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Stable Similarity Comparison of Persistent Homology Groups

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Pith's one-line read This paper defines a pseudometric on persistence barcodes that is invariant under similarity transformations, not only congruence, with an explicit eigenvalue-sorting formula.

desk verdict Interesting new barcode pseudometric with a real application, but the central operator-theoretic theorem is false; the paper needs major revision, not acceptance. read the letter →

arxiv 2411.09960 v1 pith:KBCAODM7 submitted 2024-11-15 math.AT

Jiaxing He , Bingzhe Hou , Tieru Wu , Yang Cao This is my paper · ORCID
classification math.AT MSC 55N3168T0915A1847B15
keywords persistenthomologybarcodesimilaritypseudometricGrammatrixconformallineartransformationoperatortheoryFrobeniusnormtimbreclassification
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 introduces a family of pseudometrics $d_S^{(p)}$ on persistence barcodes that compares barcodes up to similarity rather than up to congruence. Its main claim is that $d_S^{(2)}$ is a similarity invariant: replacing a barcode by another that is similar in the operator-theoretic sense leaves the distance unchanged, and the distance is exactly the Frobenius distance between the similarity orbits of the barcodes' Gram matrices. If this holds, persistent homology gains a way to compare the shape of topological features while ignoring scale, translation, rotation, and other conformal changes, which bottleneck and Wasserstein distances cannot do. The paper supports this with experiments on synthetic ellipse point clouds and on slices of piano and tuning-fork sound waves, where $d_S^{(1)}$ and $d_S^{(2)}$ separate the instruments regardless of frequency and amplitude, at computation times below bottleneck distance.

What carries the argument

The load-bearing object is the Gram matrix $G_B$ with entries $\int_{\mathbb{R}} \mathbf{1}_{(a_i,b_i]}(x)\mathbf{1}_{(a_j,b_j]}(x)\,dx$, built from the indicator functions of the bars in a barcode $B$. Dividing by the operator norm removes scale, padding with zeros makes barcodes with different numbers of bars comparable, and the sorted eigenvalue sequence $\Lambda_B$ of this normalized padded matrix is what $d_S^{(p)}$ compares. The descending-order matching formula, $d_S^{(2)} = \sqrt{\sum_i |\lambda_i-\mu_i|^2}$ after sorting, is what makes the distance easy to compute, and the similarity-orbit equality in Theorem 3.3 is the mechanism that gives the distance its invariance.

What would settle it

Run the paper's pipeline on two sound waves with identical finite bar content but different vertical offsets, so their H0 barcodes differ only in the starting point of the single infinite interval; because the definition of $d_S$ excludes infinite intervals, the distance is undefined without an added truncation rule, and the value produced will depend on the rule chosen, which can be measured directly.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that a barcode of finite bounded type can be converted into a Gram matrix whose entries are the overlaps of the indicator functions of its intervals; after normalizing by the operator norm and padding with zeros, the eigenvalues of this matrix form a sequence that describes the barcode's shape up to similarity. The pseudometric $d_S^{(2)}$ is the $\ell^2$ distance between two such eigenvalue sequences after the optimal matching, and Proposition 3.2 shows that sorting the eigenvalues in descending order makes the identity matching optimal, giving a closed-form calculation. Theorem 3.3 then states that $d_S^{(2)}$ equals the Frobenius distance between the similarity orbits of the two normalized Gram matrices, which is the sense in which the pseudometric is a similarity invariant. The paper further claims $d_S^{(1)}$ behaves analogously, and that both distances are stable under conformal linear transformations in the reported experiments.

Load-bearing premise

The entire construction requires barcodes whose intervals are all finite, so the Gram matrix has finite trace and discrete eigenvalues; the paper's sound-wave experiments use H0 sublevel-set barcodes that always contain an interval from the global minimum to $+\infty$, and the paper does not state how that unbounded bar is removed or truncated before $d_S$ is applied.

Editorial extensions

If this is right

  • If $d_S^{(2)}$ is a similarity invariant as claimed, objects related by rotation, scaling, and translation will have the same or nearly the same $d_S$ distance, so clustering can be done without alignment or normalization preprocessing.
  • The equality with the similarity-orbit Frobenius distance connects persistence barcodes to operator theory, making spectral approximation and orbit-based perturbation tools available for topological data analysis.
  • On sound waves, the claimed independence from frequency and amplitude means timbre classification can be performed directly on persistence barcodes of sublevel sets, without pitch correction or amplitude normalization.
  • Because $d_S^{(1)}$ and $d_S^{(2)}$ only require sorting eigenvalues and taking an $\ell^p$ difference, pairwise distance matrices are reported to be computable faster than bottleneck distance and comparably to accelerated Wasserstein distance.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • As an editorial extension, replacing the indicator-function Gram kernel with any positive-semidefinite kernel on intervals would produce a family of pseudometrics with the same sorted-eigenvalue formula, potentially extending the construction to barcodes with infinite intervals.
  • A testable consequence the authors do not spell out is that the choice of how to truncate the infinite bar in an H0 sublevel-set barcode will generally change the Gram matrix, so a user of the published pipeline must adopt an extra convention before the theory applies.
  • If the similarity-orbit equality extends beyond finite bounded barcodes, $d_S$ could serve as an invariance-preserving loss for representation learning where the model must be insensitive to affine transformations of the input.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper defines a pseudometric d_S^(p) on barcodes of finite bounded type by taking the eigenvalues of the normalized Gram matrix of interval characteristic functions and comparing them in l^p after optimal matching. For p=2 (and p=1) it proves that the optimal matching is the sorting of eigenvalues (Propositions 3.2 and 3.4). The central theoretical claim is Theorem 3.3, which asserts that d_S^(2) equals the Frobenius distance between the similarity orbits of the corresponding Gram matrices. The paper also reports experiments on Vietoris-Rips barcodes of synthetic ellipse data and on H0 sublevel-set barcodes of sound waves, claiming that d_S is stable under conformal linear transformations and independent of frequency and amplitude, unlike bottleneck and Wasserstein distances. The main theorem is false as stated; the proof relies on an invalid application of the Hoffman-Wielandt theorem. Furthermore, the wave experiments use infinite intervals, which are not covered by Definition 2.

Significance. If Theorem 3.3 were correct, it would establish a new bridge between operator similarity orbits and persistent-homology barcodes, and the proposed pseudometric would offer a genuinely new similarity-invariant comparison tool with computational advantages. The paper includes some sound components: Definition 7 is a genuine pseudometric (Theorem 3.1), the sorting results in Propositions 3.2 and 3.4 are correct, the code is publicly available, and the experimental comparison is clearly described. However, the central operator-theoretic equality is false, and the main application (sublevel-set H0 barcodes of sound waves) is outside the paper's own theoretical framework because it uses infinite bars. These issues invalidate the paper's primary claims.

major comments (2)
  1. [Section 3, Theorem 3.3]
  2. [Section 4.2 and Definition 2]
minor comments (4)
  1. [Examples 1 and 2]
  2. [Theorem 3.1]
  3. [Theorem 3.3 statement]
  4. [Throughout]

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: d_S is defined directly from Gram-matrix eigenvalues; the only serious defect (Theorem 3.3's Hoffman–Wielandt step) is a correctness issue, not a circular reduction.

full rationale

The paper's central construction is not circular. d_S^(p) is defined directly in Definition 7 from the eigenvalue sequences of normalized Gram matrices of interval characteristic functions; the 'similarity invariance' claim is a built-in property of that construction (eigenvalues are invariant under similarity) rather than a quantity fitted to data, imported from a self-citation, or renamed from a known result. Theorem 3.3 aims to connect d_S^(2) to the Frobenius distance between similarity orbits using the external Hoffman–Wielandt theorem; even if that step is mathematically invalid for non-normal similarity-orbit representatives, a false step is a correctness defect, not a circular reduction to the paper's own inputs. The experiments compare d_S with bottleneck and Wasserstein distances on Vietoris–Rips and sublevel-set barcodes; no parameter is tuned on the test data, and no reported 'prediction' is a renamed fit. There are no load-bearing self-citations: the cited operator-theory references [35, 36] are external monographs, and none of the cited works are by the present authors. The scope gap between Definition 2 (finite bounded bars) and the H0 sublevel-set barcodes containing an unbounded interval is an application/rigor gap, not circularity. Accordingly, no circular step can be quoted and exhibited, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The metric rests on standard operator-theoretic facts (Perron-Frobenius, Hoffman-Wielandt, rearrangement inequality) and on the domain assumption that all barcodes are finite bounded. The normalization by operator norm is a modeling choice that builds in scale-invariance. No free parameters or invented entities are introduced; the main unstated assumption is that practical H0 barcodes with infinite intervals can be truncated without changing conclusions.

assumptions (5)
  • standard math Hoffman-Wielandt theorem (Theorem 2.1): for n x n normal matrices A,B, there is a permutation with l2 matching distance at most ||B-A||_F.
    Invoked in Theorem 3.3 to bound d_S by the Frobenius distance between orbit representatives; the proof applies it beyond normality.
  • standard math Perron-Frobenius: for a non-negative symmetric matrix, the operator norm equals the largest eigenvalue.
    Used in Section 3 to define the normalized Gram matrix GB/||GB||.
  • standard math Rearrangement inequality for sorted sequences: the optimal matching for l_p between two descending sequences is the identity.
    Used in Propositions 3.2 and 3.4 to give the calculation formulas.
  • domain assumption Barcodes are of finite bounded type (Definition 2).
    The entire definition of d_S only covers finite multisets of bounded intervals (a,b]; infinite or unbounded bars are excluded.
  • domain assumption Persistent homology groups in practice induce finite bounded type barcodes.
    Remark 1 asserts this, but H0 sublevel-set barcodes of real functions contain an infinite interval, so the assumption is violated in the wave experiments.

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Cite this review

Pith. "Pith review of Stable Similarity Comparison of Persistent Homology Groups." pith.science (2026). https://pith.science/paper/KBCAODM7

@misc{pith2026241109960,
  author       = {Pith},
  title        = {Pith review of: Stable Similarity Comparison of Persistent Homology Groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KBCAODM7}},
  note         = {Machine review of arXiv:2411.09960}
}
abstract

Classification in the sense of similarity is an important issue. In this paper, we study similarity classification in Topological Data Analysis. We define a pseudometric $d_{S}^{(p)}$ to measure the distance between barcodes generated by persistent homology groups of topological spaces, and we provide that our pseudometric $d_{S}^{(2)}$ is a similarity invariant. Thereby, we establish a connection between Operator Theory and Topological Data Analysis. We give the calculation formula of the pseudometric $d_{S}^{(2)}$ $(d_{S}^{(1)})$ by arranging all eigenvalues of matrices determined by barcodes in descending order to get the infimum over all matchings. Since conformal linear transformation is one representative type of similarity transformations, we construct comparative experiments on both synthetic datasets and waves from an online platform to demonstrate that our pseudometric $d_{S}^{(2)}$ $(d_{S}^{(1)})$ is stable under conformal linear transformations, whereas the bottleneck and Wasserstein distances are not. In particular, our pseudometric on waves is only related to the waveform but is independent on the frequency and amplitude. Furthermore, the computation time for $d_{S}^{(2)}$ $(d_{S}^{(1)})$ is significantly less than the computation time for bottleneck distance and is comparable to the computation time for accelerated Wasserstein distance between barcodes.

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Works this paper leans on

42 extracted references · 31 canonical work pages

  1. [1]

    Carlsson, G.: Topology and data. Bull. Amer. Math. Soc. (N.S.) 46(2), 255–308 (2009) https://doi.org/10.1090/S0273-0979-09-01249-X

    doi:10.1090/s0273-0979-09-01249-x 2009

  2. [2]

    Cambridge Monographs on Applied and Computational Mathematics, vol

    Zomorodian, A.J.: Topology for Computing. Cambridge Monographs on Applied and Computational Mathematics, vol. 16, p. 243. Cam- bridge University Press, Cambridge (2005). https://doi.org/10.1017/CB O9780511546945 Springer Nature 2021 LATEX template Stable Similarity Comparison 25

    doi:10.1017/cb 2005

  3. [3]

    Mathematical Surveys and Monographs, vol

    Oudot, S.Y.: Persistence Theory: from Quiver Representations to Data Analysis. Mathematical Surveys and Monographs, vol. 209, p. 218. Amer- ican Mathematical Society, Providence, RI, Providence, RI (2015). https: //doi.org/10.1090/surv/209

    doi:10.1090/surv/209 2015

  4. [4]

    University Lecture Series, vol

    Polterovich, L., Rosen, D., Samvelyan, K., Zhang, J.: Topological Persis- tence in Geometry and Analysis. University Lecture Series, vol. 74, p. 128. American Mathematical Society, Providence, RI (2020)

    work page 2020

  5. [5]

    https://freesound.org/people/Teddy Frost/sounds/334536/ (2016)

    Teddy Frost: piano normal d4.wav. https://freesound.org/people/Teddy Frost/sounds/334536/ (2016)

    work page 2016

  6. [6]

    https://freesound.or g/people/jmuehlhans/sounds/220747/ (2014)

    jmuehlhans: Tuning Fork 440 Hz, Resonance Box. https://freesound.or g/people/jmuehlhans/sounds/220747/ (2014)

    work page 2014

  7. [7]

    Discrete Comput

    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28(4), 511–533 (2002) https: //doi.org/10.1007/s00454-002-2885-2

    doi:10.1007/s00454-002-2885-2 2002

  8. [8]

    Dis- crete Comput

    Zomorodian, A., Carlsson, G.: Computing persistent homology. Dis- crete Comput. Geom. 33(2), 249–274 (2005) https://doi.org/10.1007/ s00454-004-1146-y

    work page 2005

Show all 42 references
  1. [9]

    Discrete Comput

    Carlsson, G., Zomorodian, A.: The theory of multidimensional persis- tence. Discrete Comput. Geom. 42(1), 71–93 (2009) https://doi.org/10. 1007/s00454-009-9176-0

    2009

  2. [10]

    In: Computational Geometry (SCG’06), pp

    Cohen-Steiner, D., Edelsbrunner, H., Morozov, D.: Vines and vineyards by updating persistence in linear time. In: Computational Geometry (SCG’06), pp. 119–126. ACM, New York (2006). https://doi.org/10. 1145/1137856.1137877

    2006

  3. [11]

    https://github.com/rivetTDA/rivet/ (2020)

    The RIVET Developers: RIVET. https://github.com/rivetTDA/rivet/ (2020)

    2020

  4. [12]

    https://mrzv.org/software/dionysus2/

    Morozov, D.: Dionysus 2. https://mrzv.org/software/dionysus2/

  5. [13]

    Computers & Graphics 28(6), 881– 894 (2004)

    Collins, A., Zomorodian, A., Carlsson, G., Guibas, L.J.: A barcode shape descriptor for curve point cloud data. Computers & Graphics 28(6), 881– 894 (2004)

    2004

  6. [14]

    In: International Conference on Artificial Intelligence and Statistics, pp

    Br¨ uel-Gabrielsson, R., Nelson, B.J., Dwaraknath, A., Skraba, P., Guibas, L.J., Carlsson, G.: A topology layer for machine learning. In: International Conference on Artificial Intelligence and Statistics, pp. 1553–1563 (2020). PMLR

    2020

  7. [15]

    Springer Nature 2021 LATEX template 26 Stable Similarity Comparison In: Robotics: Science and Systems, pp

    Silva, V., Ghrist, R., Muhammad, A.: Blind Swarms for Coverage in 2-D. Springer Nature 2021 LATEX template 26 Stable Similarity Comparison In: Robotics: Science and Systems, pp. 335–342 (2005)

    2005

  8. [16]

    In: Handbook of Discrete and Computational Geometry, pp

    Edelsbrunner, H., Morozov, D.: Persistent homology. In: Handbook of Discrete and Computational Geometry, pp. 637–661. Chapman and Hall/CRC, Boca Raton, FL (2017)

    2017

  9. [17]

    In: Handbook of Homotopy Theory

    Carlsson, G.: Persistent homology and applied homotopy theory. In: Handbook of Homotopy Theory. CRC Press/Chapman Hall Handb. Math. Ser., pp. 297–330. CRC Press, Boca Raton, FL (2020)

    2020

  10. [18]

    Artificial Intelligence Review 55(7), 5169–5213 (2022)

    Pun, C.S., Lee, S.X., Xia, K.: Persistent-homology-based machine learn- ing: a survey and a comparative study. Artificial Intelligence Review 55(7), 5169–5213 (2022)

    2022

  11. [19]

    Journal of Machine Learning Research 18(8), 1–35 (2017)

    Adams, H., Emerson, T., Kirby, M., Neville, R., Peterson, C., Shipman, P., Chepushtanova, S., Hanson, E., Motta, F., Ziegelmeier, L.: Persistence images: A stable vector representation of persistent homology. Journal of Machine Learning Research 18(8), 1–35 (2017)

    2017

  12. [20]

    Scientific reports 10(1), 2079 (2020)

    Meng, Z., Anand, D.V., Lu, Y., Wu, J., Xia, K.: Weighted persistent homology for biomolecular data analysis. Scientific reports 10(1), 2079 (2020)

    2020

  13. [21]

    Kovacev-Nikolic, V., Bubenik, P., Nikoli´ c, D., Heo, G.: Using persis- tent homology and dynamical distances to analyze protein binding. Stat. Appl. Genet. Mol. Biol. 15(1), 19–38 (2016) https://doi.org/10.1515/sa gmb-2015-0057

    2016 doi

  14. [22]

    Wiley Online Library

    Poulenard, A., Skraba, P., Ovsjanikov, M.: Topological function optimiza- tion for continuous shape matching 37(5), 13–25 (2018). Wiley Online Library

    2018

  15. [23]

    In: Proceedings of the IEEE/CVF International Conference on Computer Vision, pp

    Ahmed, F., Nuwagira, B., Torlak, F., Coskunuzer, B.: Topo-CXR: Chest X-ray TB and Pneumonia Screening with Topological Machine Learning. In: Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 2326–2336 (2023)

    2023

  16. [24]

    Scientific reports 9(1), 8764 (2019)

    Oyama, A., Hiraoka, Y., Obayashi, I., Saikawa, Y., Furui, S., Shiraishi, K., Kumagai, S., Hayashi, T., Kotoku, J.: Hepatic tumor classification using texture and topology analysis of non-contrast-enhanced three-dimensional T1-weighted MR images with a radiomics approach. Scien...

    2019

  17. [25]

    Discrete Comput

    Cohen-Steiner, D., Edelsbrunner, H., Harer, J.: Stability of persistence diagrams. Discrete Comput. Geom. 37(1), 103–120 (2007) https://doi.or g/10.1007/s00454-006-1276-5 Springer Nature 2021 LATEX template Stable Similarity Comparison 27

    2007 doi

  18. [26]

    Acta Appl

    d’Amico, M., Frosini, P., Landi, C.: Natural pseudo-distance and optimal matching between reduced size functions. Acta Appl. Math. 109(2), 527– 554 (2010) https://doi.org/10.1007/s10440-008-9332-1

    2010 doi

  19. [27]

    Bauer, U., Lesnick, M.: Induced matchings and the algebraic stability of persistence barcodes. J. Comput. Geom. 6(2), 162–191 (2015) https: //doi.org/10.20382/jocg.v6i2a9

    2015 doi

  20. [28]

    Lesnick, M.: The theory of the interleaving distance on multidimensional persistence modules. Found. Comput. Math.15(3), 613–650 (2015) https: //doi.org/10.1007/s10208-015-9255-y

    2015 doi

  21. [29]

    Acta Appl

    Frosini, P.: Stable comparison of multidimensional persistent homology groups with torsion. Acta Appl. Math. 124, 43–54 (2013) https://doi.or g/10.1007/s10440-012-9769-0

    2013 doi

  22. [30]

    Carlsson, G., Filippenko, B.: Persistent homology of the sum metric. J. Pure Appl. Algebra 224(5), 106244–25 (2020) https://doi.org/10.1016/ j.jpaa.2019.106244

    2020

  23. [31]

    Homology Homotopy Appl

    Frosini, P., Landi, C., M´ emoli, F.: The persistent homotopy type distance. Homology Homotopy Appl. 21(2), 231–259 (2019) https://doi.org/10. 4310/HHA.2019.v21.n2.a13

    2019

  24. [32]

    Hatcher, A.: Algebraic Topology, p. 544. Cambridge University Press, Cambridge (2002)

    2002

  25. [33]

    Robinson, A., Turner, K.: Hypothesis testing for topological data analysis. J. Appl. Comput. Topol. 1(2), 241–261 (2017) https://doi.org/10.1007/ s41468-017-0008-7

    2017

  26. [34]

    arXiv preprint arXiv: 2106.13589 (2021)

    Bjerkevik, H.B., Lesnick, M.: lp-distances on multiparameter persistence modules. arXiv preprint arXiv: 2106.13589 (2021)

    2021 arXiv

  27. [35]

    Herrero, D.A.: Approximation of Hilbert Space Operators. Vol. I. Research Notes in Mathematics, vol. 72, p. 255. Pitman (Advanced Publishing Program), Boston, MA (1982)

    1982

  28. [36]

    Apostol, C., Fialkow, L.A., Herrero, D.A., Voiculescu, D.: Approximation of Hilbert Space Operators. Vol. II. Research Notes in Mathematics, vol. 102, p. 524. Pitman (Advanced Publishing Program), Boston, MA (1984)

    1984

  29. [37]

    In: Proceedings of the Statistical Data Analysis Based on the L1 Norm Conference, Neuchatel, Switzerland, vol

    Rdusseeun, L.K.P.J., Kaufman, P.: Clustering by means of medoids. In: Proceedings of the Statistical Data Analysis Based on the L1 Norm Conference, Neuchatel, Switzerland, vol. 31 (1987)

    1987

  30. [38]

    Jr.: Hierarchical grouping to optimize an objective function

    Ward, J.H. Jr.: Hierarchical grouping to optimize an objective function. Springer Nature 2021 LATEX template 28 Stable Similarity Comparison J. Amer. Statist. Assoc. 58, 236–244 (1963)

    1963

  31. [39]

    Science 315(5814), 972–976 (2007) https://doi.org/10.1126/scie nce.1136800

    Frey, B.J., Dueck, D.: Clustering by passing messages between data points. Science 315(5814), 972–976 (2007) https://doi.org/10.1126/scie nce.1136800

    2007 doi

  32. [40]

    Journal of the American Statistical Association 77(383), 553 (1983)

    Fowlkes, E.B., Mallows, C.L.: A method for comparing two hierarchical clusterings. Journal of the American Statistical Association 77(383), 553 (1983)

    1983

  33. [41]

    In: 2016 Proceedings of the Eighteenth Workshop on Algorithm Engineering and Experiments (ALENEX), pp

    Kerber, M., Morozov, D., Nigmetov, A.: Geometry helps to compare per- sistence diagrams. In: 2016 Proceedings of the Eighteenth Workshop on Algorithm Engineering and Experiments (ALENEX), pp. 103–112 (2016). SIAM

    2016

  34. [42]

    The Journal of Open Source Software3(29), 925 (2018) https://doi.org/10.21105/joss.00925

    Tralie, C., Saul, N., Bar-On, R.: Ripser.py: A lean persistent homology library for python. The Journal of Open Source Software3(29), 925 (2018) https://doi.org/10.21105/joss.00925

    2018 doi

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