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arxiv: 2605.02836 · v2 · pith:B23YWRAGnew · submitted 2026-05-04 · 💻 cs.LG · math.AT

A Closed-Form Persistence-Landmark Pipeline for Certified Point-Cloud and Graph Classification

Sushovan Majhi , Atish Mitra , \v{Z}iga Virk , Pramita Bagchi This is my paper

Pith reviewed 2026-07-01 00:20 UTC · model grok-4.3

classification 💻 cs.LG math.AT
keywords persistent homologypoint cloud classificationgraph classificationclosed-form classifiermargin boundscertificateslandmark embeddingMahalanobis ranking
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The pith

PLACE derives three guarantees for point-cloud and graph classification from training labels alone using persistent homology signatures.

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

The paper presents a pipeline called PLACE that classifies point clouds and graphs by embedding their persistent homology diagrams. It establishes three guarantees derived only from the training labels: a margin-based excess-risk rate of order O(kR/(Δ√m_min)), a closed-form rule for selecting among descriptors, and a per-prediction certificate computed at training time. The embedding adds Mitra-Virk coordinate functions evaluated at a sparse set of landmarks, and a weight rule proportional to (d_{k+1}² - d_k²)/R_k² is chosen to maximize the slope of an affine certificate under a coherence condition. A reader would care because the method supplies explicit performance bounds and certificates in regimes where data are scarce and no separate validation set is available.

Core claim

The paper claims that summing Mitra-Virk single-point coordinate functions over a sparse landmark grid, equipped with the closed-form weight rule w_k² ∝ (d_{k+1}² - d_k²)/R_k² under ν-coherence, produces an embedding for which an O(kR/(Δ√m_min)) margin bound follows directly from class-mean separation Δ and embedding radius R, the Mahalanobis margin under Ledoit-Wolf shrinkage ranks descriptors with mean Spearman ρ = +0.56 across eleven benchmarks, and three concrete radii (Pinelis, Gaussian plug-in, variance-aware Pinelis-Bernstein) can be fixed once at training time to certify individual predictions with no extra cost per test point.

What carries the argument

The PLACE embedding formed by summing Mitra-Virk single-point coordinate functions over a sparse landmark grid together with the closed-form weight rule that maximizes the distortion slope of the affine certificate under ν-coherence.

If this is right

  • An explicit O(kR/(Δ√m_min)) excess-risk bound holds whenever class separation and embedding radius are known.
  • The Ledoit-Wolf Mahalanobis margin supplies a closed-form ranking of descriptors that is strongest on chemical-graph pools.
  • A certificate radius chosen once from training data certifies every subsequent prediction at zero extra cost.
  • On the MUTAG benchmark the training-time certificate matches the population rule on every held-out example.
  • Gaps on NCI1 and NCI109 are isolated to descriptor blindness while gaps elsewhere trace to landmark-pool coverage.

Where Pith is reading between the lines

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

  • The same label-only certificate construction could be tested on other topological summaries if an analogous coherence condition can be verified.
  • Because no held-out set is required, the approach may be directly applicable to extremely small labeled collections where cross-validation is unstable.
  • The two diagnosable failure regimes suggest concrete next experiments: enlarging the descriptor pool for NCI-type graphs and increasing landmark density on Orbit5k-style data.
  • The closed-form weight rule may admit an exact consistency proof under the homogeneous protein and social graph pools mentioned in the text.

Load-bearing premise

The embedding sums Mitra-Virk single-point coordinate functions over a sparse landmark grid and the ν-coherence assumption holds so that the weight rule maximizes the distortion slope in the affine certificate.

What would settle it

A direct count showing that the empirical nearest-centroid rule and the population nearest-centroid rule disagree on any of the 940 held-out MUTAG test points, or that the Mahalanobis ranker yields negative Spearman correlation on more than one of the eleven chemical-graph benchmarks, would falsify the claimed mechanism and selection consistency.

Figures

Figures reproduced from arXiv: 2605.02836 by Atish Mitra, Pramita Bagchi, Sushovan Majhi, \v{Z}iga Virk.

Figure 1
Figure 1. Figure 1: From point cloud to persistence diagram. (a) Noisy sample from a circle. (b) Vietoris–Rips filtration at radii 𝑟1 < 𝑟2 < 𝑟3: the 1-cycle is born at 𝑟2 and dies at 𝑟3. (c) Barcode; bar length equals feature lifetime. (d) Persistence diagram; each feature becomes a point (𝑏, 𝑑), with distance 𝜏 = 𝑑 − 𝑏 to the diagonal measuring topological significance. We overlaid the 0-dim (blue) and 1-dim (red) diagrams t… view at source ↗
Figure 2
Figure 2. Figure 2: The PLACE pipeline. A point cloud or graph (left) is converted to a persistence diagram through a filtration—a growing sequence of simplicial complexes—then embedded to R ℓ by summing hat-function coordinates over a landmark grid: each diagram point (red) contributes to the coordinates indexed by the landmarks (orange) whose 𝑑B-cover squares it falls within, via Φ𝑝 (𝐴) = Í 𝑎∈𝐴 𝜑𝑅,𝑝 (𝑎). The embedded vector… view at source ↗
Figure 3
Figure 3. Figure 3: Landmark grid, hat coordinate, and summation embedding. (a) Grid G𝑅 (odd 𝑚, even 𝑛, 𝑛 ≥ 𝑚+3) with 𝑑B-cover squares of radius 3𝑅 2 ; diagram 𝐴 = {𝑎1, 𝑎2, 𝑎3} (red): 𝑎1, 𝑎2 each fall in three lattice landmarks (with 𝑝3 shared—the summation site), while the low-persistence point 𝑎3 contributes only to the diagonal landmark ∗. (b) Hat 𝜑𝑅,𝑝 (𝑥) = max{ 3𝑅 2 −𝑑B (𝑝, 𝑥), 0}: a 𝑑∞-pyramid peaking at 𝑝; its level se… view at source ↗
Figure 3
Figure 3. Figure 3: Landmark grid, hat coordinate, and summation embedding. (a) Grid G𝑅 (odd 𝑚, even 𝑛, 𝑛 ≥ 𝑚+3) with 𝑑B-cover squares of radius 3𝑅 2 ; diagram 𝐴 = {𝑎1, 𝑎2, 𝑎3} (red): 𝑎1, 𝑎2 each fall in three lattice landmarks (with 𝑝3 shared—the summation site), while the low-persistence point 𝑎3 contributes only to the diagonal landmark ∗. (b) Hat 𝜑𝑅,𝑝 (𝑥) = max{ 3𝑅 2 −𝑑B (𝑝, 𝑥), 0}: a 𝑑∞-pyramid peaking at 𝑝; its level se… view at source ↗
Figure 4
Figure 4. Figure 4: Confidence containment (Theorem 5.1). The depicted pair (𝑐, 𝑐′ ) is the worst-separated one, with ∥𝜇𝑐 − 𝜇𝑐 ′ ∥ = Δ (other pairs have distance ≥ Δ). The empirical centroid 𝜇ˆ𝑐 lies within 𝑟𝑚 of the population centroid 𝜇𝑐 (blue ball) with probability ≥ 1 − 𝛼. When 𝑟𝑚 < 1 2 Δ, any test point farther than 2𝑟𝑚 from the population Voronoi boundary (dashed) is classified identically by the empirical and populatio… view at source ↗
Figure 4
Figure 4. Figure 4: Confidence containment (Theorem 5.1). The depicted pair (𝑐, 𝑐′ ) is the worst-separated one, with ∥𝜇𝑐 − 𝜇𝑐 ′ ∥ = Δ (other pairs have distance ≥ Δ). The empirical centroid 𝜇ˆ𝑐 lies within 𝑟𝑚 of the population centroid 𝜇𝑐 (blue ball) with probability ≥ 1 − 𝛼. When 𝑟𝑚 < 1 2 Δ, any test point farther than 2𝑟𝑚 from the population Voronoi boundary (dashed) is classified identically by the empirical and populatio… view at source ↗
Figure 5
Figure 5. Figure 5: Orbit5k: point clouds (top) and 𝐻1 persistence diagrams (bottom) for each class 𝜌 ∈ {2.5, 3.5, 4.0, 4.1, 4.3}. 6.2 Graph Classification We evaluate on 11 benchmarks from (Zhao and Wang, 2019) spanning three domains: molecular graphs (MUTAG 188, NCI1 4110, NCI109 4127, PTC 344, COX2 467, DHFR 756), protein structures (PROTEINS 1113, DD 1178), and social networks (IMDB-B 1000, IMDB-M 1500, REDDIT-5K 4999). A… view at source ↗
Figure 5
Figure 5. Figure 5: Orbit5k: point clouds (top) and 𝐻1 persistence diagrams (bottom) for each class 𝜌 ∈ {2.5, 3.5, 4.0, 4.1, 4.3}. 6.2 Graph Classification We evaluate on 11 benchmarks from (Zhao and Wang, 2019) spanning three domains: molecular graphs (MUTAG 188, NCI1 4110, NCI109 4127, PTC 344, COX2 467, DHFR 756), protein structures (PROTEINS 1113, DD 1178), and social networks (IMDB-B 1000, IMDB-M 1500, REDDIT-5K 4999). A… view at source ↗
Figure 6
Figure 6. Figure 6: Graph-to-diagram pipeline on a MUTAG molecule: HKS filtration (left), view at source ↗
Figure 6
Figure 6. Figure 6: Graph-to-diagram pipeline on a MUTAG molecule: HKS filtration (left), [PITH_FULL_IMAGE:figures/full_fig_p033_6.png] view at source ↗
read the original abstract

We introduce PLACE (Persistence-Landmark Analytic Classification Engine), a closed-form pipeline for classifying point clouds and graphs through their persistent-homology signatures. Three quantitative guarantees -- a margin-based excess-risk rate, a closed-form descriptor-selection rule, and a per-prediction certificate -- are derived from training labels alone, with no learned weights or held-out calibration. The embedding sums Mitra-Virk single-point coordinate functions over a sparse landmark grid; the closed-form weight rule $w_k^2 \propto (d_{k+1}^2 - d_k^2)/R_k^2$ maximizes the distortion slope in Mitra-Virk's affine certificate under $\nu$-coherence. (i) An $O(kR/(\Delta\sqrt{m_{\min}}))$ margin bound, driven by class-mean separation $\Delta$ and embedding radius $R$, matched in the sample-starved regime $m \lesssim R/\Delta$ by a Le Cam minimax lower bound. (ii) The Mahalanobis margin under Ledoit-Wolf-shrunk covariance is the strongest closed-form ranker on a 64-descriptor chemical-graph pool (mean Spearman $\rho = +0.56$ across 11 benchmarks, positive on 10 of 11); the isotropic surrogate $\Delta/\sqrt{\ell}$ admits a closed-form selection-consistency rate on the homogeneous protein/social pools. (iii) A training-time-decided certificate, with no per-prediction overhead, in three concrete radii (Pinelis, Gaussian plug-in, and variance-aware Pinelis-Bernstein). Empirically, PLACE is the strongest diagram-based method on Orbit5k and matches the strongest topology-based baseline within statistical noise on MUTAG and COX2; remaining gaps fall into two diagnosable regimes (descriptor blindness on NCI1/NCI109; pool-coverage limits elsewhere). The Pinelis-Bernstein radius fires on 8 of the 12 benchmarks; on MUTAG the empirical and population nearest-centroid rules agree on every one of 940 held-out test predictions, validating the certificate's mechanism.

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

2 major / 2 minor

Summary. The paper introduces PLACE, a closed-form pipeline for point-cloud and graph classification via persistent-homology signatures. It asserts three guarantees derived solely from training labels with no learned weights or held-out calibration: (i) an O(kR/(Δ√m_min)) margin-based excess-risk bound driven by class-mean separation Δ and embedding radius R, matched by a Le Cam minimax lower bound in the sample-starved regime; (ii) a closed-form descriptor-selection rule w_k² ∝ (d_{k+1}² - d_k²)/R_k² that maximizes distortion slope in the Mitra-Virk affine certificate under ν-coherence; (iii) a training-time certificate in three radii (Pinelis, Gaussian plug-in, variance-aware Pinelis-Bernstein). The embedding sums Mitra-Virk coordinate functions over a sparse landmark grid. Empirically, PLACE is strongest on Orbit5k, matches top baselines within noise on MUTAG/COX2, with the Pinelis-Bernstein radius firing on 8/12 benchmarks and perfect agreement on all 940 MUTAG held-out predictions.

Significance. If the derivations hold and the ν-coherence assumption is verified on the target data, the work would be significant for delivering parameter-free, training-label-only guarantees and certificates in topological ML, improving reliability and interpretability for graphs and point clouds. Explicit credit is due for matching the margin upper bound to a Le Cam lower bound and for the concrete empirical validation of the certificate mechanism via held-out prediction agreement.

major comments (2)
  1. [Abstract] Abstract: the closed-form descriptor-selection rule and per-prediction certificate are derived under the ν-coherence assumption so that w_k² ∝ (d_{k+1}² - d_k²)/R_k² maximizes the distortion slope inside the Mitra-Virk affine certificate; no verification that ν-coherence holds on Orbit5k, MUTAG, COX2, NCI1 or the other benchmarks is indicated, and if the assumption fails the claims for (ii) and (iii) do not follow.
  2. [Abstract] Abstract: the three guarantees are presented as derived from training labels alone with no learned weights, yet the embedding and certificate rest on Mitra-Virk single-point coordinate functions and the affine certificate from the authors' prior work, with the weight rule explicitly defined to optimize that prior certificate; this dependence must be clarified for the 'closed-form from labels alone' claim to stand independently.
minor comments (2)
  1. The abstract states performance 'within statistical noise' on MUTAG and COX2 but provides no details on error-bar computation, number of runs, or statistical tests used.
  2. The abstract asserts derivations of the three guarantees but supplies no derivation steps, data-exclusion rules, or explicit conditions under which the Le Cam matching applies.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the closed-form descriptor-selection rule and per-prediction certificate are derived under the ν-coherence assumption so that w_k² ∝ (d_{k+1}² - d_k²)/R_k² maximizes the distortion slope inside the Mitra-Virk affine certificate; no verification that ν-coherence holds on Orbit5k, MUTAG, COX2, NCI1 or the other benchmarks is indicated, and if the assumption fails the claims for (ii) and (iii) do not follow.

    Authors: The selection rule and certificate are explicitly presented as holding under the ν-coherence assumption, as stated in the abstract. Consequently, the claims in (ii) and (iii) are conditional on this assumption. We did not include an empirical verification of ν-coherence for the listed benchmarks in the original submission. In the revision we will add an appendix verifying the assumption on Orbit5k, MUTAG, COX2, NCI1 and the remaining benchmarks, or provide a sensitivity discussion if mild violations occur. revision: yes

  2. Referee: [Abstract] Abstract: the three guarantees are presented as derived from training labels alone with no learned weights, yet the embedding and certificate rest on Mitra-Virk single-point coordinate functions and the affine certificate from the authors' prior work, with the weight rule explicitly defined to optimize that prior certificate; this dependence must be clarified for the 'closed-form from labels alone' claim to stand independently.

    Authors: We agree that the foundational dependence on the Mitra-Virk coordinate functions and affine certificate from prior work should be stated more explicitly. The phrase 'derived from training labels alone with no learned weights' refers to the fact that the excess-risk bound, the closed-form selection rule, and the three radii are computed directly from the training labels without additional learned parameters or held-out calibration. The embedding itself, however, is built from the Mitra-Virk functions and the weight rule is chosen to optimize the prior certificate. In the revised abstract and introduction we will clarify this dependence while preserving the distinction that no new parameters are fitted from the data. revision: yes

Circularity Check

1 steps flagged

Descriptor-selection rule and certificate rest on self-cited Mitra-Virk embedding and affine certificate under ν-coherence

specific steps
  1. self citation load bearing [Abstract]
    "The embedding sums Mitra-Virk single-point coordinate functions over a sparse landmark grid; the closed-form weight rule $w_k^2 \∝ (d_{k+1}^2 - d_k^2)/R_k^2$ maximizes the distortion slope in Mitra-Virk's affine certificate under $\nu$-coherence."

    The weight rule is defined to optimize the distortion slope inside the authors' own prior Mitra-Virk affine certificate (with Mitra and Virk as co-authors here). The closed-form descriptor-selection rule and training-time certificate therefore reduce to the self-cited construction plus the unverified ν-coherence assumption rather than being independently derived from the current paper's inputs.

full rationale

The paper presents three guarantees derived from training labels alone. The margin excess-risk bound O(kR/(Δ√m_min)) is matched to an external Le Cam lower bound and appears independent. However, the closed-form weight rule and per-prediction certificate are explicitly constructed around the Mitra-Virk single-point coordinate functions and affine certificate from prior work by overlapping authors (Mitra, Virk). The weight rule is defined precisely to maximize distortion slope inside that prior certificate under the ν-coherence assumption, with no indicated verification on the benchmark collections. This makes the selection rule and certificate load-bearing on the self-cited framework rather than independently derived, producing partial circularity (score 6). The embedding itself is imported via the same self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The pipeline depends on Mitra-Virk single-point coordinate functions as base embedding and the ν-coherence assumption to justify the closed-form weights; no free parameters are introduced because the weight rule is closed-form, and no new entities are postulated.

axioms (1)
  • domain assumption ν-coherence assumption holds for the weight rule to maximize distortion slope
    Invoked to derive the closed-form weight rule w_k² ∝ (d_{k+1}² - d_k²)/R_k² from the Mitra-Virk affine certificate.

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Reference graph

Works this paper leans on

47 extracted references · 18 canonical work pages · 1 internal anchor

  1. [1]

    Persistence images: A stable vector representation of persistent homology

    Henry Adams, Tegan Emerson, Michael Kirby, Rachel Neville, Chris Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson, Francis Motta, and Lori Ziegelmeier. Persistence images: A stable vector representation of persistent homology. Journal of Machine Learning Research, 18 0 (8): 0 1--35, 2017

    2017

  2. [2]

    A survey of vectorization methods in topological data analysis

    Dashti Ali, Aras Asaad, Maria-Jose Jimenez, Vidit Nanda, Eduardo Paluzo-Hidalgo, and Manuel Soriano-Trigueros. A survey of vectorization methods in topological data analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45 0 (12): 0 14069--14080, 2023. doi:10.1109/TPAMI.2023.3308391

    work page doi:10.1109/tpami.2023.3308391 2023

  3. [3]

    DTM -based filtrations

    Hirokazu Anai, Fr \'e d \'e ric Chazal, Marc Glisse, Yuichi Ike, Hiroya Inakoshi, Rapha \"e l Tinarrage, and Yuhei Umeda. DTM -based filtrations. In International Symposium on Computational Geometry (SoCG), pages 58:1--58:15, 2019

    2019

  4. [4]

    A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification

    Anastasios N. Angelopoulos and Stephen Bates. Conformal prediction: A gentle introduction. Foundations and Trends in Machine Learning, 16 0 (4): 0 494--591, 2023. doi:10.1561/2200000101. Originally arXiv:2107.07511

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1561/2200000101 2023

  5. [5]

    Bartlett and Marian Hristache Wegkamp

    Peter L. Bartlett and Marian Hristache Wegkamp. Classification with a reject option using a hinge loss. Journal of Machine Learning Research, 9: 0 1823--1840, 2008

    2008

  6. [6]

    V. Bentkus. On the dependence of the B erry-- E sseen bound on dimension. Journal of Statistical Planning and Inference, 113 0 (2): 0 385--402, 2003

    2003

  7. [7]

    A faster algorithm for betweenness centrality

    Ulrik Brandes. A faster algorithm for betweenness centrality. Journal of Mathematical Sociology, 25 0 (2): 0 163--177, 2001. doi:10.1080/0022250X.2001.9990249

    work page doi:10.1080/0022250x.2001.9990249 2001

  8. [8]

    Statistical topological data analysis using persistence landscapes

    Peter Bubenik. Statistical topological data analysis using persistence landscapes. Journal of Machine Learning Research, 16 0 (1): 0 77--102, 2015

    2015

  9. [9]

    Embeddings of persistence diagrams into H ilbert spaces

    Peter Bubenik and Alexander Wagner. Embeddings of persistence diagrams into H ilbert spaces. Journal of Applied and Computational Topology, 4 0 (3): 0 339--351, 2020. doi:10.1007/s41468-020-00056-w

    work page doi:10.1007/s41468-020-00056-w 2020

  10. [10]

    On the metric distortion of embedding persistence diagrams into separable H ilbert spaces

    Mathieu Carri \`e re and Ulrich Bauer. On the metric distortion of embedding persistence diagrams into separable H ilbert spaces. In Proceedings of the 35th Annual Symposium on Computational Geometry (SoCG). Schloss Dagstuhl - Leibniz-Zentrum f \"u r Informatik, June 2019

    2019

  11. [11]

    Sliced W asserstein kernel for persistence diagrams

    Mathieu Carri \`e re, Marco Cuturi, and Steve Oudot. Sliced W asserstein kernel for persistence diagrams. In Proceedings of the 34th International Conference on Machine Learning (ICML), volume 70 of Proceedings of Machine Learning Research, pages 664--673. PMLR, 2017

    2017

  12. [12]

    PersLay : A neural network layer for persistence diagrams and new graph topological signatures

    Mathieu Carri \`e re, Fr \'e d \'e ric Chazal, Yuichi Ike, Th \'e o Lacombe, Martin Royer, and Yuhei Umeda. PersLay : A neural network layer for persistence diagrams and new graph topological signatures. In International Conference on Artificial Intelligence and Statistics (AISTATS), pages 2786--2796, 2020

    2020

  13. [13]

    Guibas, and Steve Y

    Fr \'e d \'e ric Chazal, David Cohen-Steiner, Marc Glisse, Leonidas J. Guibas, and Steve Y. Oudot. Proximity of persistence modules and their diagrams. In Proceedings of the 25th Annual Symposium on Computational Geometry (SoCG), pages 237--246, 2009. doi:10.1145/1542362.1542407

    work page doi:10.1145/1542362.1542407 2009

  14. [14]

    The Structure and Stability of Persistence Modules

    Fr \'e d \'e ric Chazal, Vin de Silva, Marc Glisse, and Steve Oudot. The Structure and Stability of Persistence Modules. SpringerBriefs in Mathematics. Springer, 2016. doi:10.1007/978-3-319-42545-0

    work page doi:10.1007/978-3-319-42545-0 2016

  15. [15]

    C. K. Chow. On optimum recognition error and reject tradeoff. IEEE Transactions on Information Theory, 16 0 (1): 0 41--46, 1970. doi:10.1109/TIT.1970.1054406

    work page doi:10.1109/tit.1970.1054406 1970

  16. [16]

    Stability of persistence diagrams

    David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams. Discrete & Computational Geometry, 37 0 (1): 0 103--120, 2007

    2007

  17. [17]

    Extending persistence using Poincaré and700 Lefschetz duality.Foundations of Computational Mathematics, 9(1):79–103, April 2008

    David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Extending persistence using P oincar \'e and L efschetz duality. Foundations of Computational Mathematics, 9 0 (1): 0 79--103, 2009. doi:10.1007/s10208-008-9027-z

    work page doi:10.1007/s10208-008-9027-z 2009

  18. [18]

    Learning with rejection

    Corinna Cortes, Giulia DeSalvo, and Mehryar Mohri. Learning with rejection. In Algorithmic Learning Theory (ALT), pages 67--82, 2016

    2016

  19. [19]

    Computational Topology

    Herbert Edelsbrunner and John L Harer. Computational Topology. American Mathematical Society, Providence, RI, January 2010

    2010

  20. [20]

    Linton C. Freeman. A set of measures of centrality based on betweenness. Sociometry, 40 0 (1): 0 35--41, 1977. doi:10.2307/3033543

    work page doi:10.2307/3033543 1977

  21. [21]

    Nelson, Anjan Dwaraknath, and Primoz Skraba

    Rickard Br\"uel Gabrielsson, Bradley J. Nelson, Anjan Dwaraknath, and Primoz Skraba. A topology layer for machine learning. In International Conference on Artificial Intelligence and Statistics (AISTATS), 2020

    2020

  22. [22]

    Selective classification for deep neural networks

    Yonatan Geifman and Ran El-Yaniv. Selective classification for deep neural networks. In Advances in Neural Information Processing Systems, volume 30, pages 4878--4887, 2017

    2017

  23. [23]

    Selectivenet: A deep neural network with a reject option

    Yonatan Geifman and Ran El-Yaniv. Selectivenet: A deep neural network with a reject option. International Conference on Machine Learning (ICML), 2019

    2019

  24. [24]

    Golub and Charles F

    Gene H. Golub and Charles F. Van Loan . Matrix Computations. Johns Hopkins University Press, Baltimore, MD, 3 edition, 1996

    1996

  25. [25]

    Euler characteristic tools for topological data analysis

    Olympio Hacquard and Vadim Lebovici. Euler characteristic tools for topological data analysis. Journal of Machine Learning Research, 25: 0 1--39, 2024

    2024

  26. [26]

    Halko, P

    N. Halko, P. G. Martinsson, and J. A. Tropp. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Review, 53 0 (2): 0 217--288, 2011

    2011

  27. [27]

    Deep learning with topological signatures

    Christoph Hofer, Roland Kwitt, Marc Niethammer, and Andreas Uhl. Deep learning with topological signatures. In Advances in Neural Information Processing Systems (NeurIPS), 2017

    2017

  28. [28]

    Topological graph neural networks

    Max Horn, Edward De Brouwer, Michael Moor, Yves Moreau, Bastian Rieck, and Karsten Borgwardt. Topological graph neural networks. In International Conference on Learning Representations (ICLR), 2022

    2022

  29. [29]

    Persistence weighted G aussian kernel for topological data analysis

    Genki Kusano, Yasuaki Hiraoka, and Kenji Fukumizu. Persistence weighted G aussian kernel for topological data analysis. In Proceedings of the 33rd International Conference on Machine Learning (ICML), pages 2004--2013, 2016

    2004

  30. [30]

    Persistence F isher kernel: A R iemannian manifold kernel for persistence diagrams

    Tam Le and Makoto Yamada. Persistence F isher kernel: A R iemannian manifold kernel for persistence diagrams. In Advances in Neural Information Processing Systems (NeurIPS), volume 31, pages 10028--10039, 2018

    2018

  31. [31]

    Variational inference: A review for statisticians,

    Jing Lei, Max G'Sell, Alessandro Rinaldo, Ryan J. Tibshirani, and Larry Wasserman. Distribution-free predictive inference for regression. Journal of the American Statistical Association, 113 0 (523): 0 1094--1111, 2018. doi:10.1080/01621459.2017.1307116

    work page doi:10.1080/01621459.2017.1307116 2018

  32. [32]

    The space of persistence diagrams on n points coarsely embeds into H ilbert space

    Atish Mitra and Z iga Virk. The space of persistence diagrams on n points coarsely embeds into H ilbert space. Proceedings of the American Mathematical Society, 149 0 (6): 0 2693--2703, 2021. doi:10.1090/proc/15363

    work page doi:10.1090/proc/15363 2021

  33. [33]

    Geometric embeddings of spaces of persistence diagrams with explicit distortions

    Atish Mitra and Z iga Virk. Geometric embeddings of spaces of persistence diagrams with explicit distortions. arXiv:2401.05298, 2024. URL https://arxiv.org/abs/2401.05298

    work page arXiv 2024

  34. [34]

    Foundations of Machine Learning

    Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar. Foundations of Machine Learning. MIT Press, Cambridge, MA, 2nd edition, 2018

    2018

  35. [35]

    Ricci curvature of M arkov chains on metric spaces

    Yann Ollivier. Ricci curvature of M arkov chains on metric spaces. Journal of Functional Analysis, 256 0 (3): 0 810--864, 2009. doi:10.1016/j.jfa.2008.11.001

    work page doi:10.1016/j.jfa.2008.11.001 2009

  36. [36]

    Optimum bounds for the distributions of martingales in B anach spaces

    Iosif Pinelis. Optimum bounds for the distributions of martingales in B anach spaces. Annals of Probability, 22 0 (4): 0 1679--1706, 1994

    1994

  37. [37]

    Persformer: A transformer architecture for topological machine learning

    Raphael Reinauer, Matteo Caorsi, and Nicolas Berkouk. Persformer: A transformer architecture for topological machine learning. In arXiv preprint arXiv:2112.15210, 2021

    work page arXiv 2021

  38. [38]

    Differentiable euler characteristic transforms for shape classification

    Ernst R\"oell and Bastian Rieck. Differentiable euler characteristic transforms for shape classification. In International Conference on Learning Representations (ICLR), 2024

    2024

  39. [39]

    Computer Graphics Forum28(2), 407–416 (2009).https://doi.org/10

    Jian Sun, Maks Ovsjanikov, and Leonidas Guibas. A concise and provably informative multi-scale signature based on heat diffusion. Computer Graphics Forum, 28 0 (5): 0 1383--1392, 2009. doi:10.1111/j.1467-8659.2009.01515.x

    work page doi:10.1111/j.1467-8659.2009.01515.x 2009

  40. [40]

    Joel A. Tropp. An introduction to matrix concentration inequalities. Foundations and Trends in Machine Learning, 8 0 (1-2): 0 1--230, 2015

    2015

  41. [41]

    Tsybakov

    Alexandre B. Tsybakov. Introduction to Nonparametric Estimation. Springer Series in Statistics. Springer, 2009. doi:10.1007/b13794

    work page doi:10.1007/b13794 2009

  42. [42]

    Transductive conformal predictors

    Vladimir Vovk. Transductive conformal predictors. In Artificial Intelligence and Statistics (AISTATS), pages 1209--1217, 2013

    2013

  43. [43]

    Algorithmic Learning in a Random World

    Vladimir Vovk, Alex Gammerman, and Glenn Shafer. Algorithmic Learning in a Random World. Springer, 2005. doi:10.1007/b106715

    work page doi:10.1007/b106715 2005

  44. [44]

    How powerful are graph neural networks? In International Conference on Learning Representations (ICLR), 2019

    Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? In International Conference on Learning Representations (ICLR), 2019

    2019

  45. [45]

    Coarse and bi- L ipschitz embeddability of subspaces of the G romov-- H ausdorff space into H ilbert spaces

    Nicol \`o Zava. Coarse and bi- L ipschitz embeddability of subspaces of the G romov-- H ausdorff space into H ilbert spaces. Algebraic & Geometric Topology, 25 0 (8): 0 5153--5174, 2025. doi:10.2140/agt.2025.25.5153

    work page doi:10.2140/agt.2025.25.5153 2025

  46. [46]

    R et GK : Graph kernels based on return probabilities of random walks

    Zhen Zhang, Mianzhi Wang, Yijian Xiang, Yan Huang, and Arye Nehorai. R et GK : Graph kernels based on return probabilities of random walks. In Advances in Neural Information Processing Systems (NeurIPS), 2018

    2018

  47. [47]

    Learning metrics for persistence-based summaries and applications for graph classification

    Qi Zhao and Yusu Wang. Learning metrics for persistence-based summaries and applications for graph classification. In Advances in Neural Information Processing Systems, volume 32, pages 9855--9866, 2019. NeurIPS 2019

    2019