Pith. sign in

REVIEW 3 major objections 6 minor 47 references

Path representations in multiparameter persistent homology

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read The paper proposes path distances for multiparameter persistence modules, proves they extend the matching distance from straight slices to monotone piecewise-linear paths, and supplies software for computing them.

desk verdict A natural generalization with a genuine definitional bug: the path persistence module divides by zero on any axis-aligned segment, so the main construction is undefined on its stated domain; fixable but needs careful revision. read the letter →

arxiv 2507.23762 v1 pith:TUIUSFSY submitted 2025-07-31 math.AT

Xudong Sun , René Corbet , Carsten Marr This is my paper · ORCID
classification math.AT MSC 55N31
keywords multiparameterpersistencepersistenthomologymatchingdistancepathbottleneckWassersteinmonotonepathsmultifiltrations
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

The paper claims that distances between multiparameter persistence modules can be computed along monotone piecewise-linear paths rather than only along the straight lines used by the matching distance, and that the resulting path distance is a genuine generalization: it is always at least as large as the matching distance (Corollary 3.5). The motivation is that a single direction in the multiparameter space may miss topological differences that a bending path could reveal, so the path distance gives a more flexible way to compare data instances. The paper also introduces a Wasserstein version of the path distance, replacing the bottleneck distance, and provides software that implements the construction on bifiltrations. If the central claim is right, the path distance is the first computationally feasible extension of the matching distance from slices to paths.

What carries the argument

The central object is the path persistence module of Definition 3.3, built from a finite ordered sequence of points $p_0 \prec p_1 \prec \cdots \prec p_n$. Along each segment the module is evaluated at $M(p_i + t \cdot (p_{i+1}-p_i)/(w_{\ell_{p_i,p_{i+1}}} \|p_{i+1}-p_i\|))$, with $t$ measured from an accumulated weighted arclength coordinate; the weight $w_\ell = \min_i |v_i|$ of the segment's unit direction vector stretches the segment's contribution. The stretching lemma (Lemma 3.1) is what carries the argument: it converts the weight factor in the matching distance into a literal reparameterization of the one-parameter slice, so every straight slice is a path carried by two points. That embedding is what makes Corollary 3.5 immediate, and it is also what the software implements by projecting the multifiltration onto the chosen path.

What would settle it

Compute $d_\Pi(M,N)$ for two modules on $\mathbb{R}^2$ along the path with points $(0,0) \prec (1,0)$: the segment direction is $(1,0)$, so $w_\ell = \min(|1|,|0|)=0$, and equation (13) divides by zero. If the paper's definition is meant to cover all monotone piecewise-linear paths, this single calculation shows the construction is not defined without an added strict-increase assumption; alternatively, running the released software on such a path should reveal whether an admissibility check is enforced.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is the path persistence module $M_\pi$: a multiparameter persistence module restricted to a piecewise-linear monotone path and reparameterized segment by segment, with each segment stretched by the weight $w_\ell$ from equation (6). Applying the one-parameter bottleneck or Wasserstein distance to the resulting modules and taking the supremum over all finite monotone paths defines the path distance $d_\Pi^B$ or $d_\Pi^W$. Lemma 3.1 shows that multiplying a slice's distance by its weight is equivalent to stretching the slice module, so straight slices are exactly the two-point paths; Corollary 3.5 then gives $d_\Pi(M,N) \ge d_{\mathrm{match}}(M,N)$. The paper also reports software that computes these distances along a user-specified or optimized path.

Load-bearing premise

Whenever a leg of the path keeps one filtration parameter fixed, the weight used in the construction is zero, and the construction divides by that weight; the paper never says such legs are forbidden.

Editorial extensions

If this is right

  • The matching distance is a special case of the path distance, so the path distance is at least as discriminative: any separation visible to matching distance is visible to path distance.
  • A path can be optimized to maximize the distance between two data instances, and the achieving path marks regions of the parameter space where the two modules differ most.
  • The Wasserstein path distance accounts for all matched points in the persistence diagrams rather than a single worst pair, giving a tunable comparison with the parameter $q$.
  • The accompanying software lets a user compute bottleneck or Wasserstein distances along a specified or learned path from bifiltrations and boundary matrices, including the multicover bifiltration.
  • The path distance can in principle distinguish point clouds whose topological differences are not aligned with any single filtration direction.

Reading between the lines

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

  • A natural extension the authors leave implicit is a stability statement: if the path construction is restricted to segments whose weights stay bounded away from zero, the path distance may inherit a stability bound with respect to the interleaving distance, but the paper does not prove this.
  • The definition silently requires each path segment to strictly increase every filtration coordinate; axis-parallel segments make the weight $w_\ell$ zero and the formula in Definition 3.3 divides by zero, so a reader who wants to run the software on such paths needs an explicit admissibility condition.
  • The path distance could be compared empirically with the coherent matching distance, which transports matchings along paths rather than projecting modules; the paper notes this as a direction for future work.
  • One could use the optimized path as a data-dependent feature map for machine learning, selecting the path that best separates classes; the authors only suggest it as a heuristic to detect differing regions.
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 / 6 minor

Summary. The paper proposes to generalize the multiparameter matching distance by replacing straight-line slices with monotone piecewise-linear paths. The central construction is the path persistence module M_pi (Definition 3.3), defined by stretching each path segment by the weight w_l of its direction vector, and the corresponding path distance d_Pi, defined as the supremum over paths of the bottleneck (or Wasserstein) distance between path modules. The main theoretical claim is Corollary 3.5, which asserts that the path distance is at least the matching distance. The paper also presents algorithms for constructing paths and computing the distance between two point clouds, with an implementation in C++/Python.

Significance. The idea of comparing multiparameter persistence modules along nonlinear monotone paths is natural and potentially useful; if correctly formalized, it would genuinely extend the matching distance and offer a new tool for data comparison. The software implementation is a tangible practical contribution, and the paper connects to existing tools (Rivet, Hera, etc.). However, the mathematical development as written contains serious flaws in the definitions and in the proof of the main theorem, so the central claim is not currently established. The errors appear fixable, but the revision needs substantial care.

major comments (3)
  1. [§3.2, Definition 3.3 (Eqs. 13–16)] Definition 3.3 divides by w_l, the minimum absolute coordinate of the unit direction vector (Eq. 6). For any monotone path segment that keeps at least one coordinate unchanged, w_l = 0, so Eqs. (13) and (15) involve division by zero. Definition 3.2 allows such segments, e.g., p_i = (0,0) to p_{i+1} = (1,0), so the path persistence module M_pi is undefined on a nonempty subset of the stated domain. Consequently, the path distance d_Pi in Eq. (18) is not well-defined as written. Notably, Algorithm 2 (line 5) enforces strict increase in every coordinate, so the implementation silently uses a restricted class of paths. The authors should either restrict Definition 3.2 to paths with w_l > 0 on every segment or handle the zero-weight case explicitly.
  2. [§3.1, Lemma 3.1 and Eq. (7)] The lemma as stated is false. Equation (7) defines M^{w_l}(x) = M_l(w_l x). If a feature of M_l has birth b and death d, then in M^{w_l} the feature is born at x = b/w_l and dies at d/w_l, so the persistence diagram is scaled by 1/w_l, not w_l. The proof's claim that y^{bd} = w_l x^{bd} (after Eq. 10) reverses this relation; the correct correspondence is y^{bd} = x^{bd}/w_l. Thus the displayed equality d_B(M^{w_l},N^{w_l}) = w_l d_B(M_l,N_l) does not hold. The desired scaling factor is instead achieved by the module M_l(x/w_l), which is what the path module in Eq. (13) actually implements. Lemma 3.1 and its proof must be corrected.
  3. [§3.2, Corollary 3.5] The proof claims that the set of slices L+ is a subset of Pi_2(P), but this is incorrect: a slice is an infinite straight line, whereas Pi_2(P) consists of finite two-point paths. For a finite path starting at p0, the path module M_pi is the restriction of M to the ray from p0 along the first segment, not the full line. If the feature that maximizes the matching distance lies at parameters 'before' p0, it is absent from M_pi, so the inequality d_Pi >= d_match is not established by the given argument. The authors need to either formally allow bi-infinite paths, or prove that for every slice and every pair of modules one can choose p0 such that the ray restriction agrees with the full-line restriction on the support of the persistence diagrams (e.g., under finiteness assumptions).
minor comments (6)
  1. [§3.2, Eq. (14)] The upper summation in the right-hand inequality starts at j = 1, omitting the first segment's contribution. The upper limit should be sum_{j=0}^{i} w_{p_j,p_{j+1}} ||p_{j+1}-p_j|| so that the interval for segment i is between consecutive accumulated lengths.
  2. [§3.2, Eq. (15)] In the sum of Eq. (15), the weight is written as w_{p_j,p_{j-1}}; this should be w_{p_j,p_{j+1}} to match the segment from p_j to p_{j+1}.
  3. [§3.3, Algorithm 1] Step 4, 'rescale the two persistence diagrams', is vague: the rescaling factor is not specified, and the connection to the weights in Eq. (13) is not stated.
  4. [§3.2, Definition 3.2] The notation p0 < p1 < ... < pn in the product poset allows non-strict increase in individual coordinates. The paper should clarify whether paths must be strictly increasing in all coordinates; the subsequent definition (which divides by w_l) and the implementation (Algorithm 2, which enforces strict increase) both depend on this distinction.
  5. [Abstract and Introduction] The claim of being 'the first to deal with distances along paths' should be softened: Remark 4 already cites the coherent matching distance [10], which transports matchings along paths, so the novelty lies in the specific construction and the path-distance formulation rather than in the mere use of paths.
  6. [§3.4, Algorithm 2] Line 5 is ambiguous: the condition 'x_i(p_{t-1}) < x_i(p) < x_i(p_{t-1}) + k_i δ_i with k_i = 1,...,n_i' does not specify whether the inequality must hold for all i simultaneously with a single choice of (k_1,...,k_n), or whether for each i there exists some k_i. Please clarify the intended quantization.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the path distance is defined independently and Corollary 3.5 follows from an elementary inclusion of slices into paths.

full rationale

The paper's central construction defines a new object, the path persistence module M_pi, by evaluating a multiparameter module along a piecewise-linear monotone path, and then defines d_Pi as the supremum of one-parameter bottleneck (or Wasserstein) distances over all such paths. Corollary 3.5, asserting d_Pi >= d_match, is proved by the observation that the set of straight slices L+ is a subset of the set of 2-point paths Pi_2(P), together with Lemma 3.1, which is a direct scaling argument: stretching the module by w_l rescales all persistence-diagram coordinates by w_l, so w_l * d_B(M_l,N_l) = d_B(M^{w_l},N^{w_l}). Nothing here is fitted to data, renamed from an input, or imported from a self-citation; the cited prior work supplies standard definitions and algorithms, and the new inequality has independent mathematical content. One non-circular concern is that Definition 3.3 divides by w_{ell_{p_i,p_{i+1}}}, which by Equation (6) is the minimum absolute coordinate of the unit direction vector; for a monotone path segment that keeps a coordinate constant, e.g. p_i=(0,0) to p_{i+1}=(1,0), this weight is zero, so M_pi is undefined on part of the declared domain Pi(P). This is a correctness/domain-definition gap rather than a circularity, and it does not change the fact that the derivation chain from the definitions to Corollary 3.5 is self-contained. The paper's novelty claim about being 'the first to deal with distances along paths' is an overclaim risk, not a circular step.

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

No free parameters are fitted to a target result; the mathematical claim rests on standard persistence theory and on an unstated positivity assumption for path weights. The path persistence module M_π is a definitional construction, not a postulated physical entity, so no invented entities are listed.

free parameters (4)
  • strip size δ_i = user-specified
    Algorithm 2 uses δ_i to define admissible next points; it is a hand-chosen discretization of the parameter space and does not enter the mathematical definition of the path distance.
  • look-ahead steps n_i = user-specified
    Algorithm 2 bounds the neighborhood searched for admissible points; affects path optimization, not the distance definition.
  • max path length T = user-specified
    Algorithm 2 stops path construction at length T; limits the search space in the optimization heuristic.
  • Wasserstein exponent q = user-specified
    Definition 3.6 requires a chosen q in [1,∞); a standard parameter of the Wasserstein distance, not fitted to data.
assumptions (4)
  • standard math Persistence modules are functors to Vect_K and bottleneck/Wasserstein distances on 1-parameter persistence diagrams are well-defined and computable.
    Background in Sections 2.1 and 2.3, referencing [23, 28, 15].
  • domain assumption For every monotone path π, the restricted module M_π is q-tame or otherwise admits a persistence diagram, and the bottleneck/Wasserstein distances between M_π and N_π are finite.
    Section 3.2 defines d_π without invoking finiteness conditions; [16, 34, 36, 41] are cited in Section 2.2 but not applied to paths.
  • domain assumption Every path segment has all coordinates strictly increasing, so w_ℓ > 0 in Eq. (6).
    Definition 3.3 divides by w_ℓ in Eq. (13); the paper does not state that this restriction is needed.
  • ad hoc to paper The stretching of path modules by w_ℓ in Definition 3.3 is chosen to match the slice weights of the matching distance.
    Section 3.2 states the stretching is motivated by stability guarantees for slices and by Lemma 3.1; it is a design choice rather than a derived necessity.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Path representations in multiparameter persistent homology." pith.science (2026). https://pith.science/paper/TUIUSFSY

@misc{pith2026250723762,
  author       = {Pith},
  title        = {Pith review of: Path representations in multiparameter persistent homology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TUIUSFSY}},
  note         = {Machine review of arXiv:2507.23762}
}
read the original abstract

Multiparameter persistence module can capture more topological differences across data instances compared to using a single parameter, where the well-studied matching distance investigates the distance along a straight line in the multiparameter space that gives the biggest difference. We propose to generalize the straight line to a monotone path filtration and offer software implementations.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

47 extracted references · 43 canonical work pages

  1. [10]

    On the ge- ometrical properties of the coherent matching distance in 2D persistent homology

    Andrea Cerri, Marc Ethier, and Patrizio Frosini. “On the ge- ometrical properties of the coherent matching distance in 2D persistent homology”. In: Journal of Applied and Compu- tational Topology 3 (2019), pp. 381–422

    work page 2019

  2. [1]

    Computing the interleaving distance is NP-hard

    Håvard Bakke Bjerkevik, Magnus Bakke Botnan, and Michael Kerber. “Computing the interleaving distance is NP-hard”. In: Foundations of Computational Mathematics 2020.05 (2019), p. 6

    work page 2019

  3. [2]

    ℓ𝑝-distances on multiparameter persistence modules

    Håvard Bakke Bjerkevik and Michael Lesnick. “ℓ𝑝-distances on multiparameter persistence modules”. In: arXiv preprint arXiv:2106.13589 (2021)

    arXiv 2021

  4. [3]

    An introduc- tion to multiparameter persistence

    Magnus Bakke Botnan and Michael Lesnick. “An introduc- tion to multiparameter persistence”. In: arXiv:2203.14289 (2022)

    arXiv 2022

  5. [4]

    Exact weights, path metrics, and algebraic Wasserstein distances

    Peter Bubenik, Jonathan Scott, and Donald Stanley. “An algebraic Wasserstein distance for generalized persistence modules”. In: arXiv preprint arXiv:1809.09654 (2018)

    work page Pith review arXiv 2018

  6. [5]

    Embeddings of per- sistence diagrams into Hilbert spaces

    Peter Bubenik and Alexander Wagner. “Embeddings of per- sistence diagrams into Hilbert spaces”. In: Journal of Applied and Computational Topology 4.3 (2020), pp. 339–351

    work page 2020

  7. [6]

    The theory of mul- tidimensional persistence

    Gunnar Carlsson and Afra Zomorodian. “The theory of mul- tidimensional persistence”. In: Discrete & Computational Ge- ometry 42.1 (2009), pp. 71–93

    work page 2009

  8. [7]

    On the Metric Distortion of Embedding Persistence Diagrams into separable Hilbert spaces

    Mathieu Carrière and Ulrich Bauer. “On the metric distortion of embedding persistence diagrams into separable Hilbert spaces”. In: arXiv preprint arXiv:1806.06924 (2018)

    work page Pith review arXiv 2018

Show all 47 references
  1. [8]

    Multiparameter persistence image for topological machine learning

    Mathieu Carrière and Andrew Blumberg. “Multiparameter persistence image for topological machine learning”. In: Ad- vances in Neural Information Processing Systems 33 (2020), pp. 22432–22444

    2020

  2. [9]

    Sliced Wasserstein kernel for persistence diagrams

    Mathieu Carriére, Marco Cuturi, and Steve Oudot. “Sliced Wasserstein kernel for persistence diagrams”. In:Proceedings of the 34th International Conference on Machine Learning . Vol. 70. 2017, pp. 664–673

    2017

  3. [11]

    Betti numbers in multidimensional per- sistent homology are stable functions

    Andrea Cerri et al. “Betti numbers in multidimensional per- sistent homology are stable functions”. In: Mathematical Methods in the Applied Sciences 36.12 (2013), pp. 1543–1557

    2013

  4. [12]

    The Shift-Dimension of Multipersistence Modules

    Wojciech Chachólski, René Corbet, and Anna-Laura Sattel- berger. The Shift-Dimension of Multipersistence Modules. 2021. doi: 10.48550/ARXIV.2112.06509. url: https://arxiv.org/abs/ 2112.06509

    2021 doi arXiv

  5. [13]

    Geometric Inference for Probability Measures

    Frédéric Chazal, David Cohen-Steiner, and Quentin Mérigot. “Geometric Inference for Probability Measures”. In: Foun- dations of Computational Mathematics 11 (2011), pp. 733– 751

    2011

  6. [14]

    Proximity of Persistence Modules and Their Diagrams

    Frédéric Chazal et al. “Proximity of Persistence Modules and Their Diagrams”. In: Proceedings of the Twenty-fifth Annual Symposium on Computational Geometry . Aarhus, Denmark, 2009, pp. 237–246. isbn: 978-1-60558-501-7

    2009

  7. [15]

    Stability of Persistence Diagrams

    David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. “Stability of Persistence Diagrams”. In: Discrete and Compu- tational Geometry 37.1 (2007), pp. 103–120

    2007

  8. [16]

    The representation the- orem of persistence revisited and generalized

    René Corbet and Michael Kerber. “The representation the- orem of persistence revisited and generalized”. In: Journal of Applied and Computational Topology (2018). doi: 10.1007/ s41468-018-0015-3. url: https://doi.org/10.1007/s41468-018- 0015-3

    2018 doi

  9. [17]

    A kernel for multi-parameter persistent homology

    René Corbet et al. “A kernel for multi-parameter persistent homology”. In: Computers & Graphics: X (2019)

    2019

  10. [18]

    Computing the Multicover Bifiltration

    René Corbet et al. “Computing the Multicover Bifiltration”. In: 37th International Symposium on Computational Geometry. 2021

    2021

  11. [19]

    ToDD: Topological compound finger- printing in computer-aided drug discovery

    Andac Demir et al. “ToDD: Topological compound finger- printing in computer-aided drug discovery”. In: Advances in Neural Information Processing Systems 35 (2022), pp. 27978– 27993

    2022

  12. [20]

    Commutative Algebra: With a View Toward Algebraic Geometry

    David Eisenbud. Commutative Algebra: With a View Toward Algebraic Geometry. Graduate Texts in Mathematics. Springer, 1995

    1995

  13. [21]

    Unzerlegbare Darstellungen I

    Peter Gabriel. “Unzerlegbare Darstellungen I”. In:Manuscripta mathematica 6.1 (1972), pp. 71–103

    1972

  14. [22]

    Stable Invariants for Multidimensional Persistence

    Oliver Gäfvert and Wojciech Chachólski. “Stable Invariants for Multidimensional Persistence”. In:arXiv:1703.03632 (2017)

    2017 arXiv

  15. [24]

    Barcodes: the persistent topology of data

    Robert Ghrist. “Barcodes: the persistent topology of data”. In: Bulletin of the American Mathematical Society 45.1 (2008), pp. 61–75

    2008

  16. [25]

    Stratifying multiparameter persistent homology

    Heather A Harrington et al. “Stratifying multiparameter persistent homology”. In: arXiv:1708.07390 (2017)

    2017 arXiv

  17. [26]

    Per- sistent Homology for Virtual Screening

    Bryn Keller, Michael Lesnick, and Theodore L. Willke. “Per- sistent Homology for Virtual Screening”. In:chemRxiv preprint (2018). doi: 10 . 26434 / chemrxiv. 6969260 . v3.url: https : //chemrxiv.org/articles/PHoS_Persistent_Homology_for_ Virtual_Screening/6969260

    2018

  18. [27]

    Exact Computation of the Matching Distance on 2-Parameter Per- sistence Modules

    Michael Kerber, Michael Lesnick, and Steve Oudot. “Exact Computation of the Matching Distance on 2-Parameter Per- sistence Modules”. In: 35th International Symposium on Com- putational Geometry (SoCG 2019) . Schloss Dagstuhl-Leibniz- Zentrum fuer Informatik. 2019

    2019

  19. [28]

    Ge- ometry Helps to Compare Persistence Diagrams

    Michael Kerber, Dmitriy Morozov, and Arnur Nigmetov. “Ge- ometry Helps to Compare Persistence Diagrams”. In: 2016 Proceedings of the Eighteenth Workshop on Algorithm Engi- neering and Experiments (ALENEX) . SIAM. 2016, pp. 103– 112

    2016

  20. [29]

    Efficient Approxima- tion of the Matching Distance for 2-Parameter Persistence

    Michael Kerber and Arnur Nigmetov. “Efficient Approxima- tion of the Matching Distance for 2-Parameter Persistence”. In: 36th International Symposium on Computational Geometry (SoCG 2020). Schloss Dagstuhl-Leibniz-Zentrum für Infor- matik. 2020

    2020

  21. [30]

    Per- sistence weighted Gaussian kernel for topological data anal- ysis

    Genki Kusano, Yasuaki Hiraoka, and Kenji Fukumizu. “Per- sistence weighted Gaussian kernel for topological data anal- ysis”. In: International Conference on Machine Learning . 2016, pp. 2004–2013. Xudong Sun, Rene Corbet, and Carsten Marr

    2016

  22. [31]

    The rank invariant stability via interleav- ings

    Claudia Landi. “The rank invariant stability via interleav- ings”. In: arXiv:1412.3374 (2014)

    2014 arXiv

  23. [32]

    The theory of the interleaving distance on multidimensional persistence modules

    Michael Lesnick. “The theory of the interleaving distance on multidimensional persistence modules”. In: Foundations of Computational Mathematics 15.3 (2015), pp. 613–650

    2015

  24. [33]

    Computing Minimal Presentations and Betti Numbers of 2-Parameter Persistent Homology

    Michael Lesnick and Matthew Wright. “Computing Minimal Presentations and Betti Numbers of 2-Parameter Persistent Homology”. In: arXiv:1902.05708 (2019)

    2019 arXiv

  25. [34]

    Interactive visualiza- tion of 2-D persistence modules

    Michael Lesnick and Matthew Wright. “Interactive visualiza- tion of 2-D persistence modules”. In: arXiv:1512.00180 (2015)

    2015 arXiv

  26. [35]

    Proba- bility measures on the space of persistence diagrams

    Yuriy Mileyko, Sayan Mukherjee, and John Harer. “Proba- bility measures on the space of persistence diagrams”. In: Inverse Problems 27.12 (2011), p. 124007

    2011

  27. [36]

    Data structures for real multiparameter persis- tence modules

    Ezra Miller. “Data structures for real multiparameter persis- tence modules”. In: arXiv:1709.08155 (2017)

    2017 arXiv

  28. [37]

    Ezra Miller and Bernd Sturmfels.Combinatorial commutative algebra. Vol. 227. Springer Science & Business Media, 2005

    2005

  29. [38]

    Topological autoencoders

    Michael Moor et al. “Topological autoencoders”. In: Interna- tional conference on machine learning . PMLR. 2020, pp. 7045– 7054

    2020

  30. [39]

    A stable multi-scale kernel for topo- logical machine learning

    Jan Reininghaus et al. “A stable multi-scale kernel for topo- logical machine learning”. In: IEEE Conference on Computer Vision and Pattern Recognition . 2015, pp. 4741–4748

    2015

  31. [40]

    Generalized per- sistence analysis based on stable rank invariant

    Henri Riihimäki and Wojciech Chachólski. “Generalized per- sistence analysis based on stable rank invariant”. In: arXiv preprint arXiv:1807.01217 (2018)

    2018 arXiv

  32. [41]

    Multidimensional persistence and noise

    Martina Scolamiero et al. “Multidimensional persistence and noise”. In: Foundations of Computational Mathematics 17.6 (2017), pp. 1367–1406

    2017

  33. [42]

    A Multicover Nerve for Geometric In- ference

    Donald R. Sheehy. “A Multicover Nerve for Geometric In- ference”. In: Proceedings of the 24th Canadian Conference on Computational Geometry. 2012

    2012

  34. [43]

    ReinBo: Machine learning pipeline conditional hierarchy search and configu- ration with Bayesian optimization embedded reinforcement learning

    Xudong Sun, Jiali Lin, and Bernd Bischl. “ReinBo: Machine learning pipeline conditional hierarchy search and configu- ration with Bayesian optimization embedded reinforcement learning”. In: Machine Learning and Knowledge Discovery in Databases: International Workshops of ECML ...

    2019

  35. [44]

    Reinforcement learning

    Richard S Sutton, Andrew G Barto, et al. “Reinforcement learning”. In: Journal of Cognitive Neuroscience 11.1 (1999), pp. 126–134

    1999

  36. [45]

    Invariants and Metrics for Multiparame- ter Persistent Homology

    Ashleigh Thomas. “Invariants and Metrics for Multiparame- ter Persistent Homology”. PhD thesis. Duke University, 2019

    2019

  37. [46]

    Fréchet means for distributions of persistence diagrams

    Katharine Turner et al. “Fréchet means for distributions of persistence diagrams”. In: Discrete & Computational Geome- try 52 (2014), pp. 44–70

    2014

  38. [47]

    Multiparameter Persistence Landscapes

    Oliver Vipond. “Multiparameter Persistence Landscapes”. In: arXiv:1812.09935 (2018)

    2018 arXiv

  39. [48]

    Nonembeddability of persistence dia- grams with p> 2 Wasserstein metric

    Alexander Wagner. “Nonembeddability of persistence dia- grams with p> 2 Wasserstein metric”. In: Proceedings of the American Mathematical Society 149.6 (2021), pp. 2673–2677. APPENDIX A APPROXIMATE PATH OPTIMIZATION WITH REINFORCMENT LEANRING In Algorithm 3, we use reinforceme...

    2021

Pith tools

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