = 0.10, both methods give… view at source ↗
Figure 5
Figure 5. Figure 5: Stanford Bunny dataset. score is higher for SFCM (0.346 vs. 0.307), while ARI and matching score are not available because the point cloud has no class labels. These results suggest that GK-Mapper retains a more connected and cycle-rich graph on the Bunny point cloud while also reducing the observed edge variation under the selected perturbation. 6.4. UCI Handwritten Digits Dataset 1797 grayscale 8 × 8 dig… view at source ↗
Figure 6
Figure 6. Figure 6: UCI Digits dataset. 6.5. Wisconsin Breast Cancer Dataset 569 samples with 30 features describing cell nuclei from digitised fine￾needle aspirates [32, 30, 31]; binary malignant/benign labels. To probe the high-resolution regime (1/c≈0.01), we use c = 100, t = 0.015, h = 0.10, with m0 = 1.65 (SFCM) and m0 = 2.40 (GK-Mapper). GK-Mapper substantially increases tcrit from 0.0291 to 0.2478 and improves the empi… view at source ↗
Figure 7
Figure 7. Figure 7: Wisconsin Breast Cancer dataset. to 0.4511), the UCI Digits dataset (0.1000 to 0.4386), and the Breast Cancer dataset (0.0291 to 0.2478). This indicates that GK Mapper allows the graph to remain nontrivial over a wider range of threshold values. In Theorem 1, a threshold satisfying t > tcrit makes the graph edgless and discrete. Thus, finding tcrit is an essential first step before interpreting the resulti… view at source ↗
read the original abstract

Topological Data Analysis uses tools from algebraic topology to study the shape and structure of data. The Mapper algorithm provides a graph-based summary of high-dimensional datasets by combining a filter function, a cover of the filter range, and clustering on the corresponding pullback sets. Several variants of Mapper have been proposed, including Conventional Mapper, F-Mapper, and Shape Fuzzy C-Means Mapper. In this article, we introduce Gustafson-Kessel Fuzzy Mapper Graphs, a geometry-adaptive extension of Shape Fuzzy C-Means Mapper. The proposed method replaces spherical fuzzy covers with ellipsoidal covers induced by the Gustafson-Kessel fuzzy clustering framework, making it more suitable for high-dimensional datasets with anisotropic and non-spherical geometry. We develop a stability framework for the graphs produced by Gustafson-Kessel Mapper and Shape Fuzzy C-Means Mapper. We prove that the membership functions depend smoothly on the fuzzifier, establish a precise condition for the existence of edges, and show that the graph is locally stable under small perturbations of the fuzzifier. We further describe the critical-event structure of graph changes in terms of threshold crossings of the membership functions and show that the graph is constant between consecutive critical events. When the threshold-crossing set is finite, this yields an eventual freezing threshold. Finally, we empirically show that Gustafson-Kessel Mapper can produce more stable graphs than Shape Fuzzy C-Means Mapper on high-dimensional and geometrically complex datasets.

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

1 major / 0 minor

Summary. The paper introduces Gustafson-Kessel Fuzzy Mapper Graphs (GK-Mapper) as a geometry-adaptive extension of Shape Fuzzy C-Means Mapper, replacing spherical covers with ellipsoidal ones induced by Gustafson-Kessel fuzzy clustering. It develops a stability framework proving that membership functions depend smoothly on the fuzzifier, establishing a precise condition for edge existence, showing local stability of the graph under small fuzzifier perturbations, describing graph changes via threshold crossings of membership functions with constancy between consecutive critical events, and noting that finiteness of the crossing set yields an eventual freezing threshold. Empirical comparisons indicate GK-Mapper produces more stable graphs than prior variants on high-dimensional, anisotropic datasets.

Significance. If the stability results hold, the work supplies a useful parameter-sensitivity analysis for fuzzy Mapper constructions in TDA, clarifying how graph structure evolves with the fuzzifier and offering a mechanism to identify stable regimes. The local stability and critical-event description are potentially valuable for practitioners working with non-spherical data geometries.

major comments (1)
  1. [Abstract] Abstract: the global claim that the graph is eventually constant (via an eventual freezing threshold) rests on the hypothesis that the threshold-crossing set is finite. No argument, bound, or genericity condition is supplied showing this holds for finite point clouds or for continuous distributions equipped with the Gustafson-Kessel distance; without it only local stability follows.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on the stability framework. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the global claim that the graph is eventually constant (via an eventual freezing threshold) rests on the hypothesis that the threshold-crossing set is finite. No argument, bound, or genericity condition is supplied showing this holds for finite point clouds or for continuous distributions equipped with the Gustafson-Kessel distance; without it only local stability follows.

    Authors: We agree that the eventual-freezing statement is conditional on finiteness of the threshold-crossing set and that the manuscript supplies no general proof or genericity argument establishing this finiteness for arbitrary finite point clouds or continuous distributions. The abstract and theorems already qualify the claim with the clause “When the threshold-crossing set is finite.” To make the conditional nature fully explicit and to respond to the referee’s concern, we will revise the abstract to foreground the hypothesis, add a short remark in the stability section noting that finiteness is an assumption (with isolated crossings expected for smooth membership functions on finite data, though a rigorous bound is not provided), and clarify that unconditional results are limited to local stability. These changes will be incorporated in the revised manuscript. revision: partial

Circularity Check

0 steps flagged

No circularity: stability claims rest on independent smoothness and crossing analysis

full rationale

The paper's core results—smooth dependence of membership functions on the fuzzifier, edge-existence conditions, local stability under perturbations, and constancy between critical events—are derived from standard properties of Gustafson-Kessel clustering and Mapper constructions. The finiteness assumption for the threshold-crossing set is stated explicitly as a hypothesis yielding eventual constancy, without any reduction of the stability statements to fitted parameters or self-referential definitions. No load-bearing step equates a claimed prediction or uniqueness result to its own inputs by construction, and no self-citation chain is invoked to justify the central theorems. The derivation therefore remains self-contained against external fuzzy-clustering and topological-data-analysis benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract: the central claims rest on standard properties of fuzzy clustering (smooth membership functions) and the finiteness of threshold crossings for the freezing result. No new entities are postulated.

free parameters (1)
  • fuzzifier parameter
    The fuzzifier is the variable whose perturbations are analyzed for stability; its specific value is chosen per dataset but treated as a continuous parameter in the proofs.
axioms (2)
  • domain assumption Membership functions depend smoothly on the fuzzifier
    Invoked as the basis for local stability under small perturbations; stated as proved but location in abstract is the stability framework description.
  • ad hoc to paper Threshold-crossing set is finite
    Conditional assumption used to obtain eventual freezing threshold; appears in the final sentence of the stability claims.

pith-pipeline@v0.9.1-grok · 5791 in / 1431 out tokens · 29965 ms · 2026-06-26T12:31:28.160814+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

32 extracted references

  1. [1]

    A sober look at clustering stability,

    S. Ben-David, U. von Luxburg, and D. Pál, “A sober look at clustering stability,” inProc. 19th Annu. Conf. Learning Theory (COLT), 2006, pp. 5-19

    2006

  2. [2]

    J. C. Bezdek,Pattern Recognition with Fuzzy Objective Function Algo- rithms. New York, NY, USA: Plenum Press, 1981

    1981

  3. [3]

    FCM: The fuzzyC-Mean clus- tering algorithm,

    J. C. Bezdek, R. Ehrlich, and W. Full, “FCM: The fuzzyC-Mean clus- tering algorithm,”Comput. Geosci., vol. 10, no. 2-3, pp. 191-203, 1984

    1984

  4. [4]

    F- Mapper: A fuzzy mapper clustering algorithm,

    Q.-T. Bui, B. Vo, H.-A. N. Do, N. Q. V. Hung, and V. Snasel, “F- Mapper: A fuzzy mapper clustering algorithm,”Knowl.-Based Syst., vol. 189, p. 105097, 2020

    2020

  5. [5]

    SFCM: A fuzzy clustering algorithm of extracting the shape information of data,

    Q.-T. Bui, B. Vo, V. Snasel, W. Pedrycz, T.-P. Hong, N.-T. Nguyen, and M.-Y. Chen, “SFCM: A fuzzy clustering algorithm of extracting the shape information of data,”IEEE Trans. Fuzzy Syst., vol. 29, no. 1, pp. 75-89, Jan. 2021

    2021

  6. [6]

    Structure and stability of the 1-dimensional Mapper,

    M. Carrière and S. Oudot, “Structure and stability of the 1-dimensional Mapper,”Found. Comput. Math., vol. 18, no. 6, pp. 1333-1396, 2018

    2018

  7. [7]

    Statistical analysis and parame- ter selection for Mapper,

    M. Carrière, B. Michel, and S. Oudot, “Statistical analysis and parame- ter selection for Mapper,”J. Mach. Learn. Res., vol. 19, no. 1, pp. 1-39, 2018

    2018

  8. [8]

    Chazal, V

    F. Chazal, V. de Silva, M. Glisse, and S. Oudot,The Structure and Stability of Persistence Modules. Cham, Switzerland: Springer, 2016. 23

    2016

  9. [9]

    An introduction to topological data analy- sis: Fundamental and practical aspects for data scientists,

    F. Chazal and B. Michel, “An introduction to topological data analy- sis: Fundamental and practical aspects for data scientists,”Front. Artif. Intell., vol. 4, p. 667963, 2021

    2021

  10. [10]

    Stability of persis- tence diagrams,

    D. Cohen-Steiner, H. Edelsbrunner, and J. Harer, “Stability of persis- tence diagrams,”Discrete Comput. Geom., vol. 37, no. 1, pp. 103-120, 2007

    2007

  11. [11]

    The Ma- halanobis distance,

    R. De Maesschalck, D. Jouan-Rimbaud, and D. L. Massart, “The Ma- halanobis distance,”Chemom. Intell. Lab. Syst., vol. 50, no. 1, pp. 1-18, 2000

    2000

  12. [12]

    Multiscale Mapper: Topologi- cal summarisation via codomain covers,

    T. K. Dey, F. Mémoli, and Y. Wang, “Multiscale Mapper: Topologi- cal summarisation via codomain covers,” inProc. ACM-SIAM Symp. Discrete Algorithms (SODA), 2016, pp. 997-1013

    2016

  13. [13]

    Topological persis- tence and simplification,

    H. Edelsbrunner, D. Letscher, and A. Zomorodian, “Topological persis- tence and simplification,” inProc. IEEE Symp. Found. Comput. Sci., 2000, pp. 454-463

    2000

  14. [14]

    Barcodes: The persistent topology of data,

    R. Ghrist, “Barcodes: The persistent topology of data,”Bull. Amer. Math. Soc., vol. 45, no. 1, pp. 61-75, 2008

    2008

  15. [15]

    Fuzzy clustering with a fuzzy covari- ance matrix,

    D. E. Gustafson and W. C. Kessel, “Fuzzy clustering with a fuzzy covari- ance matrix,” inProc. IEEE Conf. Decision Control, 1979, pp. 761-766

    1979

  16. [16]

    Deconstructing the Mapper algorithm to extract richer topological and temporal features from functional neuroimaging data,

    D. Haşeganet al., “Deconstructing the Mapper algorithm to extract richer topological and temporal features from functional neuroimaging data,”Netw. Neurosci., vol. 8, no. 4, pp. 1355-1382, 2024

    2024

  17. [17]

    A possibilistic approach to cluster- ing,

    R. Krishnapuram and J. M. Keller, “A possibilistic approach to cluster- ing,”IEEE Trans. Fuzzy Syst., vol. 1, no. 2, pp. 98-110, May 1993

    1993

  18. [18]

    The theory of the interleaving distance on multidimensional persistence modules,

    M. Lesnick, “The theory of the interleaving distance on multidimensional persistence modules,”Found. Comput. Math., vol. 15, no. 3, pp. 613-650, 2015

    2015

  19. [19]

    Identification of type 2 diabetes subgroups through topo- logical analysis of patient similarity,

    L. Liet al., “Identification of type 2 diabetes subgroups through topo- logical analysis of patient similarity,”Sci. Transl. Med., vol. 7, no. 311, p. 311ra174, 2015. 24

    2015

  20. [20]

    A comprehensive review of the Mapper algorithm and its applications across various fields (2007–2025),

    V. N. Madukpeet al., “A comprehensive review of the Mapper algorithm and its applications across various fields (2007–2025),”Int. J. Data Sci. Anal., vol. 21, 2025

    2007

  21. [21]

    Nicolau, A

    M. Nicolau, A. J. Levine, and G. Carlsson, Topology-based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival,”Proc. Natl. Acad. Sci. USA, vol. 108, no. 17, pp. 7265-7270, 2011

    2011

  22. [22]

    A topological approach for cancer subtyping from gene expression data,

    O. Rafique and A. H. Mir, “A topological approach for cancer subtyping from gene expression data,”J. Biomed. Inf., vol. 102, p. 103357, 2020

    2020

  23. [23]

    Simulating mixtures of mul- tivariate data with fixed cluster overlap in FSDA library,

    M. Riani, A. Cerioli, D. Perrottaet al., “Simulating mixtures of mul- tivariate data with fixed cluster overlap in FSDA library,”Adv. Data Anal. Classif., vol. 9, pp. 461–481, 2015

    2015

  24. [24]

    Topological methods for the analysis of high-dimensional data sets and 3D object recognition,

    G. Singh, F. Mémoli, and G. Carlsson, “Topological methods for the analysis of high-dimensional data sets and 3D object recognition,” in Eurographics Symp. Point-Based Graphics, 2007, pp. 91-100

    2007

  25. [25]

    Kepler Mapper: A flexible Python implementation of the Mapper algorithm,

    H. J. van Veen, N. Saul, D. Eargle, and S. Mangham, “Kepler Mapper: A flexible Python implementation of the Mapper algorithm,”J. Open Source Softw., vol. 4, no. 42, p. 1315, 2019

    2019

  26. [26]

    Distance metric learning with application to clustering with side-information,

    E. P. Xing, A. Y. Ng, M. I. Jordan, and S. Russell, “Distance metric learning with application to clustering with side-information,” inAdv. Neural Inf. Process. Syst. (NeurIPS), vol. 15, 2002

    2002

  27. [27]

    A novel kernelized fuzzyC-Mean algo- rithm with application in medical image segmentation,

    D. Q. Zhang and S. C. Chen, “A novel kernelized fuzzyC-Mean algo- rithm with application in medical image segmentation,”Artif. Intell. Med., vol. 32, no. 1, pp. 37-50, 2004

    2004

  28. [28]

    Mapper Interactive: A scalable, extendable, and inter- active toolbox for the visual exploration of high-dimensional data,

    Y. Zhouet al., “Mapper Interactive: A scalable, extendable, and inter- active toolbox for the visual exploration of high-dimensional data,” in Proc. IEEE Pacific Vis. Symp. (PacificVis), 2021, pp. 101-110

    2021

  29. [29]

    H. J. Zimmermann,Fuzzy Set Theory-and Its Applications, 4th ed. Dor- drecht, Netherlands: Springer, 2001

    2001

  30. [30]

    UCI Machine Learning Repository,

    D. Dua and C. Graff, “UCI Machine Learning Repository,” University of California, Irvine, School of Information and Computer Sciences, 2019. 25

    2019

  31. [31]

    Scikit-learn: Machine Learning in Python,

    F. Pedregosaet al., “Scikit-learn: Machine Learning in Python,”J. Mach. Learn. Res., vol. 12, pp. 2825-2830, 2011

    2011

  32. [32]

    Breast cancer Wisconsin (diagnostic) data set,

    W. H. Wolberg, W. N. Street, and O. L. Mangasarian, “Breast cancer Wisconsin (diagnostic) data set,” UCI Machine Learning Repository, 1995. 26 Algorithm 1Gustafson-Kessel FuzzyC-Mean (GK-FCM) Require:DatasetX={x 1, . . . , xn} ⊂R p, number of clustersc, fuzzifier m >1, volume parametersq j >0, tolerancetol Ensure:Membership matrixU= [u ij], cluster centre...

    1995