pith. machine review for the scientific record. sign in

arxiv: 2605.09866 · v1 · submitted 2026-05-11 · 💻 cs.CG · math.AT· math.FA

Recognition: 2 theorem links

· Lean Theorem

Higher-order Persistence Diagrams

Charles Fanning, Mehmet Aktas

Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:53 UTC · model grok-4.3

classification 💻 cs.CG math.ATmath.FA
keywords higher-order persistence diagramspersistence diagramscontainment relationstopological data analysiszeta transformsdiagram aggregationharmonic analysis
0
0 comments X

The pith

Higher-order persistence diagrams arise recursively from containment relations among intervals, enabling direct aggregation via zeta transforms.

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

TDA pipelines often generate many persistence diagrams, but common vectorizations discard the containment relations between intervals that carry structural information. The paper defines higher-order persistence diagrams by recursively treating one interval's containment of another as a new higher-order interval. This keeps the relational structure available for comparison and aggregation of entire collections. Harmonic analysis reduces the evaluation of these aggregated diagrams to zeta transforms, which can be performed in nearly linear time instead of quadratic pair enumeration. The result matters because it lets pipelines retain interval-level details while scaling to larger collections.

Core claim

Higher-order persistence diagrams are obtained from a recursive construction in which containment relations among ordinary persistence intervals define higher-order persistence intervals. This construction supports direct comparison and aggregation on the diagrams themselves while preserving interval-level structure. Harmonic analysis reduces frequency-space evaluations of the aggregated diagrams to zeta transforms, avoiding explicit construction of the higher-order objects and replacing quadratic pair enumeration with nearly linear-time computation.

What carries the argument

Recursive containment relation among persistence intervals that defines higher-order intervals, reduced to zeta transforms for aggregation.

If this is right

  • Aggregations of persistence diagrams retain which intervals contain others rather than flattening them away.
  • Comparison and kernel methods can operate directly on the preserved interval relations.
  • Frequency-space zeta transforms replace explicit quadratic enumeration with faster evaluation for large collections.
  • Existing TDA pipelines can incorporate the construction without changing the input diagrams.

Where Pith is reading between the lines

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

  • The recursive structure could extend naturally to other hierarchical topological summaries that possess containment or nesting relations.
  • Linear-time aggregation might enable new statistical or machine-learning procedures on diagram collections that treat containment as a first-class feature.
  • If the transform equivalence holds, closed-form expressions for certain summary statistics of higher-order diagrams may become available.

Load-bearing premise

The zeta-transform reduction in frequency space exactly reproduces the higher-order diagrams obtained from the recursive containment construction.

What would settle it

Compute the explicit higher-order diagram for a small set of intervals by enumerating all containments, then compare every entry against the result of the zeta-transform evaluation on the same set.

Figures

Figures reproduced from arXiv: 2605.09866 by Charles Fanning, Mehmet Aktas.

Figure 1
Figure 1. Figure 1: (A) A weighted simplicial complex defining the first filtration. (B) The barcode computed [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The virtual persistence diagram of order [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Algorithm 1 explicitly constructs the aggregate, whereas Algorithm 2 computes a scalar [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Aggregation runtime for A1(D(G1), . . . , D(Gm)) across ten graph models (m = 30). Runtime measures aggregation only. Plots show runtime vs. N for naïve and harmonic methods, with shaded bands showing the range over 30 runs [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Pipeline for GNN training drift. At each epoch, a weighted parameter graph is extracted [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
read the original abstract

Many topological data analysis (TDA) pipelines compute large collections of persistence diagrams, yet vectorizations and kernel methods discard the rank-induced implication relations among persistence intervals that are essential for faithful structural comparison and interpretability. We introduce higher-order persistence diagrams, a recursive construction in which containment relations among persistence intervals define higher-order persistence intervals. This construction performs comparison and aggregation directly on persistence diagrams and preserves interval-level structure. We use harmonic analysis to reduce frequency-space evaluations of aggregated diagrams to zeta transforms. This reduction avoids explicit construction of higher-order diagrams and replaces quadratic pair enumeration with nearly linear-time evaluation. Experiments on random network models show substantial speedups over explicit aggregation. Anonymized code is available at https://anonymous.4open.science/r/higher-order-persistence-8201.

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

3 major / 2 minor

Summary. The paper introduces higher-order persistence diagrams via a recursive construction that defines new intervals from containment relations among intervals in standard persistence diagrams. It claims that harmonic analysis reduces frequency-space evaluations of aggregated higher-order diagrams to zeta transforms, avoiding explicit quadratic pair enumeration and enabling nearly linear-time computation while preserving interval-level structure. Experiments on random network models demonstrate substantial speedups over explicit aggregation, with anonymized code provided.

Significance. If the zeta-transform reduction is shown to be faithful to the recursive containment definition, the approach could enable scalable structural comparison and aggregation of persistence diagrams in TDA without discarding rank-induced relations, addressing a practical bottleneck in pipelines that handle large diagram collections. The emphasis on preserving interval structure and the provision of code for reproducibility are strengths.

major comments (3)
  1. [Abstract and method description] The central claim that harmonic analysis reduces aggregated higher-order diagram evaluations exactly to zeta transforms (avoiding information loss relative to the recursive containment construction) lacks an explicit derivation, equivalence proof, or invertibility argument. This equivalence is load-bearing for the nearly-linear-time claim and the assertion that the method computes the stated higher-order diagrams.
  2. [Harmonic analysis reduction] No verification is provided that the zeta convolution on the poset of intervals recovers the exact containment-induced higher-order intervals (e.g., via Möbius inversion or direct comparison to the recursive definition). Without this, it is unclear whether the computed object matches the introduced higher-order diagrams.
  3. [Experiments] The experiments report speedups on random networks but provide no quantitative validation that the zeta-transform outputs match explicit higher-order diagram aggregation on even small instances, nor error analysis for the approximation.
minor comments (2)
  1. [Introduction] The abstract and introduction would benefit from a brief statement of the poset structure on intervals and a small illustrative example of the recursive containment construction.
  2. [Method] Notation for the frequency-space representation and the precise definition of the zeta transform in this context should be introduced with equations for clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and constructive suggestions. The comments correctly identify areas where additional rigor and validation would strengthen the manuscript. We will revise the paper to include an explicit derivation of the harmonic analysis reduction, a verification of the zeta transform equivalence, and quantitative experimental validation on small instances.

read point-by-point responses

  1. Referee: [Abstract and method description] The central claim that harmonic analysis reduces aggregated higher-order diagram evaluations exactly to zeta transforms (avoiding information loss relative to the recursive containment construction) lacks an explicit derivation, equivalence proof, or invertibility argument. This equivalence is load-bearing for the nearly-linear-time claim and the assertion that the method computes the stated higher-order diagrams.

    Authors: We agree that the manuscript would benefit from an explicit derivation. In the revised version we will add a dedicated subsection deriving the reduction from the incidence algebra of the interval poset. The argument will show that the frequency-space aggregation via harmonic analysis is precisely the zeta transform over containment relations, with the inverse given by Möbius inversion on the same poset. This establishes exact equivalence to the recursive definition and confirms that no structural information is lost. revision: yes

  2. Referee: [Harmonic analysis reduction] No verification is provided that the zeta convolution on the poset of intervals recovers the exact containment-induced higher-order intervals (e.g., via Möbius inversion or direct comparison to the recursive definition). Without this, it is unclear whether the computed object matches the introduced higher-order diagrams.

    Authors: We will address this by adding both a theoretical verification and a small-scale direct comparison. The new material will explicitly apply Möbius inversion to recover the higher-order intervals from the zeta convolution and will include a side-by-side computation on a synthetic collection of intervals, confirming that the outputs coincide with those obtained from the recursive containment construction. revision: yes

  3. Referee: [Experiments] The experiments report speedups on random networks but provide no quantitative validation that the zeta-transform outputs match explicit higher-order diagram aggregation on even small instances, nor error analysis for the approximation.

    Authors: We acknowledge the absence of direct numerical validation. The revised experimental section will include a controlled study on small persistence diagrams (fewer than 20 intervals) for which explicit enumeration remains tractable. We will report the exact numerical agreement between the zeta-transform outputs and the direct aggregation, together with an error analysis demonstrating that discrepancies are limited to floating-point precision and that the method is exact in exact arithmetic. revision: yes

Circularity Check

0 steps flagged

No circularity detected; recursive construction and zeta-transform reduction presented as independent computational shortcut

full rationale

The abstract defines higher-order persistence diagrams via a new recursive containment-based construction on intervals, then separately invokes harmonic analysis to reduce aggregated evaluations to zeta transforms for efficiency. No quoted equations or self-citations in the provided text show the zeta transform being defined in terms of the higher-order diagrams themselves, nor any fitted parameter renamed as a prediction, nor load-bearing self-citation chains. The derivation chain remains self-contained against external benchmarks, with the reduction framed as an algorithmic optimization rather than a definitional equivalence. This matches the default expectation of no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the validity of the recursive containment definition and the equivalence of the zeta-transform reduction to explicit higher-order aggregation. No free parameters are mentioned. The paper assumes standard properties of persistence intervals and harmonic analysis from prior TDA literature.

axioms (2)
  • domain assumption Persistence intervals are well-defined and their containment relations can be used to recursively define higher-order intervals without additional structure.
    Invoked in the introduction of the higher-order construction.
  • domain assumption Harmonic analysis applies to aggregated persistence diagrams such that frequency-space evaluation reduces exactly to zeta transforms.
    Central to the efficiency claim in the abstract.
invented entities (1)
  • higher-order persistence diagrams no independent evidence
    purpose: To encode containment relations among standard persistence intervals for direct comparison and aggregation.
    New recursive construction introduced in the paper; no independent evidence provided in abstract.

pith-pipeline@v0.9.0 · 5418 in / 1475 out tokens · 51568 ms · 2026-05-12T04:53:51.718762+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

72 extracted references · 72 canonical work pages

  1. [1]

    Journal of Applied and Computational Topology , year =

    Bubenik, Peter and Elchesen, Alex , title =. Journal of Applied and Computational Topology , year =

    work page

  2. [2]

    Computational Geometry , year =

    Bubenik, Peter and Elchesen, Alex , title =. Computational Geometry , year =

    work page

  3. [3]

    Foundations of Computational Mathematics , year =

    Bubenik, Peter and de Silva, Vin and Scott, Jonathan , title =. Foundations of Computational Mathematics , year =

    work page

  4. [4]

    Journal of Machine Learning Research , year =

    Bubenik, Peter , title =. Journal of Machine Learning Research , year =

    work page

  5. [5]

    Journal of Applied and Computational Topology , year =

    Bubenik, Peter and Scott, Jonathan and Stanley, Donald , title =. Journal of Applied and Computational Topology , year =

    work page

  6. [6]

    Exact weights and path metrics for triangulated categories and the derived category of persistence modules , journal =

    Bubenik, Peter and V. Exact weights and path metrics for triangulated categories and the derived category of persistence modules , journal =. 2025 , volume =

    work page 2025

  7. [7]

    Proceedings 41st Annual Symposium on Foundations of Computer Science , year =

    Edelsbrunner, Herbert and Letscher, David and Zomorodian, Afra , title =. Proceedings 41st Annual Symposium on Foundations of Computer Science , year =

    work page

  8. [8]

    Discrete & Computational Geometry , year =

    Zomorodian, Afra and Carlsson, Gunnar , title =. Discrete & Computational Geometry , year =

    work page

  9. [9]

    , title =

    Oudot, Steve Y. , title =. 2015 , series =

    work page 2015

  10. [10]

    Discrete & Computational Geometry , year =

    Cohen-Steiner, David and Edelsbrunner, Herbert and Harer, John , title =. Discrete & Computational Geometry , year =

    work page

  11. [11]

    Foundations of Computational Mathematics , year =

    Cohen-Steiner, David and Edelsbrunner, Herbert and Harer, John , title =. Foundations of Computational Mathematics , year =

    work page

  12. [12]

    Proceedings of the 22nd Annual Symposium on Computational Geometry , year =

    Cohen-Steiner, David and Edelsbrunner, Herbert and Morozov, Dmitriy , title =. Proceedings of the 22nd Annual Symposium on Computational Geometry , year =

    work page

  13. [13]

    The Structure and Stability of Persistence Modules , year =

    Chazal, Fr. The Structure and Stability of Persistence Modules , year =. 1207.3674 , archivePrefix =

    work page arXiv

  14. [14]

    2008 , eprint =

    Carlsson, Gunnar and de Silva, Vin , title =. 2008 , eprint =

    work page 2008

  15. [15]

    Algorithms and Computation , editor =

    Carlsson, Gunnar and Singh, Gurjeet and Zomorodian, Afra , title =. Algorithms and Computation , editor =. 2009 , volume =

    work page 2009

  16. [16]

    Foundations of Computational Mathematics , year =

    Lesnick, Michael , title =. Foundations of Computational Mathematics , year =

    work page

  17. [17]

    Journal of Applied and Computational Topology , year =

    Patel, Amit , title =. Journal of Applied and Computational Topology , year =

    work page

  18. [18]

    Generalized Persistence Diagrams for Persistence Modules over Posets , journal =

    Kim, Woojin and M. Generalized Persistence Diagrams for Persistence Modules over Posets , journal =. 2021 , volume =

    work page 2021

  19. [19]

    , title =

    Betthauser, Leo and Bubenik, Peter and Edwards, Parker B. , title =. Discrete & Computational Geometry , year =

    work page

  20. [20]

    Journal of Applied and Computational Topology , year =

    Divol, Vincent and Lacombe, Thibaut , title =. Journal of Applied and Computational Topology , year =

    work page

  21. [21]

    Persistence Stability for Geometric Complexes , journal =

    Chazal, Fr. Persistence Stability for Geometric Complexes , journal =. 2014 , volume =

    work page 2014

  22. [22]

    Journal of Computational Geometry , year =

    Bauer, Ulrich and Lesnick, Michael , title =. Journal of Computational Geometry , year =

    work page

  23. [23]

    and Tillmann, Ulrike and Grindrod, Peter and Harrington, Heather A

    Otter, Nina and Porter, Mason A. and Tillmann, Ulrike and Grindrod, Peter and Harrington, Heather A. , title =. EPJ Data Science , year =

    work page

  24. [24]

    Advances in Neural Information Processing Systems 20 , year =

    Rahimi, Ali and Recht, Benjamin , title =. Advances in Neural Information Processing Systems 20 , year =

    work page

  25. [25]

    2004 , doi =

    Berlinet, Alain and Thomas-Agnan, Christine , title =. 2004 , doi =

    work page 2004

  26. [26]

    Transactions of the American Mathematical Society , year =

    Aronszajn, Nachman , title =. Transactions of the American Mathematical Society , year =

    work page

  27. [27]

    2025 , eprint =

    Fanning, Charles and Aktas, Mehmet , title =. 2025 , eprint =

    work page 2025

  28. [28]

    2026 , eprint =

    Fanning, Charles and Aktas, Mehmet , title =. 2026 , eprint =

    work page 2026

  29. [29]

    , title =

    Folland, Gerald B. , title =. 2015 , doi =

    work page 2015

  30. [30]

    1984 , volume =

    Berg, Christian and Christensen, Jens Peter Reus and Ressel, Paul , title =. 1984 , volume =

    work page 1984

  31. [31]

    Lecture Notes in Mathematics , year =

    Jorgensen, Palle and Pedersen, Steen and Tian, Feng , title =. Lecture Notes in Mathematics , year =

    work page

  32. [32]

    , title =

    Liggett, Thomas M. , title =. 2010 , volume =

    work page 2010

  33. [33]

    1994 , volume =

    Fukushima, Masatoshi and Oshima, Yoichi and Takeda, Masayoshi , title =. 1994 , volume =

    work page 1994

  34. [34]

    2009 , volume =

    Applebaum, David , title =. 2009 , volume =

    work page 2009

  35. [35]

    1975 , volume =

    Berg, Christian and Forst, Gunnar , title =. 1975 , volume =

    work page 1975

  36. [36]

    Morris, S. A. , title =. Proceedings of the American Mathematical Society , year =

    work page

  37. [37]

    , title =

    Morris, Sidney A. , title =. 2003 , publisher =

    work page 2003

  38. [38]

    and Morris, Sidney A

    Hofmann, Karl H. and Morris, Sidney A. , title =. 2023 , volume =

    work page 2023

  39. [39]

    arXiv preprint arXiv:0911.3788 , year =

    van Neerven, Jan , title =. arXiv preprint arXiv:0911.3788 , year =

    work page arXiv

  40. [40]

    Weaver, Nik , title =

    work page

  41. [41]

    Vakhania, N. N. and Tarieladze, V. I. and Chobanyan, S. A. , title =. 1987 , volume =

    work page 1987

  42. [42]

    , title =

    Parlett, Beresford N. , title =

    work page

  43. [43]

    Journal of Machine Learning Research , year =

    Adams, Henry and Emerson, Tegan and Kirby, Michael and Neville, Rachel and Peterson, Chris and Shipman, Patrick and Chepushtanova, Sofya and Hanson, Eric and Motta, Francis and Ziegelmeier, Lori , title =. Journal of Machine Learning Research , year =

    work page

  44. [44]

    Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition , year =

    Reininghaus, Jan and Huber, Stefan and Bauer, Ulrich and Kwitt, Roland , title =. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition , year =

    work page

  45. [45]

    Proceedings of the 33rd International Conference on Machine Learning , year =

    Kusano, Genki and Hiraoka, Yasuaki and Fukumizu, Kenji , title =. Proceedings of the 33rd International Conference on Machine Learning , year =

    work page

  46. [46]

    Sliced Wasserstein Kernel for Persistence Diagrams , journal =

    Carri. Sliced Wasserstein Kernel for Persistence Diagrams , journal =. 2017 , volume =

    work page 2017

  47. [47]

    Advances in Neural Information Processing Systems , year =

    Le, Tam and Yamada, Makoto , title =. Advances in Neural Information Processing Systems , year =

    work page

  48. [48]

    Advances in Neural Information Processing Systems , year =

    Hofer, Christoph and Kwitt, Roland and Niethammer, Marc and Uhl, Andreas , title =. Advances in Neural Information Processing Systems , year =

    work page

  49. [49]

    Artificial Intelligence Review , year =

    Pun, Chi Seng and Lee, Si Xian and Xia, Kelin , title =. Artificial Intelligence Review , year =

    work page

  50. [50]

    and Byrne, Nicholas and Oksuz, Ilkay and Zimmer, Veronika A

    Clough, James R. and Byrne, Nicholas and Oksuz, Ilkay and Zimmer, Veronika A. and Schnabel, Julia A. and King, Andrew P. , title =. IEEE Transactions on Pattern Analysis and Machine Intelligence , year =

    work page

  51. [51]

    Medical Image Analysis , year =

    Qaiser, Talha and Tsang, Yee-Wah and Taniyama, Daiki and Sakamoto, Naoya and Nakane, Kazuaki and Epstein, David and Rajpoot, Nasir , title =. Medical Image Analysis , year =

    work page

  52. [52]

    Proceedings of the IEEE/CVF International Conference on Computer Vision , year =

    Qi, Yaolei and He, Yuting and Qi, Xiaoming and Zhang, Yuan and Yang, Guanyu , title =. Proceedings of the IEEE/CVF International Conference on Computer Vision , year =

    work page

  53. [53]

    and Tillmann, Ulrike and Pugh, Christopher W

    Vipond, Oliver and Bull, Joshua Adam and Macklin, Philip S. and Tillmann, Ulrike and Pugh, Christopher W. and Byrne, Helen M. and Harrington, Heather A. , title =. Proceedings of the National Academy of Sciences , year =

    work page

  54. [54]

    European Congress of Mathematics , year =

    Edelsbrunner, Herbert and Morozov, Dmitriy , title =. European Congress of Mathematics , year =

    work page

  55. [55]

    1963 , volume =

    Milnor, John , title =. 1963 , volume =

    work page 1963

  56. [56]

    Riehl, Emily , title =

    work page

  57. [57]

    1971 , volume =

    Mac Lane, Saunders , title =. 1971 , volume =

    work page 1971

  58. [58]

    2022 , volume =

    Riehl, Emily and Verity, Dominic , title =. 2022 , volume =

    work page 2022

  59. [59]

    Hatcher, Allen , title =

    work page

  60. [60]

    Peter , title =

    May, J. Peter , title =

    work page

  61. [61]

    Daniel and Dwyer, William G

    Christensen, J. Daniel and Dwyer, William G. and Isaksen, Daniel C. , title =. Advances in Mathematics , year =

    work page

  62. [62]

    Annals of Mathematics , year =

    Hashimoto, Junji , title =. Annals of Mathematics , year =

    work page

  63. [63]

    Feuilletages sur les vari

    Haefliger, Andr. Feuilletages sur les vari. Topology , year =

    work page

  64. [64]

    Contemporary Mathematics , year =

    Hurder, Steven , title =. Contemporary Mathematics , year =

    work page

  65. [65]

    Journal of Noncommutative Geometry , year =

    Accornero, Luca and Crainic, Marius , title =. Journal of Noncommutative Geometry , year =

    work page

  66. [66]

    1966 , volume =

    Husemoller, Dale , title =. 1966 , volume =

    work page 1966

  67. [67]

    and Strogatz, Steven H

    Watts, Duncan J. and Strogatz, Steven H. , title =. Nature , year =

    work page

  68. [68]

    Inventiones mathematicae , volume=

    Pointwise theorems for amenable groups , author=. Inventiones mathematicae , volume=. 2001 , publisher=

    work page 2001

  69. [69]

    Journal of Machine Learning Research , volume=

    Topology of deep neural networks , author=. Journal of Machine Learning Research , volume=

    work page

  70. [70]

    arXiv preprint arXiv:2211.13706 , year=

    Fast M " obius and Zeta Transforms , author=. arXiv preprint arXiv:2211.13706 , year=

    work page arXiv

  71. [71]

    Correlation with Molecular Orbital Energies and Hydrophobicity , author=

    Structure-Activity Relationship of Mutagenic Aromatic and Heteroaromatic Nitro Compounds. Correlation with Molecular Orbital Energies and Hydrophobicity , author=. Journal of Medicinal Chemistry , volume=

    work page

  72. [72]

    TUDataset: A collection of benchmark datasets for learning with graphs.arXiv preprint arXiv:2007.08663,

    TUDataset: A collection of benchmark datasets for learning with graphs , author=. arXiv preprint arXiv:2007.08663 , year=

    work page arXiv 2007