Pith. sign in

REVIEW 3 major objections 3 minor 42 references

On the Spectral Synthesis of Lipschitz Persistence Diagram Vectorizations

T0 review · 3 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Lipschitz persistence diagram vectorizations are, modulo constants, exactly bounded 1-cocycles on the Grothendieck completion; when scalarizations are additive plus Fourier–Stieltjes, they generate synthesizable varieties.

desk verdict Serious paper, clean cocycle reformulation, but the Fourier machinery rests on a self-cited local-compactness theorem and the abstract overstates the separable-pair result. read the letter →

arxiv 2607.16957 v2 pith:4DC2STR6 submitted 2026-07-18 math.FA math.AT

Charles Fanning , Mehmet Emin Aktas This is my paper · ORCID
classification math.FAmath.AT MSC 43A4546B2055N31
keywords persistencediagramvectorizationsspectralsynthesisLipschitzextensionscocyclerepresentationFourier–Stieltjestransformsvirtualdiagramskernelsharmonicanalysis
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

Persistence diagram vectorizations are functions from the space of persistence diagrams into a Banach space. This paper shows that if such a vectorization admits a Lipschitz extension to the Grothendieck completion of the diagram space, then that extension is completely described, modulo constants, by its translation increments, which form a bounded 1-cocycle. It then proves a spectral-synthesis criterion: whenever every bounded linear scalarization of the extension splits into an additive homomorphism plus a Fourier–Stieltjes transform, the variety generated by all scalar cocycles is synthesizable, meaning its exponential monomials are dense in the topology of pointwise convergence. The criterion is carried to separable metric pairs by restricting to uniformly discrete subpairs and imposing a uniform total-variation bound on pushed-forward representing measures. A final conjecture proposes that Lipschitz continuity alone is sufficient for this uniformly discrete spectral synthesis, supported by a classification of common vectorizations.

What carries the argument

The increment cocycle: for a Lipschitz f on K(X,A), the family b_h(x)=f(x+h)-f(x) satisfies the cocycle identity b_{h+k}=τ_k b_h+b_k, and the reconstruction f_b(x)=b_x(0) recovers f modulo constants; this turns the extension problem into a Banach-space cocycle boundary value problem. The spectral-synthesis criterion then uses the decomposition of λ(b_x(0)) into an additive homomorphism plus a Fourier–Stieltjes transform (an integral of characters against a finite complex regular Borel measure on the Pontryagin dual); the multiplier χ(h)-1 exposes characters in the support of the measure as exponential monomials in the variety. For separable pairs, the machinery includes uniformly discrete su

What would settle it

Compute, for a uniformly discrete subpair Z with K(Z,A)≅ℤ, the variety generated by all scalarizations λ(f∘i_Z) for a Lipschitz extension f that does not satisfy the additive-plus-Fourier–Stieltjes hypothesis. If that variety contains a point-mass function such as 1_a but exponential monomials fail to be pointwise dense, the synthesis claim fails. Alternatively, test the hypothesis of Theorem 3.19 directly: for the Betti curve or persistence landscape extension on X=[0,1], A=∂Ω, verify whether sup_n ||(P^Z_n)#μ_{n,λ}||_TV is finite for every uniformly discrete subpair Z; finding a Z and λ wher

Watch

Extended reading notes

Core claim

The central discovery is an isometric isomorphism between Lipschitz functions on the virtual persistence diagram group K(X,A), taken modulo constants, and the space of bounded 1-cocycles for the translation action of K(X,A) on bounded E-valued functions: a Lipschitz f is recovered from its cocycle b by f(x)=b_x(0), and the cocycle norm equals the Lipschitz seminorm. On top of this, the paper proves that if every λ∈E* makes the primitive x↦λ(b_x(0)) a sum of an additive homomorphism and a Fourier–Stieltjes transform of a finite complex measure on the Pontryagin dual, then the functions λ(b_h), h∈K(X,A), λ∈E*, generate a synthesizable variety—equivalently, exponential monomials span a pointwis

Load-bearing premise

The Fourier-analytic half of the paper assumes the virtual persistence diagram group K(X,A) is locally compact—equivalently, by the paper's own infrastructure, that the pointed metric quotient (X/A,d1,[A]) is uniformly discrete—and the separable-pair extension additionally assumes a uniform total-variation bound on pushed-forward measures that is not verified for the example vectorizations.

Editorial extensions

If this is right

  • Lipschitz vectorizations modulo constants are exactly bounded 1-cocycles; the Lipschitz seminorm equals the cocycle norm, so extension questions become cocycle questions.
  • Under the additive-plus-Fourier–Stieltjes hypothesis, the variety generated by all scalar cocycles is synthesizable: exponential monomials are pointwise dense.
  • For separable metric pairs, the uniform total-variation bound on pushed-forward measures implies uniformly discrete spectral synthesis on every uniformly discrete subpair.
  • The classification tables show that every studied vectorization or kernel feature map that admits a Lipschitz extension—Betti curves, L∞ landscapes and silhouettes, persistence images, template functions, Schauder basis, persistence weighted Gaussian and scale-space kernels—also has uniformly discrete spectral synthesis; those without Lipschitz extensions (persistent entropy, tropical coordinates,
  • For translation-invariant kernels, the family of increment kernels determines the representing measure away from the trivial character, so spectral information is recoverable from increment feature maps.

Reading between the lines

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

  • If the paper's conjecture holds, Lipschitz continuity becomes a complete characterization of uniformly discrete spectral synthesis for separable metric pairs; then any 1-Wasserstein-Lipschitz vectorization would automatically be approximable by exponential monomials on finite subsets, checkable by a Lipschitz estimate alone.
  • The sufficient condition in Theorem 3.5 is stated for every bounded linear functional; a testable weakening is whether a generating set of scalarizations (for example, coordinate functionals for finite-dimensional E) suffices, which would reduce verification cost for concrete vectorizations.
  • The uniform total-variation bound in Theorem 3.19 is not verified for the example vectorizations; computing it for Betti curves or L∞ landscapes on a non-discrete separable pair would decide whether the extension theorem covers the conjecture's intended scope or needs a sharper hypothesis.
  • The cocycle reconstruction f_b(x)=b_x(0) suggests a constructive path to new vectorizations: design bounded 1-cocycles directly, since any such cocycle yields a Lipschitz extension modulo constants.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper studies persistence diagram vectorizations D(X,A)→E and their Lipschitz extensions to the Grothendieck completion K(X,A). The main structural theorem (Theorem 3.1) gives an isometric isomorphism between Lip(K(X,A),E)/E and the space of bounded 1-cocycles Z^1(K(X,A),ℓ^∞(K(X,A),E)). Theorem 3.5 shows that if, for every bounded linear functional λ on E, the primitive x↦λ(b_x(0)) belongs to Hom(K(X,A),ℂ)+B(K(X,A)), then the scalar cocycles λ(b_h), h∈K(X,A), λ∈E*, generate a synthesizable variety. Section 3.3 extends the notion to separable metric pairs via restrictions to uniformly discrete subpairs: Theorem 3.19 asserts uniformly discrete spectral synthesis under a B-representability condition on ε_n-nets plus a uniform total-variation bound on pushed-forward measures. The final section classifies many standard vectorizations and kernel feature maps (Tables 1–4), with proofs in Appendices A–C, and proposes Conjecture 4.1.

Significance. Assuming the infrastructural Theorem 2.9, the paper offers a genuine and interesting bridge between persistence diagram vectorizations and abstract harmonic analysis: the cocycle representation of Lipschitz extensions is clean, and Theorem 3.5 is a substantial sufficient condition for synthesizability of the generated variety. The paper also provides detailed, checkable proofs for a wide range of examples, including negative results for entropy, tropical coordinates, and several kernel feature maps, which is valuable for the TDA community. Its main limitation is that the Fourier-analytic machinery rests on an unproved, self-cited local-compactness theorem and on strong uniformity conditions that are not verified in the examples; the abstract claims more than the theorem statements deliver.

major comments (3)
  1. [§3.2, Theorem 3.5 (also Theorem 1.2)] The Fourier-analytic notions used in Theorem 3.5—B(K(X,A)), the dual Ĝ, Bochner's theorem, Pontryagin duality, and Stone–Weierstrass on C(Ĝ)—are well-defined only when K(X,A) is a locally compact abelian group. Section 3.2 begins with 'Since (K(X,A),ρ) is discrete,' which is imported from the self-cited Theorem 2.9 (Fanning and Aktas 2026c, Theorem 3.2) and is not proved or independently verified in this manuscript. The statements of Theorems 1.2 and 3.5 omit such a hypothesis. If Theorem 2.9 is false or does not apply to the stated metric pairs, B(K(X,A)) is undefined and Theorem 3.5 loses its foundation. Please state the discreteness/local-compactness hypothesis explicitly in Theorem 3.5 and its corollaries, and either prove Theorem 2.9 or supply an independent verification for the metric pairs used.
  2. [Abstract, Definition 3.14, Theorem 3.19] The abstract claims to 'extend spectral synthesis to ... separable metric pairs,' but Theorem 3.19 only establishes uniformly discrete spectral synthesis: it synthesizes V_Z(f) on each uniformly discrete subpair Z, not a variety on K(X,A) itself when X/A is not uniformly discrete. Moreover, the hypotheses of Theorem 3.19/Lemma 3.18—λ∘f∘i_{Y_n}∈B(K(Y_n,A)) and sup_n ||ν^Z_{n,λ}||_{TV}<∞ for every uniformly discrete Z and every λ—are strong and are not verified for any of the examples in Section 4. The classification is instead obtained through Lemmas A.2, A.3, A.9, B.9, and C.4, which do not check those bounds. Please qualify the abstract and either weaken the claim or show how the conditions can be met.
  3. [§4 and Appendix A–C] Because the examples bypass Theorem 3.19, the paper's advertised 'extension to separable metric pairs' remains an existence theorem whose hypotheses are not operationalized. In particular, the uniform total-variation condition in Lemma 3.18(2) is a priori as hard to check as the desired conclusion. This is not an internal inconsistency, but it limits the significance of the extension theorem. A discussion of how the condition might be verified, or a comparison with the elementary routes used in the appendix, would strengthen the paper.
minor comments (3)
  1. [Appendix B, Proposition B.3] The title reads 'The of the persistence VLAD vectorizations'; it should read 'The subclass of the persistence VLAD vectorizations' or similar.
  2. [Table 1 and Proposition A.18] In Table 1, the Persistence Landscape row is labeled L^p(ℕ×ℝ), but Proposition A.18 shows that the Lipschitz case occurs only for p=∞. The table convention permits this, but the row could confuse readers; consider adding 'p=∞ for the Lipschitz case' or a footnote.
  3. [Definition 3.14 and Theorem 3.19] The term 'uniformly discrete spectral synthesis' is introduced in Definition 3.14, which appears after the reader has already met the concept in the introduction and Theorems 3.19–3.20. Moving the definition before the theorem statements would improve readability.

Circularity Check

1 steps flagged · score 4.0 of 10

Central spectral-synthesis results rely on a load-bearing self-cited local-compactness theorem, but the core derivation is otherwise not circular.

  1. self citation load bearing [Section 2.2 (Theorem 2.9); used in §3.2 and Lemma 3.18]
    "The following are equivalent (Fanning and Aktas, 2026c, Theorem 3.2 and Corollary 3.3): 1. The pointed metric quotient (X/A, d1, [A]) is uniformly discrete. 2. The metric group (K(X,A), ρ) is discrete. 3. The metric group (K(X,A), ρ) is locally compact."

    The Fourier-analytic core of the paper (Theorem 3.5, Corollary 3.7, Lemma 3.18, Theorem 3.19) requires K(X,A) to be a locally compact abelian group. Section 3.2 asserts 'Since (K(X,A), ρ) is discrete' and Lemma 3.18 repeats 'By (Fanning and Aktas, 2026c, Theorem 3.2 and Corollary 3.3), the group G is a discrete locally compact abelian group.' The only support for this infrastructure is a self-cited prior theorem by the same authors, stated in Theorem 2.9 and not proved or independently verified in this manuscript. If that theorem fails or does not apply to the stated metric pairs, the Fourier–Stieltjes and Pontryagin arguments are undefined. This makes the central claim depend on a load-bearing self-citation, though not on a definitional equivalence.

full rationale

The main derivation is not circular by construction. Theorem 3.1 is proved directly from the cocycle identity and translation invariance. Theorem 3.5 is a genuine sufficient condition: the hypothesis that each scalarization λ(b_x(0)) lies in Hom(K(X,A),C)+B(K(X,A)) is not the same as the conclusion that the functions λ(b_h) generate a synthesizable variety; the proof uses Fourier–Stieltjes representation, Stone–Weierstrass density of characters, and Riesz duality to establish density of exponential monomials. Lemma 3.17 similarly gives a real proof that varieties generated by Fourier–Stieltjes transforms are synthesizable. The example classifications are independent checks against specific vectorizations. The one significant circularity concern is the paper's reliance on Fanning and Aktas 2026c for local compactness / discreteness of K(X,A), which is neither proved nor independently verified here; this is load-bearing for all Fourier-analytic statements. However, that is an imported self-citation supporting infrastructure, not an equivalence between the input and the predicted result, so a moderate score of 4 is appropriate. Unverified assumptions such as the TV-bound condition in Theorem 3.19 are correctness risks, not circularity.

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

The paper introduces no fitted parameters and no new physical or categorical entities. Its load-bearing inputs are standard published frameworks (Bubenik–Elchesen, Székelyhidi), two self-cited infrastructure theorems, and two strong conditional hypotheses that are not discharged for the main examples. The honest contribution is a conditional framework plus a classification of known vectorizations.

assumptions (5)
  • standard math The Bubenik–Elchesen framework: D(X,A) is the quotient of the free commutative monoid by the diagonal, K(X,A) is its Grothendieck completion, and the 1-Wasserstein metric extends uniquely to a translation-invariant metric ρ (Bubenik and Elchesen 2022, Sections 2.3–2.5, Prop 3.11, Def 4.10).
    The entire paper operates inside this framework; it is a published, externally cited foundation.
  • domain assumption Uniform discreteness of (X/A,d1,[A]) is equivalent to discreteness/local compactness of K(X,A) (Fanning and Aktas 2026c, Theorem 3.2 and Corollary 3.3).
    This self-cited result is the doorway for Pontryagin duality and Fourier–Stieltjes theory used in Theorems 3.5, 3.19 and the examples. It is not proven in this paper and has no external verification.
  • standard math Székelyhidi's spectral synthesis framework: varieties, exponential monomials, synthesizability, Theorem 15.6 equivalence with density of exponential monomials, and the finite torsion-free rank criterion (Székelyhidi 2014, 2015, 2024).
    Background theory used to define the goal of the paper.
  • ad hoc to paper Theorem 3.5 hypothesis: for every λ ∈ E*, the primitive function x ↦ λ(b_x(0)) belongs to Hom(K(X,A), C) + B(K(X,A)).
    This is a strong sufficient condition assumed rather than proved; it is not verified for most vectorizations except through the appendix constructions.
  • ad hoc to paper Theorem 3.19 hypotheses: for each ε_n-net subpair Y_n and every λ, λ∘f∘i_{Y_n} ∈ B(K(Y_n,A)), and for every uniformly discrete subpair Z the pushed-forward measures satisfy sup_n ||ν^Z_{n,λ}||_TV < ∞.
    These uniformity conditions are load-bearing for the separable metric pair extension but are not shown to hold for the example vectorizations; the examples are verified by separate direct constructions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the Spectral Synthesis of Lipschitz Persistence Diagram Vectorizations." pith.science (2026). https://pith.science/paper/4DC2STR6

@misc{pith2026260716957,
  author       = {Pith},
  title        = {Pith review of: On the Spectral Synthesis of Lipschitz Persistence Diagram Vectorizations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4DC2STR6}},
  note         = {Machine review of arXiv:2607.16957}
}
abstract

A persistence diagram represents the birth and death of homology classes along a filtration as a multiset of intervals, and we consider persistence diagram vectorizations to be maps $D(X,A) \to E$ sending persistence diagrams over a metric pair $(X,A)$ to values in a Banach space $E$. We prove an isometric isomorphism between Lipschitz extensions of vectorizations, modulo constants, on the Grothendieck completion $K(X,A)$ of $D(X,A)$ and the bounded $1$-cocycles for the translation action of $K(X,A)$ on $\ell^\infty(K(X,A),E)$. We then prove that if every bounded linear functional on $E$ maps a Lipschitz extension of a vectorization to the sum of an additive homomorphism from $K(X,A)$ to $\mathbb{C}$ and a Fourier--Stieltjes transform of a finite complex regular Borel measure on $\widehat{K(X,A)}$, then the associated cocycle generates a synthesizable variety. We extend spectral synthesis to Lipschitz vectorizations of persistence diagrams on separable metric pairs and show whether examples of persistence diagram vectorizations in the literature have Lipschitz extensions with spectral synthesis.

Figures

Figures reproduced from arXiv: 2607.16957 by the authors.

Figure 1
Figure 1. The 𝜀𝑛 -nets 𝑌𝑛∕𝐴 ⊆ 𝑋∕𝐴 show how 𝑃 𝑍 𝑛 approximates finite supports in 𝑍∕𝐴 by points of 𝑌𝑛∕𝐴. The overlap represents compatible restrictions from larger uniformly discrete subpairs to the 𝑍-level, whose restricted translates generate 𝑍(𝑓). Theorem 3.13. Let 𝑍 ↦ 𝑍 assign a variety 𝑍 ⊆ 𝐶(𝐾(𝑍, 𝐴)) to each uniformly discrete subpair 𝐴 ⊆ 𝑍 ⊆ 𝑋. Suppose that 𝑉𝑍(𝑓) ⊆ 𝑍 for every uniformly discrete subpair 𝐴 ⊆ 𝑍 ⊆ 𝑋, an… view at source ↗

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 8 linked inside Pith

  1. [1]

    Journal of Applied and Computational Topology , year =

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

  2. [2]

    Computational Geometry , volume =

    Peter Bubenik and Alex Elchesen , title =. Computational Geometry , volume =. 2022 , issn =. doi:10.1016/j.comgeo.2022.101882 , url =

    arXiv 2022

  3. [3]

    2024 , eprint =

    Relative Optimal Transport , author =. 2024 , eprint =. doi:10.48550/arXiv.2411.05678 , url =

    arXiv doi:10.48550/arxiv.2411.05678 2024

  4. [4]

    2025 , eprint =

    Charles Fanning and Mehmet Aktas , title =. 2025 , eprint =. doi:10.48550/arXiv.2512.07282 , url =

    arXiv doi:10.48550/arxiv.2512.07282 2025

  5. [5]

    2026 , eprint =

    Charles Fanning and Mehmet Aktas , title =. 2026 , eprint =. doi:10.48550/arXiv.2602.15153 , url =

    doi:10.48550/arxiv.2602.15153 2026

  6. [6]

    2026 , eprint =

    Charles Fanning and Mehmet Aktas , title =. 2026 , eprint =. doi:10.48550/arXiv.2603.02117 , url =

    doi:10.48550/arxiv.2603.02117 2026

  7. [7]

    2026 , eprint =

    Charles Fanning and Mehmet Aktas , title =. 2026 , eprint =. doi:10.48550/arXiv.2605.09866 , url =

    arXiv doi:10.48550/arxiv.2605.09866 2026

  8. [8]

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

    Herbert Edelsbrunner and David Letscher and Afra Zomorodian , title =. Proceedings 41st Annual Symposium on Foundations of Computer Science , year =. doi:10.1109/SFCS.2000.892133 , url =

    arXiv

Show all 42 references
  1. [9]

    Discrete & Computational Geometry , year =

    David Cohen-Steiner and Herbert Edelsbrunner and John Harer , title =. Discrete & Computational Geometry , year =. doi:10.1007/s00454-006-1276-5 , url =

    doi

  2. [10]

    Oudot , title =

    Steve Y. Oudot , title =. 2015 , publisher =. doi:10.1090/surv/209 , url =

    2015 doi

  3. [11]

    Discrete & Computational Geometry , year =

    Afra Zomorodian and Gunnar Carlsson , title =. Discrete & Computational Geometry , year =. doi:10.1007/s00454-004-1146-y , url =

    doi

  4. [12]

    Kalton , title =

    Gilles Godefroy and Nigel J. Kalton , title =. Studia Mathematica , volume =. 2003 , publisher =. doi:10.4064/sm159-1-6 , url =

    2003 doi

  5. [13]

    2005 , publisher =

    Juha Heinonen , title =. 2005 , publisher =

    2005

  6. [14]

    Folland , title =

    Gerald B. Folland , title =. 2016 , pages =. doi:10.1201/b19172 , url =

    2016 doi

  7. [15]

    New Results on Spectral Synthesis , journal =

    L. New Results on Spectral Synthesis , journal =. 2024 , volume =. doi:10.1007/s00605-024-01950-6 , url =

    2024 doi

  8. [16]

    Harmonic and Spectral Analysis , publisher =

    L. Harmonic and Spectral Analysis , publisher =. 2014 , isbn =. doi:10.1142/8924 , url =

    2014 doi

  9. [17]

    Spectral Synthesis on Special Varieties , journal =

    L. Spectral Synthesis on Special Varieties , journal =. 2015 , url =

    2015

  10. [18]

    2004 , publisher =

    Alain Berlinet and Christine Thomas-Agnan , title =. 2004 , publisher =. doi:10.1007/978-1-4419-9096-9 , url =

    2004 doi

  11. [19]

    IEEE Transactions on Pattern Analysis and Machine Intelligence , volume =

    Dashti Ali and Aras Asaad and Maria-Jose Jimenez and Vidit Nanda and Eduardo Paluzo-Hidalgo and Manuel Soriano-Trigueros , title =. IEEE Transactions on Pattern Analysis and Machine Intelligence , volume =. 2023 , doi =

    2023

  12. [20]

    Frontiers in Artificial Intelligence , volume =

    Felix Hensel and Michael Moor and Bastian Rieck , title =. Frontiers in Artificial Intelligence , volume =. 2021 , doi =

    2021

  13. [21]

    Mathematics , volume =

    Aras Asaad and Dashti Ali and Taban Majeed and Rasber Rashid , title =. Mathematics , volume =. 2022 , doi =

    2022

  14. [22]

    Advances in Computational Mathematics , volume =

    Yu-Min Chung and Austin Lawson , title =. Advances in Computational Mathematics , volume =. 2022 , doi =

    2022

  15. [23]

    2018 , eprint =

    Chi Seng Pun and Kelin Xia and Si Xian Lee , title =. 2018 , eprint =. doi:10.48550/arXiv.1811.00252 , url =

    2018 doi arXiv

  16. [24]

    On the Stability of Persistent Entropy and New Summary Functions for Topological Data Analysis , journal =

    Nieves Atienza and Roc. On the Stability of Persistent Entropy and New Summary Functions for Topological Data Analysis , journal =. 2020 , issn =. doi:10.1016/j.patcog.2020.107509 , url =

    2020

  17. [25]

    An Entropy-Based Persistence Barcode , journal =

    Harish Chintakunta and Thanos Gentimis and Roc. An Entropy-Based Persistence Barcode , journal =. 2015 , issn =. doi:10.1016/j.patcog.2014.06.023 , url =

    2015 doi

  18. [26]

    Journal of Applied and Computational Topology , year =

    Eric Berry and Yen-Chi Chen and Jessi Cisewski-Kehe and Brittany Terese Fasy , title =. Journal of Applied and Computational Topology , year =. doi:10.1007/s41468-020-00048-w , url =

    doi

  19. [27]

    Journal of Machine Learning Research , volume =

    Peter Bubenik , title =. Journal of Machine Learning Research , volume =. 2015 , issn =. doi:10.5555/2789272.2789275 , url =

    2015

  20. [28]

    Topological Data Analysis , series =

    Peter Bubenik , title =. Topological Data Analysis , series =. 2020 , publisher =. doi:10.1007/978-3-030-43408-3_4 , url =

    2020 doi

  21. [29]

    Stochastic Convergence of Persistence Landscapes and Silhouettes , booktitle =

    Fr. Stochastic Convergence of Persistence Landscapes and Silhouettes , booktitle =. 2014 , publisher =. doi:10.1145/2582112.2582128 , url =

    2014

  22. [30]

    Journal of Machine Learning Research , volume =

    Henry Adams and Tegan Emerson and Michael Kirby and Rachel Neville and Chris Peterson and Patrick Shipman and Sofya Chepushtanova and Eric Hanson and Francis Motta and Lori Ziegelmeier , title =. Journal of Machine Learning Research , volume =. 2017 , issn =. doi:10.5555/31220...

    2017

  23. [31]

    Perea and Elizabeth Munch and Firas A

    Jose A. Perea and Elizabeth Munch and Firas A. Khasawneh , title =. Foundations of Computational Mathematics , volume =. 2023 , issn =. doi:10.1007/s10208-022-09567-7 , url =

    2023 doi

  24. [32]

    Tropical Coordinates on the Space of Persistence Barcodes , journal =

    Sara Kali. Tropical Coordinates on the Space of Persistence Barcodes , journal =. 2019 , issn =. doi:10.1007/s10208-018-9379-y , url =

    2019 doi

  25. [33]

    arXiv preprint arXiv:2510.10347 , year =

    Peter Bubenik and Zachariah Ross , title =. arXiv preprint arXiv:2510.10347 , year =. doi:10.48550/arXiv.2510.10347 , url =

    doi

  26. [34]

    IEEE Transactions on Pattern Analysis and Machine Intelligence , volume =

    Ilya Chevyrev and Vidit Nanda and Harald Oberhauser , title =. IEEE Transactions on Pattern Analysis and Machine Intelligence , volume =. 2020 , issn =. doi:10.1109/TPAMI.2018.2885516 , url =

    2020

  27. [35]

    SIAM Journal on Applied Algebra and Geometry , volume =

    Chad Giusti and Darrick Lee , title =. SIAM Journal on Applied Algebra and Geometry , volume =. 2023 , issn =. doi:10.1137/22M1528471 , url =

    2023 doi

  28. [36]

    Persistence Codebooks for Topological Data Analysis , journal =

    Bartosz Zieli. Persistence Codebooks for Topological Data Analysis , journal =. 2021 , issn =. doi:10.1007/s10462-020-09897-4 , url =

    2021 doi

  29. [37]

    ATOL: Measure Vectorization for Automatic Topologically-Oriented Learning , booktitle =

    Martin Royer and Fr. ATOL: Measure Vectorization for Automatic Topologically-Oriented Learning , booktitle =. 2021 , publisher =. doi:10.48550/arXiv.1909.13472 , url =

    2021 doi arXiv

  30. [38]

    PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures , booktitle =

    Mathieu Carri. PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures , booktitle =. 2020 , publisher =. doi:10.48550/arXiv.1904.09378 , url =

    2020 doi arXiv

  31. [39]

    Mathieu Carri. Sliced. Proceedings of the 34th International Conference on Machine Learning , series =. 2017 , publisher =. doi:10.48550/arXiv.1706.03358 , url =

    2017 doi arXiv

  32. [40]

    Advances in Neural Information Processing Systems 31 , editor =

    Tam Le and Makoto Yamada , title =. Advances in Neural Information Processing Systems 31 , editor =. 2018 , doi =

    2018

  33. [41]

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

    Genki Kusano and Yasuaki Hiraoka and Kenji Fukumizu , title =. Proceedings of the 33rd International Conference on Machine Learning , series =. 2016 , publisher =. doi:10.48550/arXiv.1601.01741 , url =

    2016 doi arXiv

  34. [42]

    Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) , pages =

    Jan Reininghaus and Stefan Huber and Ulrich Bauer and Roland Kwitt , title =. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) , pages =. 2015 , doi =

    2015

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.