A Hilbert space embedding of persistence diagrams and barcodes

Peter Bubenik

REVIEW 3 major objections 4 minor 24 references

A Hilbert space embedding of persistence diagrams and barcodes

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read For every $1\le p\le\infty$, the persistence landscape is a 1-Lipschitz embedding of p-finite countable barcodes and persistence diagrams into $L^p(\mathbb{N}\times\mathbb{R})$.

desk verdict A correct and clean finite-case inequality; the countable embedding claim is asserted rather than proved. read the letter →

arxiv 2608.08858 v1 pith:HLKMRYXQ submitted 2026-08-09 math.AT

Peter Bubenik This is my paper · ORCID

classification math.AT MSC 55N31

keywords persistencelandscapediagramsbarcodesp-WassersteindistanceL^pspace1-LipschitzembeddingHilberttopologicaldataanalysis

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 proves that persistence landscapes are not merely stable summaries but genuine metric embeddings of barcodes and persistence diagrams into function spaces. For each $1\le p\le\infty$, it shows that the landscape map sends the metric space of p-finite countable barcodes with the p-Wasserstein distance $W^\triangle_p$ into $L^p(\mathbb{N}\times\mathbb{R})$, and does so with Lipschitz constant one: the $L^p$ distance between two landscapes is never larger than the Wasserstein distance between the original barcodes or diagrams. The same statement holds for persistence diagrams, using the distance between ordered pairs induced by the $L^p$ difference of their triangle functions. When $p=2$ the target is a separable Hilbert space, which is the setting where statistical and machine-learning tools for topological data are usually developed.

What carries the argument

The central mechanism is the triangle function and the persistence landscape built from it. For an interval $I=[b,d)$, the triangle function $\triangle_I(t)$ is the distance from $t$ to the complement of $I$, a tent-shaped function of height half the interval's length; for a barcode $B$, the landscape is $\Lambda_B(k,t)=\operatorname{kmax}_j \triangle_{I_j}(t)$, the kth largest of these tent values at $t$. The ground metric for Wasserstein distance between intervals is $d_p(I,J)=\|\triangle_I-\triangle_J\|_p$, and the p-Wasserstein distance $W^\triangle_p$ is the optimal matching built on that ground metric. The load-bearing inequality is Lemma 3.3: for two sequences, replacing each by its order statistics cannot increase the $\ell^p$ norm of their difference. That inequality turns the kth-largest construction into a 1-Lipschitz map, and the passage to countable p-finite barcodes is handled by completeness of the Wasserstein spaces.

What would settle it

Take a p-finite countable barcode $B$, let $B_n$ be the barcode of its first $n$ intervals, and compute $\|\Lambda_B-\Lambda_{B_n}\|_p$: if this quantity does not converge to zero, then the pointwise landscape is not the continuous 1-Lipschitz extension required by Theorems 4.5 and 4.6. Alternatively, find two distinct p-finite countable barcodes with identical persistence landscapes; any such pair would refute the embedding claim directly.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.5 and Theorem 4.6: for every $1\le p\le\infty$, the persistence landscape gives a 1-Lipschitz embedding $\Lambda:(D^\triangle_p(\mathrm{cInt}(\mathbb{R})_{bco}), W^\triangle_p)\to L^p(\mathbb{N}\times\mathbb{R})$ of p-finite countable barcodes, and likewise $\Lambda:(D^\triangle_p(\mathbb{R}^2_<), W^\triangle_p)\to L^p(\mathbb{N}\times\mathbb{R})$ of p-finite countable persistence diagrams. The finite case is proved by a rearrangement inequality: sorting the interval-distance evaluations by size before taking an $\ell^p$ norm can only shrink the distance, so the kth-largest landscape operation is 1-Lipschitz with respect to $W^\triangle_p$. The countable case is obtained by declaring the landscape to be the 1-Lipschitz extension to the completion of the finite barcode space, relying on earlier completeness results for these Wasserstein spaces. Injectivity on finite barcodes is cited from an earlier paper, and the same injectivity is assumed to persist on the completed space.

Load-bearing premise

The proof that the landscape extends from finite to countable p-finite barcodes assumes, rather than verifies, that the pointwise landscape of a countable barcode is the $L^p$ limit of the landscapes of its finite truncations and that this extension remains injective on the completion; if two distinct countable barcodes shared one landscape, the map would no longer be an embedding.

Editorial extensions

If this is right

For $p=2$, countable barcodes and diagrams sit explicitly inside a separable Hilbert space, so Hilbert-space methods such as means, PCA, and kernel evaluations apply directly to persistence summaries.
The inequality $\|\Lambda_B-\Lambda_{B'}\|_p \le W^\triangle_p(B,B')$ gives a computable lower bound on Wasserstein distance: whenever two landscapes differ, the barcodes must differ by at least that amount in Wasserstein distance.
For $p=\infty$, the result contains the bottleneck stability statement, since $W^\triangle_\infty$ is the bottleneck distance; for $p=1$, it contains the rank-based Wasserstein case.
The p-finite condition on barcodes is expressed by a summability condition on interval lengths (Proposition 5.2), so the embedding applies exactly to barcodes whose interval-length data have the right moment.
The same statements hold for persistence diagrams as for barcodes, because the map sending an interval $[b,d)$ to the point $(b,d)$ is an isometry between the two metric spaces.

Reading between the lines

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

Editorial inference: the theorem is one-sided—it guarantees that Wasserstein distance is at least $L^p$ landscape distance, but not the reverse. Statistical pipelines that approximate $W^\triangle_p$ by $L^p$ landscape distance will compress distances and may fail to separate barcodes that are actually far apart in Wasserstein metric.
Editorial inference: the only part of the proof that is not fully explicit is the injectivity and continuity of the extension from finite to countable barcodes. If a counterexample pair of distinct countable barcodes with identical landscapes exists, the title result would still give a 1-Lipschitz map but not an embedding; checking truncation convergence is the natural next test.
Editorial inference: because the proof flows from order statistics and triangle functions, the same argument likely generalizes to other summaries built from sorted interval functions—for instance weighted landscapes or rank-transformed landscape variants—as long as the ground metric is the $L^p$ difference of the underlying interval functions.

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.

Desk Editor's Note

read the letter

Colleague—

The paper does one thing cleanly and then reaches for more than it proves. The finite-case inequality (Theorem 3.1) is correct, and the proof via monotone rearrangement is economical; the p=∞ limit and the reduction from diagrams to barcodes are fine. What's genuinely new is the extension to 1

The soft spot is Section 4. The proofs of Theorems 4.5 and 4.6 are each a single sentence saying the pointwise-defined landscape 'is the 1-Lipschitz extension.' That hides two things. First, to identify the pointwise landscape with the completion extension you need to show that for a p-finite countable barcode B with truncations B_n, Λ_{B_n} → Λ_B in L^p. This is true—Λ_{B_n} increases pointwise to Λ_B and the norms are bounded by the p-finite tail sums, so monotone convergence gives it—but the paper doesn't state or prove it. Second, and more load-bearing: an injective 1-Lipschitz map on a dense subspace need not extend to an injective map on the completion. The paper never proves injectivity of the pointwise landscape on D^Δ_p. Without that, 'embedding' is not established. Finite-case injectivity doesn't automatically carry over. I don't have a counterexample, and I suspect injectivity holds, but it needs an argument or a citation.

The citation pattern is not circular: the earlier self-citations carry the completion setup and the p=1,∞ cases, and the new claim is proved independently for finite barcodes. No invented entities, no free parameters.

Bottom line: a solid incremental paper with a repairable gap. The finite result is publishable on its own. The countable embedding theorem should not be stated as proved until the missing convergence and injectivity arguments are supplied. A serious editor should send it out; a referee should ask for the missing details. For readers who care about embeddings of barcode spaces into L^p or Hilbert space, this is worth having after revision.

Referee Report

3 major / 4 minor

Summary. The paper proves that the persistence landscape gives a 1-Lipschitz embedding of spaces of countable barcodes and persistence diagrams, equipped with the p-Wasserstein distance W^△_p, into L^p(N×R) for 1≤p≤∞. For finite barcodes, Theorem 3.1 establishes the key Lipschitz inequality via a monotone-rearrangement lemma and Tonelli's theorem; injectivity is cited from earlier work. The countable case (Theorems 4.5 and 4.6) is treated by identifying the pointwise-defined landscape on p-finite countable barcodes with the 1-Lipschitz extension of the finite landscape map to the metric completion. The paper also relates W^△_1 and W^△_∞ to the rank-based distance and the bottleneck distance, and compares p-finiteness for d_p with p-finiteness for ℓ_p.

Significance. The finite-case inequality is clean and appears correct, and the overall strategy of extending by completion is natural and potentially valuable: for p=2 the result would give an explicit embedding into a separable Hilbert space, with resulting statistical and machine-learning applications. The paper is also careful to compare the new distances with existing ones. However, the passage from the finite to the countable case is not rigorously justified in the manuscript: the proof of Theorems 4.5 and 4.6 is a single sentence that asserts, rather than demonstrates, that the pointwise landscape is the L^p limit of truncated landscapes and that this limit map is injective on the completed space. In addition, Theorem 3.6 is stated for a domain that, by the paper's own definition, includes diagrams with infinite coordinates, for which the landscape is not L^p-valued. These are load-bearing issues for the main claims, so the manuscript needs a major revision.

major comments (3)

[Section 4, Theorems 4.5 and 4.6] The proof of Theorems 4.5 and 4.6 consists of the single assertion that the pointwise-defined landscape of a p-finite countable barcode is the 1-Lipschitz extension of the finite landscape map. This identification needs proof. For p<∞, order statistics of an infinite sequence are not continuous in ℓ^p pointwise, so one must show that for truncations B_n of B, Λ_{B_n} converges to Λ_B in L^p. An estimate such as ||Λ_B−Λ_{B_n}||_p ≤ (∑_{j>n} ||△_{I_j}||_p^p)^{1/p} (and the analogous sup-norm estimate for p=∞) is required but is not stated or proved.
[Section 4, Theorems 4.5 and 4.6] Injectivity of the extended map on the completed space is not established. An injective 1-Lipschitz map on a dense subspace need not extend to an injective map on the completion, so the finite-case injectivity cited in Theorem 3.5 does not automatically carry over. The proof must separately show that two distinct p-finite countable barcodes (or diagrams) have distinct persistence landscapes; otherwise the term 'embedding' in the main theorems is not justified.
[Theorem 3.6 and Section 2.3] The domain D(R^2_<) is defined in Section 2.3 to include diagrams with coordinates in [−∞,∞], for example the single point (0,∞). For such a diagram α, Λ_α equals △_{[0,∞)}, whose L^p norm is infinite for every 1≤p≤∞, so Λ does not map D(R^2_<) into L^p(N×R). The proof of Theorem 3.6 only treats diagrams corresponding to bounded intervals. The theorem should be restricted to finite-coordinate diagrams (or to p-finite diagrams), and the notation should consistently distinguish R^2_< from its extended-coordinate counterpart.

minor comments (4)

[Throughout] The notation R^2_< is used for both finite-coordinate and extended-coordinate diagrams, for example in Definition 2.3 and Theorem 3.6; please introduce a clearly distinct symbol, such as an overline, and use it consistently.
[Lemma 3.3] The proof invokes monotone rearrangement for the convex function |t|^p and then passes to p=∞ by a limit; a direct proof or a specific reference for the rearrangement inequality would make the argument more self-contained.
[Definition 4.1] The footnote defining kth maxima for infinite sequences is essential for the p=∞ case; consider moving that discussion into the main text.
[Section 4] The sentence 'From this construction, we obtain the following two results' is stronger than the construction alone provides; the construction gives a candidate extension, but the embedding property requires the additional convergence and injectivity arguments requested above.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the 1-Lipschitz inequality is proved directly from a rearrangement lemma, and no prediction or definition reduces to its own input.

full rationale

The central derivation for finite barcodes, Theorem 3.1, is self-contained: it follows from Proposition 3.2, Lemma 3.3 (monotone rearrangement), and Tonelli's theorem, giving ||Λ_B - Λ_{B'}||_p ≤ W^△_p(B,B'). The countable results in Theorems 4.5 and 4.6 are intended to follow by extending this 1-Lipschitz map to the completion identified in Lemma 4.2, which is cited from prior work [8]. This is not circular: the cited completion statement is independent of the landscape embedding and does not assume the target theorem. The paper's heavy self-citation (e.g., [5] for injectivity, [8] for completion, [12] for p=1) is not load-bearing in the sense of assuming the conclusion, because those results are parameter-free theorems proved elsewhere and are not defined in terms of the embedding being established. The main correctness concern, namely that the pointwise landscape of a p-finite countable barcode is asserted to equal the completion extension without proof of L^p convergence or of injectivity on the completed space, is a proof gap about existence and identification, not a circularity: the inequality being proved is not assumed as an input. Therefore no step reduces by construction to its own input.

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

No free parameters or invented entities. The proof relies on the standard monotone rearrangement theorem, the product measure/L^p framework, the completion result imported from [8], and the standard extension of a 1-Lipschitz map to a completion. The paper's own definitions of d_q and W^△_p are not fitted to data.

assumptions (4)

standard math Monotone rearrangement minimizes the L^p cost of matching two sequences on the real line (Lemma 3.3).
Used to bound the sorted landscape differences by the optimal matching cost in Theorem 3.1; the proof invokes the standard optimal transportation fact for convex cost.
domain assumption D^△_p with W^△_p is the completion of the finite barcode space (Lemma 4.2 from [8]).
Imported from prior work by the same group to justify extending the finite embedding to countable p-finite barcodes.
standard math Any 1-Lipschitz map from a metric space to a complete metric space extends to its completion.
Standard metric-space result invoked before Theorems 4.5 and 4.6.
domain assumption The persistence landscape is injective on finite barcodes.
Cited from [5]; needed for the word 'embedding' and assumed to persist on the completion, though the extension of injectivity is not proved.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A Hilbert space embedding of persistence diagrams and barcodes." pith.science (2026). https://pith.science/paper/HLKMRYXQ

@misc{pith2026260808858,
  author       = {Pith},
  title        = {Pith review of: A Hilbert space embedding of persistence diagrams and barcodes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HLKMRYXQ}},
  note         = {Machine review of arXiv:2608.08858}
}

abstract

For p in $[1,\infty]$, we show that the persistence landscape gives 1-Lipschitz embeddings of metric spaces of countable persistence diagrams and barcodes with p-Wasserstein distances into an $L^p$ space.

Figures

Figures reproduced from arXiv: 2608.08858 by the authors.

**Figure 1.** Graphs of triangle functions △I . Left: I = [b, d). Right: I = [b, ∞). d}. Let ∆ = {(b, b) ∈ [−∞,∞] 2}. We have a commutative diagram as follows. (2.1) Int(R)bco Int(R)co Int(R) R 2 ≤/∆ R 2 ≤/∆ ∼= The left vertical arrow sends [b, d) to (b, d) and the empty interval to ∆. The right vertical arrow sends a nonempty interval I to (inf I,sup I) and sends the empty interval to ∆. The bottom horizontal arrow sends ∆ to ∆ … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 22 canonical work pages

[1]

Bi-Lipschitz embeddings of the space of unorderedm-tuples with a partial transportation metric

David Bate and Ana Luc ´ ıa Garcia Pulido. “Bi-Lipschitz embeddings of the space of unorderedm-tuples with a partial transportation metric”.Math. Ann.390.2 (2024), pp. 3109–3131

work page 2024
[2]

The space of persistence diagrams fails to have Yu’s property A

Greg Bell, Austin Lawson, Neil Pritchard, and Dan Yasaki. “The space of persistence diagrams fails to have Yu’s property A”.Topology Proc.58 (2021), pp. 279–288

work page 2021
[3]

Graded Persistence Dia- grams and Persistence Landscapes

Leo Betthauser, Peter Bubenik, and Parker B. Edwards. “Graded Persistence Dia- grams and Persistence Landscapes”.Discrete & Computational Geometry67.1 (2022), pp. 203–230

work page 2022
[4]

Robust statistics, hypothesis testing, and confidence intervals for persistent homology on metric measure spaces

Andrew J. Blumberg, Itamar Gal, Michael A. Mandell, and Matthew Pancia. “Robust statistics, hypothesis testing, and confidence intervals for persistent homology on metric measure spaces”.Found. Comput. Math.14.4 (2014), pp. 745–789

work page 2014
[5]

Statistical Topological Data Analysis Using Persistence Landscapes

Peter Bubenik. “Statistical Topological Data Analysis Using Persistence Landscapes”. Journal of Machine Learning Research16 (2015), pp. 77–102

work page 2015
[6]

The persistence landscape and some of its properties

Peter Bubenik. “The persistence landscape and some of its properties”.Topological data analysis—the Abel Symposium 2018. Vol. 15. Abel Symp. Springer, Cham, 2020, pp. 97–117

work page 2018
[7]

Universality of persistence diagrams and the bottle- neck and Wasserstein distances

Peter Bubenik and Alex Elchesen. “Universality of persistence diagrams and the bottle- neck and Wasserstein distances”.Computational Geometry105-106 (2022), p. 101882

work page 2022
[8]

Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces

Peter Bubenik and Alex Elchesen. “Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces”.J. Appl. Comput. Topol.6.4 (2022), pp. 429– 474

work page 2022

Show all 24 references

[9]

Relative optimal transport

Peter Bubenik and Alex Elchesen. “Relative optimal transport”.Studia Math.(2026, in press), 46pp. arXiv:2411.05678 [math.MG]

2026 arXiv
[10]

Topological and metric properties of spaces of generalized persistence diagrams

Peter Bubenik and Iryna Hartsock. “Topological and metric properties of spaces of generalized persistence diagrams”.J. Appl. Comput. Topol.8.2 (2024), pp. 347–399

2024
[11]

Embeddings of persistence diagrams into Hilbert spaces

Peter Bubenik and Alexander Wagner. “Embeddings of persistence diagrams into Hilbert spaces”.J. Appl. Comput. Topol.4.3 (2020), pp. 339–351

2020
[12]

Peter Bubenik and Wanchen Zhao.A rank-based distance for interval modules and Wasserstein stability of persistence landscapes. 2026. arXiv:2509.20921 [math.AT]. 9

2026 arXiv
[13]

On the metric distortion of embedding persis- tence diagrams into separable Hilbert spaces

Mathieu Carri` ere and Ulrich Bauer. “On the metric distortion of embedding persis- tence diagrams into separable Hilbert spaces”.35th International Symposium on Com- putational Geometry. Vol. 129. LIPIcs. Leibniz Int. Proc. Inform. Schloss Dagstuhl. Leibniz-Zent. Inform., Wad...

2019
[14]

Sliced Wasserstein Kernel for Persistence Diagrams

Mathieu Carri` ere, Marco Cuturi, and Steve Oudot. “Sliced Wasserstein Kernel for Persistence Diagrams”.Proceedings of the 34th International Conference on Machine Learning - Volume 70. ICML’17. JMLR.org, 2017, pp. 664–673

2017
[15]

Stability for Persistence Diagrams

David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. “Stability for Persistence Diagrams”.Discrete & Computational Geometry37 (2007), pp. 263–271

2007
[16]

Decomposition of pointwise finite-dimensional persistence modules

William Crawley-Boevey. “Decomposition of pointwise finite-dimensional persistence modules”.Journal of Algebra and Its Applications14 (Mar. 2015)

2015
[17]

Understanding the topology and the geometry of the space of persistence diagrams via optimal partial transport

Vincent Divol and Th´ eo Lacombe. “Understanding the topology and the geometry of the space of persistence diagrams via optimal partial transport”.J. Appl. Comput. Topol.5.1 (2021), pp. 1–53

2021
[18]

Probability measures on the space of persistence diagrams

Yuriy Mileyko, Sayan Mukherjee, and John Harer. “Probability measures on the space of persistence diagrams”.Inverse Problems - INVERSE PROBL27 (Dec. 2011)

2011
[19]

Atish Mitra and Ziga Virk.Geometric embeddings of spaces of persistence diagrams with explicit distortions. 2025. arXiv:2401.05298 [math.MG]

2025
[20]

The space of persistence diagrams onnpoints coarsely embeds into Hilbert space

Atish Mitra and ˇZiga Virk. “The space of persistence diagrams onnpoints coarsely embeds into Hilbert space”.Proc. Amer. Math. Soc.149.6 (2021), pp. 2693–2703

2021
[21]

Coarse Embeddability of Wasserstein Space and the Space of Persistence Diagrams

Neil Pritchard and Thomas Weighill. “Coarse Embeddability of Wasserstein Space and the Space of Persistence Diagrams”.Discrete & Computational Geometry74.2 (2024), pp. 358–373

2024
[22]

Fr´ echet means for distributions of persistence diagrams

Katharine Turner, Yuriy Mileyko, Sayan Mukherjee, and John Harer. “Fr´ echet means for distributions of persistence diagrams”.Discrete Comput. Geom.52.1 (2014), pp. 44– 70

2014
[23]

Zaloˇ zba UL FRI, University of Ljubl- jana, 2022

ˇZiga Virk.Introduction to Persistent Homology. Zaloˇ zba UL FRI, University of Ljubl- jana, 2022

2022
[24]

Nonembeddability of persistence diagrams withp >2 Wasserstein metric

Alexander Wagner. “Nonembeddability of persistence diagrams withp >2 Wasserstein metric”.Proc. Amer. Math. Soc.149.6 (2021), pp. 2673–2677. University of Florida, Department of Mathematics Email address:peter.bubenik@ufl.edu 10

2021

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Bi-Lipschitz embeddings of the space of unorderedm-tuples with a partial transportation metric

David Bate and Ana Luc ´ ıa Garcia Pulido. “Bi-Lipschitz embeddings of the space of unorderedm-tuples with a partial transportation metric”.Math. Ann.390.2 (2024), pp. 3109–3131

work page 2024

[2] [2]

The space of persistence diagrams fails to have Yu’s property A

Greg Bell, Austin Lawson, Neil Pritchard, and Dan Yasaki. “The space of persistence diagrams fails to have Yu’s property A”.Topology Proc.58 (2021), pp. 279–288

work page 2021

[3] [3]

Graded Persistence Dia- grams and Persistence Landscapes

Leo Betthauser, Peter Bubenik, and Parker B. Edwards. “Graded Persistence Dia- grams and Persistence Landscapes”.Discrete & Computational Geometry67.1 (2022), pp. 203–230

work page 2022

[4] [4]

Robust statistics, hypothesis testing, and confidence intervals for persistent homology on metric measure spaces

Andrew J. Blumberg, Itamar Gal, Michael A. Mandell, and Matthew Pancia. “Robust statistics, hypothesis testing, and confidence intervals for persistent homology on metric measure spaces”.Found. Comput. Math.14.4 (2014), pp. 745–789

work page 2014

[5] [5]

Statistical Topological Data Analysis Using Persistence Landscapes

Peter Bubenik. “Statistical Topological Data Analysis Using Persistence Landscapes”. Journal of Machine Learning Research16 (2015), pp. 77–102

work page 2015

[6] [6]

The persistence landscape and some of its properties

Peter Bubenik. “The persistence landscape and some of its properties”.Topological data analysis—the Abel Symposium 2018. Vol. 15. Abel Symp. Springer, Cham, 2020, pp. 97–117

work page 2018

[7] [7]

Universality of persistence diagrams and the bottle- neck and Wasserstein distances

Peter Bubenik and Alex Elchesen. “Universality of persistence diagrams and the bottle- neck and Wasserstein distances”.Computational Geometry105-106 (2022), p. 101882

work page 2022

[8] [8]

Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces

Peter Bubenik and Alex Elchesen. “Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces”.J. Appl. Comput. Topol.6.4 (2022), pp. 429– 474

work page 2022