Kan Approximations of the Persistent Homology Transform

Justin Curry; Shreya Arya

REVIEW 3 major objections 5 minor 33 references

Kan Approximations of the Persistent Homology Transform

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read The persistent homology transform can be interpolated from finitely many sampled directions, with an explicitly bounded error.

desk verdict Main theorem false as stated because the left-Kan colimit omits the special top-parallel morphisms; the gap is concrete and fixable, and the paper deserves a careful revision. read the letter →

arxiv 2507.22816 v1 pith:G3RIMGVA submitted 2025-07-30 math.AT cs.CGmath.CT

Shreya Arya , Justin Curry This is my paper · ORCID

classification math.ATcs.CGmath.CT MSC 55N3118A40

keywords persistenthomologytransformKanextensionsinterleavingdistanceshapeinterpolationtopologicaldataanalysisLipschitzstabilityo-minimalstructures

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

This paper asks whether the persistent homology transform (PHT) of a shape can be interpolated from finitely many sampled directions and whether the interpolation error can be bounded. The authors answer both questions affirmatively by working at the module level, where each sampled direction contributes not just a persistence diagram but the homology maps that connect sublevel sets across directions. Adapting the Kan-extension interpolation theory of [24,25], they construct left, right, and center extensions of the PHT and prove these extensions are 1-Lipschitz. Their main approximation results are Theorem 4.6, bounding the interleaving distance between the true PHT and the extension by twice the Hausdorff distance from the sampled direction set to the sphere, and Theorem 4.12, bounding the fully discrete case by $2\varepsilon_A + \varepsilon_T$. A correct proof would give the first provable scheme for approximating the PHT from finite data, with the error shrinking linearly as the directional sampling becomes denser.

What carries the argument

The load-bearing object is the space-time category $A_X$. Its objects are pairs $(v, \alpha)$ consisting of a direction $v$ in the sample set $A$ and a filtration value $\alpha$ in an interval $X$, and it has a morphism $(v, \alpha) \to (w, \beta)$ whenever $\alpha + d(v, w) \le \beta$, together with a top morphism to $(w, \sup X)$ for every pair; Lemma 2.12 and Lemma 2.13 turn these inequalities into inclusions of sublevel sets, so the PHT data assemble into a functor $G : A_X \to \mathbf{Vect}$. The left, right, and center Kan extensions of $G$ along the inclusion $A_X \to \mathbb{S}^{d-1}_X$ then provide the interpolation formulas: colimit over the past light cone, limit over the future light cone, and the image of the natural map from the former to the latter. The coherence condition of [24,25] is exactly the requirement that the data form such a functor, and Proposition 3.3 proves that every PHT satisfies it automatically.

What would settle it

Compute the left Kan extension for the single-point shape $M = \{(0, 1)\}$ with one sampled direction $v = (1, 0)$, query the south pole $w = (0, -1)$, and compare the birth positions of the true and extended $\Theta$-persistence modules: the paper's Example 4.5 predicts an interleaving distance of exactly $\pi$, matching the bound $2 d_g(v, w)$. More generally, a decisive test would evaluate, for a range of constructible shapes inside the unit disk and direction sets $A$, the true interleaving distance between $\mathrm{PHT}^X_n(M)$ and its center Kan extension and check whether it ever exceeds $2 d_H(A, \mathbb{S}^{d-1})$ (or $2\varepsilon_A + \varepsilon_T$ for fully discrete data).

Watch

Extended reading notes

Core claim

The paper's central claim is that the persistent homology transform of a constructible set $M$ inside the unit disk can be interpolated from finitely many sampled directions, and the interpolation error can be controlled, provided the samples include the maps that homology induces between sublevel sets in different directions. The authors model the sampled data as a functor $G : A_X \to \mathbf{Vect}$ from a space-time pre-order category $A_X$ to vector spaces, and they construct three canonical extensions of $G$ to the whole sphere of directions: the left Kan extension (a colimit over past light cones), the right Kan extension (a limit over future light cones), and the center Kan extension (the image of the natural map between them). Theorem 4.6 states that for every direction and every homological degree, the interleaving distance between the true transform and any of these extensions is at most twice the Hausdorff distance from the sample direction set $A$ to the sphere. Theorem 4.12 extends the bound to fully discrete data, giving $2\varepsilon_A + \varepsilon_T$ when directions form an $\varepsilon_A$-net and filtration values form an $\varepsilon_T$-net. The example of the unit disk shows that the left and right extensions can introduce spurious bars where the center extension does not, and Example 4.5 shows the factor of two in the pointwise bound is sharp.

Load-bearing premise

The approximation only exists if the sample carries the actual homology maps between sublevel sets in different directions—a functor $G$, not just the persistence diagrams—and these maps must be estimated heuristically in practice.

Editorial extensions

If this is right

If the sample direction set is an $\varepsilon$-net of the sphere, the Kan-extended PHT is within interleaving distance $2\varepsilon$ of the true PHT in every homological degree, so denser directional sampling provably improves the approximation.
The extension is 1-Lipschitz, so nearby directions produce nearby persistence modules and the approximated PHT varies continuously over the sphere.
When both directions and height values are sampled, the error is at most $2\varepsilon_A + \varepsilon_T$, so the approximation degrades only linearly in the two sampling radii.
The construction works for any constructible shape contained in the unit disk, with no further tameness assumptions beyond o-minimality.

Reading between the lines

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

A practical implementation of this scheme would need to estimate the inter-direction homology maps from persistence diagrams via coherent matchings; the paper explicitly leaves that estimation to future work, so it is the main bottleneck between the theorem and a software pipeline.
The same space-time category construction should transfer to the Euler characteristic transform and Betti curve transforms, since the only geometric input needed is the nesting of sublevel sets; that would give analogous interpolation bounds for those signatures.
Lemma 4.10's use of angular coordinates hints that angular sampling grids, whose spacings are measured geodesically, yield strictly better constants than Euclidean grids of the same cardinality; a direct experimental comparison of the two sampling schemes on non-convex shapes would settle which is preferable in practice.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Here's the take on 2507.22816. The paper's central result, Theorem 4.6, is false as stated. The displayed definition of L^-(w,β) in Section 3.2 omits the special top morphisms (v,α)→(w,sup X) that were added to the space-time category in Section 3.1. The under-comma slice at β=sup X contains all sampled objects, not just those satisfying α+d(v,w)≤β. The paper explicitly says L^- is the projection of that slice, but the formula is a strict subset at the top. The stress-test counterexample is correct: take M={p=(1,0)}, A={v=(1,0)}, X=I, and query direction w at small angle δ. The displayed L^- gives the zero module everywhere, while the true PHT has a bar [cosδ,1). The interleaving distance is infinite, so the 2·d_H bound fails. This is not a notation quibble; Lemma 4.1 and Theorem 4.6 are proved for the truncated colimit, and the proof's construction of Ψ silently relies on the special morphism exactly at β+d(v,w)=sup X. Theorem 4.12 inherits the same problem.

The fix is simple: define L^- as the actual comma slice, or explicitly add the top objects to the colimit. The stress-test notes that with the corrected definition, the counterexample gives a finite interleaving distance (1-cosδ) below 2d(v,w), so the theorem is likely salvageable. That is the key referee demand.

Credit where due: this is the first application of the Bubenik–de Silva–Nanda interpolation machinery to the PHT, and the new angular coordinate system (Definition 2.7) is a real contribution. The paper is clearly written and the categorical setup is appropriate. The Lipschitz extension theorem (Theorem 3.10) is a nice conceptual step. The idea that module-level sample data, including inter-direction maps, can yield provable interpolation is genuinely useful for TDA shape analysis.

Softer spots: the right and center Kan extensions are not covered by any error bound, and the examples for them look under-verified; the reader is right to flag possible inconsistencies with the formal definitions. The practical barrier of needing inter-direction homology maps is acknowledged in Remark 3.11 but remains a serious limitation.

Bottom line: this paper is worth a serious referee, but the current version cannot be accepted as is. The main theorem must be corrected and re-proved. If the fix goes through, the result is a solid contribution to the PHT literature.

Referee Report

3 major / 5 minor

Summary. The paper proposes a method to interpolate the persistent homology transform (PHT) from finitely many sampled directions by viewing the direction-indexed PHT as a coherent functor on a space-time category and applying the Kan-extension theory of Bubenik, de Silva, and Nanda. Section 3 constructs left, right, and center Kan extensions in both Euclidean and angular coordinates, and Section 4 proves pointwise and global interleaving-distance bounds, including a fully discrete bound when both directions and filtration values are sampled. The stated main result, Theorem 4.6, is that the interleaving distance between the true PHT and its Kan-extension approximation is at most twice the Hausdorff distance from the sample direction set to the sphere, for every homological degree.

Significance. If correct, this is a genuinely useful interpolation theorem: it turns a finite sample of the PHT at the module level into a Lipschitz approximation with explicit, parameter-free error control, and it appears to be the first such approximation guarantee for the PHT from finite directional data. The derivation is parameter-free, the main machinery is credited to prior work by Bubenik, de Silva, and Nanda, and the paper honestly records the key practical limitation in Remark 3.11, namely that computing the extension requires inter-direction homology maps, not merely per-direction persistence diagrams. The examples are instructive, and the distinction between Euclidean and angular coordinates is thoughtfully developed.

major comments (3)

[§3.2, Lemma 4.1, Theorem 4.6] The displayed definition of the past light cone, L^-(w,β) = {(v,α)∈A_X | α+d(v,w)≤β}, is not the projection of the under-comma category ι↓(w,β) when β = sup X, because A_X contains the always-morphism (v,α)→(w,sup X) for every (v,α). At β = sup X the comma slice contains all objects, while the displayed inequality omits those with α+d(v,w)>sup X, notably α = sup X whenever d(v,w)>0. This is load-bearing: Lemma 4.1 constructs the map Σ_{v,w} by placing (v,α) in the diagram that defines C_w(T_{d(v,w)}(α)), and when T_{d(v,w)}(α)=sup X this requires exactly the omitted special morphism. Concretely, take M={p=(1,0)}, A={v=(1,0)}, X=[-1,1], and w=(cos δ,sin δ). Under the displayed formula, C_w(β)=0 for all β, because the only nonzero H_0(M_{v,α}) occurs at α=1, which never satisfies 1+d(v,w)≤β; the true PHT has a bar [cos δ,1), so the interleaving distance is infinite and Theorem 4.6 is false for the extension as defined by the displayed set. The fix is to define L^- as the projection of the actual comma slice, or to add the special top morphisms explicitly, and then re-verify the proof and the examples; with that corrected definition the argument in Lemma 4.1 appears to go through, but the current text is internally inconsistent.
[Theorem 4.6 and §4.1–4.2] Remark 4.2 restricts the proof to the left Kan extension, and the proof of Lemma 4.1 only constructs the interleaving for that case. Theorem 4.6, however, is stated for "the Kan extension approximation" without qualification, Section 3.3 presents the right and center extensions as equally valid alternatives, and Example 4.3 asserts the same bound for "any of the three Kan extensions." No argument is supplied for the right or center extensions; duality is not automatic under the capped shift, and the center extension is an image rather than a limit or colimit. The theorem should either be explicitly restricted to the left Kan extension or supplemented with proofs for the other two.
[§3.3, Example 3.9] The right- and center-extension computations use the future light cone L^+(w,β) = {(v,α)∈A_X | β+d(v,w)≤α}, which is subject to the same top-parameter truncation as L^- once the always-morphisms to sup X are included in A_X. When β+d(v,w)≤sup X fails, the formal over-comma category still contains (v,sup X) by the special morphism, so the displayed L^+ omits objects that the categorical definition includes. The barcodes in Example 3.9 and Figure 4 should therefore be recomputed against the formal comma-category definitions, or the definition of L^+ should be corrected consistently. This matters because the paper never proves the error bound for these extensions, so the examples are the only evidence that they behave as claimed.

minor comments (5)

[§2.6 and Corollary 3.4] The heading "Lipshitz Stability" and the word "Lipshitz" in Corollary 3.4 should be corrected to "Lipschitz."
[Theorem 3.10] The text "described aboove" contains a typo and should read "described above."
[§3.1] In the definition of A_X, the always-morphism is declared for any v,w∈S^{d-1}, but the objects of A_X are pairs (v,α) with v∈A; the declaration should be for v,w∈A, or the category should be defined on all of S^{d-1} and then restricted.
[§4.4, Theorem 4.12] The proof uses a one-sided notion of an ε_T-net, requiring that for every y there exists t'∈T with y≤t'≤y+ε_T, whereas the theorem statement says only that T is an ε_T-net of X. This stronger condition should be stated in the theorem, since a standard symmetric net near sup X need not provide such a t'.
[Example 4.5] This example is the tightness example and is correct in spirit, but after the correction of the L^- definition it should be re-annotated to show which step relies on the special morphism to sup X in the space-time category.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the approximation bound is parameter-free and rests on independent Kan-extension machinery; the sole self-citation is non-load-bearing.

full rationale

The central derivation (Lemma 4.1, Theorems 4.6 and 4.12) contains no fitted constants and does not define its target quantity in terms of its inputs. The extension is defined by a Kan colimit/limit over a past/future light cone, and the error bound is proved directly from geometric nesting inclusions M_{v,t} subset M_{w,s} (Lemmas 2.12 and 2.13) plus the Hausdorff-net condition; the only comparison quantity entering the theorem is the given sample set A, and the result is stated for arbitrary A. The coherence result (Proposition 3.3) and the existence of a Lipschitz extension rest on the independent earlier work of de Silva-Nanda and Bubenik-de Silva-Nanda [24,25], with no author overlap in the load-bearing role. Remark 3.11 openly concedes that in practice the internal maps between directions are not given and must be estimated by coherent matchings; this is a real limitation of applicability, but it is not a circular step because the bounds are conditional on the stated module-level input and do not smuggle the target PHT into that input. The sole self-citation, [30], appears in Remark 2.10 only to identify the PHT interleaving distance with the bottleneck distance of [30], and it is not needed for either the construction or the error bound. The mathematical concern that the displayed formula for L^-(w,beta) omits the top morphisms at beta = sup X is a correctness issue rather than a circularity issue, and does not change this verdict.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central derivation relies on the coherence of the PHT (Proposition 3.3), which is proven. No free parameters are fitted. The main external input is the Kan extension machinery from [24,25].

assumptions (4)

standard math Constructible (o-minimal) sets are triangulable and have finite-dimensional homology over a field.
Used to define the PHT and ensure persistence modules decompose into interval modules; Sections 2.1 and 2.2.
domain assumption The shape M is rescaled to lie in the unit disk so that heights range in [-1,1] and the Lipschitz constant is 1.
Remark 2.6; this normalization is without loss of generality up to scaling.
standard math The interpolation theorems of Bubenik, de Silva, and Nanda [24,25]: coherent maps extend to 1-Lipschitz maps and Kan extensions compute them.
Theorem 3.1 and Lemma 3.2 adopt these as black boxes.
ad hoc to paper The space-time category AX is a valid pre-order with the 'always morphism' to sup X added.
Section 3.1; this construction is adapted from [25] but the specific ordering is used to prove coherence and interleaving bounds. Its treatment of top-height morphisms is a source of ambiguity in the right/center examples.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Kan Approximations of the Persistent Homology Transform." pith.science (2026). https://pith.science/paper/G3RIMGVA

@misc{pith2026250722816,
  author       = {Pith},
  title        = {Pith review of: Kan Approximations of the Persistent Homology Transform},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G3RIMGVA}},
  note         = {Machine review of arXiv:2507.22816}
}

abstract

The persistent homology transform (PHT) of a subset $M \subset \mathbb{R}^d$ is a map $\text{PHT}(M):\mathbb{S}^{d-1} \to \mathbf{Dgm}$ from the unit sphere to the space of persistence diagrams. This map assigns to each direction $v\in \mathbb{S}^{d-1}$ the persistent homology of the filtration of $M$ in direction $v$. In practice, one can only sample the map $\text{PHT}(M)$ at a finite set of directions $A \subset \mathbb{S}^{d-1}$. This suggests two natural questions: (1) Can we interpolate the PHT from this finite sample of directions to the entire sphere? If so, (2) can we prove that the resulting interpolation is close to the true PHT? In this paper we show that if we can sample the PHT at the module level, where we have information about how homology from each direction interacts, a ready-made interpolation theory due to Bubenik, de Silva, and Nanda using Kan extensions can answer both of these questions in the affirmative. A close inspection of those techniques shows that we can infer the PHT from a finite sample of heights from each direction as well. Our paper presents the first known results for approximating the PHT from finite directional and scalar data.

Figures

Figures reproduced from arXiv: 2507.22816 by the authors.

**Figure 1.** Given directions A = {vk = e 2πik/6 | k = 0, . . . , 5} ⊂ S 1 , marked on the horizontal axis, our goal is to calculate the value of the PHT at the unknown direction w = e iπ/2 and Θ-value −π/6. To do this, we look at the past light cone, whose boundary is drawn in red. The red boxes indicate the largest Θ-values for each direction that can include into Mw,cos(−π/6). The blue lines indicate the past light cones of e… view at source ↗

**Figure 2.** Continuing the example from [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** The past light cone at the value w = e iπ/2 and β = −2π/3 is that part of AΘ that is below the red cone. This corresponds to the two intervals [(vj , −π),(vj , −5π/6)] for j = 1, 2. Because dg(v1, v2) = π/3 there can be no internal morphisms between these sub-categories, so the past light cone is disconnected. This will cause the Kan extension of any constant functor to be non-constant. The cokernel, and the predict… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: From left to right: (1) the true PHTΘ 0 at direction w = e iπ/2 ∈ S 1 , (2) the left-Kan extension of PHTΘ 0 at w using A = {vk = e 2πik/6 | k = 0, . . . , 5}, (3) the right-Kan extension for the same w and A, finally (4) the center-Kan extension. See Example 3.9 for a…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 29 canonical work pages

[1]

& Boyer, D

Turner, K., Mukherjee, S. & Boyer, D. M. Persistent homology t ransform for modeling shapes and surfaces. Inf. Inference 3, 310–344 (2014). URL https://doi.org/10.1093/imaiai/iau011

doi:10.1093/imaiai/iau011 2014
[2]

¨Uber diebestimmung von funktionen durch ihre integralwerte langs b esonderen manifolden

Radon, J. ¨Uber diebestimmung von funktionen durch ihre integralwerte langs b esonderen manifolden. ber. s¨ achsische akadamie wissenschaften 69, 262–267 (1917)

work page 1917
[3]

On the determination of functions from their integral values along certain manifolds

Radon, J. On the determination of functions from their integral values along certain manifolds. IEEE transactions on medical imaging 5, 170–176 (1986)

work page 1986
[4]

Cycles lagrangiens, fonctions constructibles et ap plications

Schapira, P. Cycles lagrangiens, fonctions constructibles et ap plications. S´ eminaire´Equations aux d´ eriv´ ees partielles (Polytechnique) dit aussi” S´ eminaire Goulaouic-Schwartz” 1–9 (1988)

work page 1988
[5]

Some integral calculus based on euler characteristic

Viro, O. Some integral calculus based on euler characteristic. vo lume 1346 of lecture notes in mathe- matics (1989)

work page 1989
[6]

Operations on constructible functions

Schapira, P. Operations on constructible functions. Journal of pure and applied algebra 72, 83–93 (1991)

work page 1991
[7]

Tomography of constructible functions

Schapira, P. Tomography of constructible functions. In International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes , 427–435 (Springer, 1995)

work page 1995
[8]

& Robinson, M

Curry, J., Ghrist, R. & Robinson, M. Euler calculus with applications to signals and sensing. In Pro- ceedings of Symposia in Applied Mathematics , vol. 70, 75–146 (2012)

work page 2012

Show all 33 references

[9]

MacPherson, R. D. Chern classes for singular algebraic varieties . Annals of Mathematics 100, 423–432 (1974)

1974
[10]

Index theorem for constructible sheaves

Kashiwara, M. Index theorem for constructible sheaves. Ast´ erisque130, 4 (1985)

1985
[11]

& Stolﬁ, J

Guibas, L., Ramshaw, L. & Stolﬁ, J. A kinetic framework for comp utational geometry. In 24th Annual Symposium on Foundations of Computer Science (sfcs 1983) , 100–111 (IEEE Computer Society, 1983)

1983
[12]

Topological persisten ce and simpliﬁcation

Edelsbrunner, Letscher & Zomorodian. Topological persisten ce and simpliﬁcation. Discrete & computa- tional geometry 28, 511–533 (2002)

2002
[13]

Topological signal processing, vol

Robinson, M. Topological signal processing, vol. 81 (Springer, 2014)

2014
[14]

& Uhl, A

Hofer, C., Kwitt, R., Niethammer, M. & Uhl, A. Deep learning with to pological signatures. Advances in neural information processing systems 30 (2017)

2017
[15]

X., Mukherjee, S

Crawford, L., Monod, A., Chen, A. X., Mukherjee, S. & Rabad´ a n, R. Predicting clinical outcomes in glioblastoma: an application of topological and functional data ana lysis. Journal of the American Statistical Association 115, 1139–1150 (2020)

2020
[16]

Am´ ezquita, E. J. et al. Measuring hidden phenotype: quantifying the shape of barley seed s using the euler characteristic transform. in silico Plants 4, diab033 (2022)

2022
[17]

& Rieck, B

R¨ oell, E. & Rieck, B. Diﬀerentiable euler characteristic transforms for shape classiﬁcation. In The Twelfth International Conference on Learning Representations . URL https://arxiv.org/abs/2310.07630

arXiv
[18]

Marsh, L. et al. Detecting temporal shape changes with the euler characteristic t ransform. Transactions of Mathematics and Its Applications 8, tnae002 (2024)

2024
[19]

An invitation to the euler characteristic transform

Munch, E. An invitation to the euler characteristic transform. arXiv preprint arXiv:2310.10395 (2023)

2023 arXiv
[20]

& Turner, K

Curry, J., Mukherjee, S. & Turner, K. How many directions det ermine a shape and other suﬃciency results for two topological transforms. Trans. Amer. Math. Soc. Ser. B 9, 1006–1043 (2022). URL https://doi.org/10.1090/btran/122

2022 doi
[21]

& Mai, H

Ghrist, R., Levanger, R. & Mai, H. Persistent homology and euler integral transforms. Journal of Applied and Computational Topology 2, 55–60 (2018)

2018
[22]

& Harer, J

Mileyko, Y., Mukherjee, S. & Harer, J. Probability measures on t he space of persistence diagrams. Inverse Problems 27, 124007 (2011)

2011
[23]

Munch, E. et al. Probabilistic fr´ echet means for time varying persistence diagrams . Electronic Journal of Statistics 9, 1173–1204 (2015). 18 S. ARYA AND J. CURRY

2015
[24]

& Nanda, V

de Silva, V. & Nanda, V. Geometry in the space of persistence mo dules. In Proceedings of the twenty- ninth annual symposium on Computational geometry , 397–404 (2013)

2013
[25]

& Nanda, V

Bubenik, P., de Silva, V. & Nanda, V. Higher interpolation and exte nsion for persistence modules. SIAM J. Appl. Algebra Geom. 1, 272–284 (2017). URL https://doi.org/10.1137/16M1100472

2017 doi
[26]

& Scott, J

Bubenik, P., de Silva, V. & Scott, J. Metrics for generalized pers istence modules. Found. Comput. Math. 15, 1501–1531 (2015). URL https://doi.org/10.1007/s10208-014-9229-5

2015 doi
[27]

Tame topology and o-minimal structures , vol

Van den Dries, L. Tame topology and o-minimal structures , vol. 248 (Cambridge university press, 1998)

1998
[28]

Decomposition of pointwise ﬁnite-dimension al persistence modules

Crawley-Boevey, W. Decomposition of pointwise ﬁnite-dimension al persistence modules. Journal of Algebra and its Applications 14, 1550066 (2015)

2015
[29]

& Oudot, S

Chazal, F., de Silva, V., Glisse, M. & Oudot, S. The structure and stability of persistence modules . SpringerBriefs in Mathematics (Springer, [Cham], 2016). URL https://doi.org/10.1007/978-3-319-42545-0

2016 doi
[30]

& Mukherjee, S

Arya, S., Curry, J. & Mukherjee, S. A sheaf-theoretic const ruction of shape space. Foundations of Computational Mathematics 1–51 (2024)

2024
[31]

The theory of the interleaving distance on multidimen sional persistence modules

Lesnick, M. The theory of the interleaving distance on multidimen sional persistence modules. Founda- tions of Computational Mathematics 15, 613–650 (2015)

2015
[32]

& Lesnick, M

Bauer, U. & Lesnick, M. Induced matchings of barcodes and th e algebraic stability of persistence. In Proceedings of the thirtieth annual symposium on Computati onal geometry , 355–364 (2014)

2014
[33]

topological

Riehl, E. Category theory in context . Aurora Dover Modern Math Originals (Dover Publications, Inc., Mineola, NY, 2016). Appendix A. Kan Extensions We provide a brief review of Kan extensions. For a comprehensive tr eatment, see [ 33]. Deﬁnition A.1 (Left Kan Extension) . LetC...

2016

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

& Boyer, D

Turner, K., Mukherjee, S. & Boyer, D. M. Persistent homology t ransform for modeling shapes and surfaces. Inf. Inference 3, 310–344 (2014). URL https://doi.org/10.1093/imaiai/iau011

doi:10.1093/imaiai/iau011 2014

[2] [2]

¨Uber diebestimmung von funktionen durch ihre integralwerte langs b esonderen manifolden

Radon, J. ¨Uber diebestimmung von funktionen durch ihre integralwerte langs b esonderen manifolden. ber. s¨ achsische akadamie wissenschaften 69, 262–267 (1917)

work page 1917

[3] [3]

On the determination of functions from their integral values along certain manifolds

Radon, J. On the determination of functions from their integral values along certain manifolds. IEEE transactions on medical imaging 5, 170–176 (1986)

work page 1986

[4] [4]

Cycles lagrangiens, fonctions constructibles et ap plications

Schapira, P. Cycles lagrangiens, fonctions constructibles et ap plications. S´ eminaire´Equations aux d´ eriv´ ees partielles (Polytechnique) dit aussi” S´ eminaire Goulaouic-Schwartz” 1–9 (1988)

work page 1988

[5] [5]

Some integral calculus based on euler characteristic

Viro, O. Some integral calculus based on euler characteristic. vo lume 1346 of lecture notes in mathe- matics (1989)

work page 1989

[6] [6]

Operations on constructible functions

Schapira, P. Operations on constructible functions. Journal of pure and applied algebra 72, 83–93 (1991)

work page 1991

[7] [7]

Tomography of constructible functions

Schapira, P. Tomography of constructible functions. In International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes , 427–435 (Springer, 1995)

work page 1995

[8] [8]

& Robinson, M

Curry, J., Ghrist, R. & Robinson, M. Euler calculus with applications to signals and sensing. In Pro- ceedings of Symposia in Applied Mathematics , vol. 70, 75–146 (2012)

work page 2012