Linearized Optimal Transport pyLOT Library: A Toolkit for Machine Learning on Point Clouds

Alexander Cloninger; Jun Linwu; Nicholas Karris; Varun Khurana

REVIEW 5 major objections 4 minor 35 references

Linearized Optimal Transport pyLOT Library: A Toolkit for Machine Learning on Point Clouds

T0 review · 5 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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

Pith's one-line read This paper introduces pyLOT, a Python library that embeds point clouds into a Hilbert space via optimal transport maps, so that classification, clustering, dimension reduction, and generation reduce to linear operations.

desk verdict A software paper whose artifact isn't available and whose headline accuracy claim isn't tested, but with a genuinely useful iterative-barycenter demonstration buried inside. read the letter →

arxiv 2502.03439 v1 pith:KBE2ENV3 submitted 2025-02-05 stat.ML cs.LGcs.MSstat.CO

Jun Linwu , Varun Khurana , Nicholas Karris , Alexander Cloninger This is my paper · ORCID

classification stat.MLcs.LGcs.MSstat.CO MSC 49Q2262R1062-0468T09

keywords linearizedoptimaltransportpointcloudsWassersteinbarycentermachinelearningdimensionreductionclassificationtoothdatasetPythonlibrary

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 introduces pyLOT, a Python library that embeds probability distributions or point clouds into the Hilbert space $L^2(\sigma)$ by computing the optimal transport map from a fixed reference measure $\sigma$ to each data point. The central claim is that after this linearized optimal transport (LOT) embedding, standard linear machine learning methods suffice for classification, clustering, dimension reduction, and data generation on measure-valued data. To support this, the authors run a case study on 58 high-resolution 3D scans of primate molars, reporting near-perfect classification accuracy and LOT barycenters that closely approximate true Wasserstein barycenters while cutting computation time from days to minutes. The paper argues that LOT is especially valuable when the number of measure-valued data points is small, since the embedding is non-parametric and requires no extensive training.

What carries the argument

The load-bearing object is the LOT embedding $\mu \mapsto T^\mu_\sigma$, where $T^\mu_\sigma$ is the optimal transport map from a fixed reference measure $\sigma$ to $\mu$, solved in practice as the barycentric projection of the discrete optimal transport plan. This map lives in $L^2(\sigma)$, so a point cloud becomes a vector; the Wasserstein manifold is replaced by its tangent space at $\sigma$, and distances become $\|T^{\mu_1}_\sigma - T^{\mu_2}_\sigma\|_{L^2(\sigma)}$. The second mechanism is the reference measure itself: starting from a Gaussian reference and iteratively replacing it with the LOT barycenter of the data moves the tangent space closer to the data, which the paper shows reduces relative barycenter error.

What would settle it

Take a point-cloud dataset whose classes differ by a transformation that violates the compatibility condition $h \circ T^\mu_\sigma = T^{h_\sharp\mu}_\sigma$ (for example, large rotations about a fixed axis), embed with a fixed Gaussian reference, and check whether linear classifiers on the LOT embeddings separate the classes. If accuracy is near chance while exact Wasserstein distances separate them, the reference-choice assumption fails.

Watch

Extended reading notes

Core claim

The central discovery is that replacing the Wasserstein distance with the $L^2(\sigma)$ distance between optimal transport maps — the LOT distance — preserves enough geometry for downstream linear methods to match the accuracy of full optimal transport at a fraction of the cost. Concretely, each tooth point cloud $\mu_i$ is mapped to the barycentric projection $T^{\mu_i}_\sigma$ of the optimal transport plan from a reference $\sigma$, and these maps are used as feature vectors. On the tooth dataset the LOT embeddings give high classification accuracy with KNN, linear SVM, and RBF SVM, and the LOT barycenter $\left(\sum_k w_k T^{(k)}\right)_\sharp x_r$ closely tracks the true Wasserstein barycenter, with relative error decreasing as the reference is iterated toward the data barycenter. The authors conclude that the LOT approach performs similarly to pure Wasserstein methods while being much faster.

Load-bearing premise

The fixed Gaussian reference, or the iterated barycenter reference, is close enough to the data that the LOT tangent-space embedding preserves the distances that matter for classification and barycenter approximation.

Editorial extensions

If this is right

Any collection of point clouds of varying sample size can be registered to one reference and compared with Euclidean distances, so order-dependent matching between point sets is avoided.
Linear classifiers, PCA, LDA, and linear barycenters become applicable to measure-valued data without training a neural network.
The LOT barycenter approximates the true Wasserstein barycenter with relative error around 0.002–0.01 after two reference iterations on this dataset, at a small fraction of the computational cost.
Iterating the reference toward class barycenters improves downstream clustering and barycenter sharpness.
Linear-program OT embeddings preserve sharp tooth features better than Sinkhorn embeddings for this data, so the choice of solver matters for feature-rich point clouds.

Reading between the lines

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

Not in the paper: the near-perfect linear-SVM scores likely reflect the small 58-sample, oversampled setting; a larger multi-species dataset would test whether the accuracy persists or whether the reference choice degrades.
Not in the paper: the sharp drop in relative error at iteration 2 in Table 6 suggests the iterated-barycenter reference is doing real geometric work, so one could test whether a few Sinkhorn iterations toward a reference achieve similar gains at lower cost than full LP solves.
Not in the paper: because LOT embeds each point cloud by solving one OT problem per sample, the library should scale best when each point cloud is moderately sized; for very large point clouds the LP solver becomes the bottleneck, making the Sinkhorn path worth benchmarking against data with sharp features.

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

Short version: this is a software paper whose software isn't available for inspection and whose headline claim isn't tested by any experiment in the manuscript. The underlying math is standard and correct, and the iterative-barycenter relative-error results are a genuinely nice demonstration that the method works; but as it stands, the paper asks the reader to take the central empirical claims on faith.

What's new: pyLOT wraps known LOT methods (Wang et al., Kuang and Tabak, Werenski et al.) and adds an iterative reference-refinement loop. No new theory, and the paper doesn't claim any. The library design is sensible: embed via barycentric projection, then use off-the-shelf linear ML. The tooth dataset case study is new. The relative-error table (Table 6) shows the iterated LOT barycenter getting much closer to the true Wasserstein barycenter, with tiny standard deviations after two iterations. That's evidence the iterative reference trick does what it claims, at least on this data.

Where it's soft: - No repository, no commit hash, no data link. For a library paper, that's the artifact itself. The code snippets don't let anyone reproduce the experiments. This alone would force major revision in any venue. - The abstract says LOT has 'similar accuracy with a fraction of the computational cost.' No classifier is ever run on the original point clouds with Wasserstein distance as the metric. The only cost comparison is barycenter runtime (Table 3), and the only accuracy comparison is internal to LOT embeddings. So the comparative accuracy claim is unsupported by the experiments. That's a load-bearing gap, not a minor omission. - Classification numbers come from a single train/test split, with the test set oversampled after splitting. 24 test samples. The authors do say the perfect Linear SVM scores 'may be attributed to the very small dataset,' which is honest, but the tables still can't be read as evidence for the abstract's claim. Repeated cross-validation on the original class distribution would be the minimal fix. - Table 6 lists 'Sample Size' as 11, 5, 9, 33 (the class counts) while the text says M=20 random weight experiments. It's not wrong, but the label is confusing.

The math in Sections 2-3 is standard and correctly presented; the citations point to the right prior work, including the papers that actually introduced these methods. I don't see circularity or data-fitting-as-prediction here. The authors are not overselling their theoretical novelty, just the empirical evidence.

Who it's for: a practitioner deciding whether to use LOT would find the library description useful once the code is released. As it stands, I'd want to see it again after the code and data are out and the classification section is redone. But it deserves referee time — there's a real artifact here, the iterative reference experiments are worth checking, and the application is a good test case.

Recommendation: send to peer review, with the clear expectation of major revision. Don't desk reject; it's not a vacuous submission. But the authors need to release the library and fix the evaluation before I'd take the empirical claims seriously.

Referee Report

5 major / 4 minor

Summary. The paper introduces pyLOT, a Python library for linearized optimal transport (LOT) embeddings, and demonstrates it on a dataset of 58 primate tooth point clouds. Sections 2 and 3 review standard OT and LOT background and then describe modules for embedding, dimensionality reduction, classification, barycenter generation, and iterative reference updates. The paper claims that LOT achieves accuracy comparable to 'pure Wasserstein' approaches at a fraction of the computational cost, and that iterative reference updates improve barycenter approximation.

Significance. If the computational and empirical claims are substantiated, the library could be a practically useful tool: LOT embeddings into L^2(σ) allow standard linear machine learning on measure-valued data, and the paper correctly situates this within the existing theory (Brenier's theorem, compatibility conditions, finite-sample concentration results). The exposition of the embedding and barycenter modules is clear, and the distinction between exact LP-based and Sinkhorn-based transport is well presented. However, the paper's central comparative claim is not directly tested, no code or data is made accessible, and the classification evaluation uses an oversampled 24-sample test set. The significance is therefore conditional on substantial empirical and reproducibility improvements.

major comments (5)

[§1, §3.3.1, Tables 1-2] The abstract's central claim that LOT has 'similar accuracy with a fraction of the computational cost' is not tested: no classifier is run on the original point clouds using Wasserstein distances as the metric, so there is no Wasserstein classification baseline. The only runtime comparison in Table 3 concerns barycenter generation, not classification, and Tables 1-2 report accuracies of LOT-embedded features only. Please add a Wasserstein-distance baseline (e.g., kNN with pairwise W2 on the raw point clouds) and compare runtimes for the same downstream task.
[§3.3.1, Tables 1-2] The reported classification results are based on a test set of 24 samples that was oversampled after the train/test split to equalize class sizes. This changes the evaluation distribution and can inflate accuracy, especially for a dataset with class sizes 5, 9, 11, and 33. The perfect Linear SVM score on the test set is acknowledged by the authors to be attributable to the small dataset, but the tables cannot be read as evidence for the abstract's comparative claim. Please report results on the original imbalanced test set, or use stratified cross-validation, and include per-class metrics with confidence intervals.
[§1, §3.3.1 (Listings 1-5)] As a software paper, the manuscript does not provide a repository URL, commit hash, installation instructions, or a link to the tooth dataset. The code snippets in the listings are not sufficient to run the library or verify the reported numbers, and the dataset is described only by reference to prior work. Please include a DOI or repository URL and a documented release of the data preprocessing pipeline.
[§3.4.2, Table 6] The 'relative error' metric δ in Section 3.4.2 is not a valid relative error: it normalizes the distance between the LOT barycenter and the true barycenter by the maximum pairwise Wasserstein distance among the LOT-embedded point clouds, a quantity that depends on the reference measure and changes across iterations. Consequently the decrease in the mean of δ from iteration 0 to iteration 2 in Table 6 may reflect a change in the denominator rather than an improvement in barycenter approximation. Please report the unnormalized W2 error to the true barycenter, or provide a normalization justified by a guaranteed upper bound.
[§3.1.1, §3.5] The case study uses a single 5000-point Gaussian reference (Section 3.1.1) and the iterative scheme of Section 3.5, but no sensitivity analysis with respect to the reference distribution is reported. Since the paper itself emphasizes that the quality of the LOT embedding depends on the choice of σ (Section 2.2), the robustness of the reported classification accuracies and barycenter errors to the reference is load-bearing for the general usefulness of the library. Please add experiments with different references (e.g., varying variance, number of points, or an alternative reference such as one class's barycenter).

minor comments (4)

[§2.1, Eq. (6) and surrounding notation] The notation for the transport plan constraints is inconsistent: Section 2.1 uses P1_m=α and P^T1_n=β with α∈R^k, β∈R^k, while Eq. (6) uses G1_n=a and G^T1_m=b with a∈Δ^m, b∈Δ^n. Please align the notation and dimensions.
[Table 3] The reported 'True Barycenter' elapsed time of 408,835 seconds (about 4.7 days) seems implausibly large for a 58-sample dataset unless the algorithm is run at very high resolution; if this is a typo (e.g., milliseconds), please correct it and specify the computational environment.
[§2.2] There are several typos, e.g., 'upmost importance' should be 'utmost importance', 'meausre-valued' should be 'measure-valued', and 'interpretted' should be 'interpreted'. Please proofread.
[§3.5] The formula in the iterative-embedding description, '1/|Cc| ∑_{i∈Cc} T^(i) = min ...', is a definition rather than an equation to be minimized; please rewrite it so the barycenter is defined by an argmin expression.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: pyLOT implements established LOT embeddings and barycenter updates; the unsupported accuracy comparison and test oversampling are evidence gaps, not circular steps.

full rationale

pyLOT is a software and case-study paper: the LOT embedding is defined by the standard optimal transport map construction (Eqs. 1, 8-10), and the iterative reference update is the known Wasserstein-barycenter fixed-point iteration from [14] and [33], explicitly credited in Sections 2.2 and 2.3.1. No parameter is fitted to a subset of the data and then renamed as a prediction: the reference measure is either a fixed Gaussian or updated through a stated algorithm, and post-embedding classifiers are evaluated on a held-out test set. The theoretical guarantees cited in Section 2.2 (Brenier's theorem, compatibility isometry from [13], concentration from [9], barycenter convergence from [14]) are published results with stated assumptions; although several are self-citations, under the review rules these are independent support and do not make the present claim circular. The abstract's comparative statement that LOT has 'similar accuracy with a fraction of the computational cost' is not actually tested against a Wasserstein-distance classifier, and Section 3.3.1 oversamples the test set before evaluation, which can inflate the reported accuracies. These are correctness/evidence limitations, not instances of a derivation reducing to its own inputs. No equation in the paper defines a target quantity in terms of the quantity it purports to predict, and no known result is presented as a new derivation without attribution. Therefore the circularity score is 0.

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

The paper does not introduce new theoretical entities. Its central empirical results rest on free parameters such as the Gaussian reference size and the iteration count, and on background mathematical theorems from the optimal transport literature. The main domain assumption is that the tooth point clouds are compatible enough with the reference measure for the LOT embedding to preserve class geometry; this is not verified.

free parameters (4)

Reference measure (5000-point Gaussian) = 5000 points sampled from a Gaussian, later centered and scaled using the data
Chosen by hand; the embedding dimension and the geometry preservation depend on this choice. Used in Section 3.1.1 and Section 3.5.
Sinkhorn regularization lambda = 0.05
Used only for the comparison in Figure 2; the main experiments use the linear programming solver, so this parameter does not affect the primary results.
Iteration count for iterative embeddings = 3
Chosen by the user; it affects the computational cost and the degree of reference refinement in Section 3.5.
Number of random weight draws M = 20
Used for barycenter generation and relative error experiments in Section 3.4.2 and Table 6.

assumptions (4)

standard math Brenier's theorem guarantees existence and uniqueness of optimal transport maps from absolutely continuous reference measures.
Invoked in Theorem 2.1 and used throughout to justify the LOT embedding as a map.
domain assumption The Wasserstein space has a formal Riemannian structure and the LOT embedding behaves like a logarithm map locally near the reference measure.
Section 2.2 uses this heuristic to argue that the choice of reference matters; it is not proven in the paper.
domain assumption The tooth point clouds satisfy (epsilon-)compatibility with the chosen reference measure, so the LOT embedding preserves class structure.
The paper cites [13] for conditions but does not verify them on the tooth data; the classification and barycenter results rely on this assumption.
standard math The iterative barycenter update converges to the true Wasserstein barycenter.
Section 2.2 cites [14, Theorem 2.12] for convergence; no general rates are claimed.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Linearized Optimal Transport pyLOT Library: A Toolkit for Machine Learning on Point Clouds." pith.science (2026). https://pith.science/paper/KBE2ENV3

@misc{pith2026250203439,
  author       = {Pith},
  title        = {Pith review of: Linearized Optimal Transport pyLOT Library: A Toolkit for Machine Learning on Point Clouds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KBE2ENV3}},
  note         = {Machine review of arXiv:2502.03439}
}

read the original abstract

The pyLOT library offers a Python implementation of linearized optimal transport (LOT) techniques and methods to use in downstream tasks. The pipeline embeds probability distributions into a Hilbert space via the Optimal Transport maps from a fixed reference distribution, and this linearization allows downstream tasks to be completed using off the shelf (linear) machine learning algorithms. We provide a case study of performing ML on 3D scans of lemur teeth, where the original questions of classification, clustering, dimension reduction, and data generation reduce to simple linear operations performed on the LOT embedded representations.

Figures

Figures reproduced from arXiv: 2502.03439 by the authors.

**Figure 1.** LOT interpretations. Another interpretation of LOT arises in the finite sample case. Consider a collection of data sets Xi ∈ R ni×d where the number of samples ni may vary. A common problem is defining distances between these data sets that are independent of the order in which the samples are drawn. This commonly arises when: the order of the samples collected is arbitrary, there’s no a priori registration between … view at source ↗

**Figure 2.** Tooth Comparison 3.2 Dimension Reduction The LOTDimensionalityReduction submodule provides tools for reducing the dimensionality of Linearized Optimal Transport (LOT) embeddings derived from discrete measure-valued (or point-cloud-valued) data. It includes methods for balancing imbalanced datasets and applying dimensionality reduction techniques such as Linear Discriminant Analysis (LDA) and Principal Component Anal… view at source ↗

**Figure 3.** PCA VS. LDA Cluster • y_train (np.array): A 1D array of labels corresponding to the training data features. Shape: (number_of_training_samples,). • X_test (np.array): A 2D array representing the test data features. Shape: (number_of_test_samples, number_of_features). • y_test (np.array): A 1D array of labels corresponding to the test data features. Shape: (number_of_test_samples,). The outputs are • best_classifier … view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Fixed Weight Tooth Samples (Tarsius) 3 # fixed weights for the three teeth 4 WEIGHTS_FIXED = np . array ([ 5 [1 ,0 ,0] , [0 ,1 ,0] , [0 ,0 ,1] , 6 [0.9 , 0.1 , 0] , [0.9 , 0 , 0.1] , [0.1 , 0.9 , 0] , 7 [0 , 0.9 , 0.1] , [0.1 , 0 , 0.9] , [0 , 0.1 , 0.9] , 8 [0.6 , 0.2…

**Figure 5.** Figure 5: Barycenter Comparision (Fixed Weight) Wasserstein distance between the LOT-embedded point clouds, which represents the upper bound of potential error in barycenter approximation by considering the diameter of the embedded class. Subsequently, we compute the ratio of th…

**Figure 6.** Figure 6: PCA VS. LDA Cluster Being careful to only generate iterated references from the training data, we repeated the classification problem and saw the cross-validation performance of these models in [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]

**Figure 7.** Figure 7: Barycenter Comparison (Iterative) Similar to the single Gaussian reference case, it is useful to check if the iterative embeddings gain smaller relative error with each iteration. Including the original single Gaussian reference, we show in [PITH_FULL_IMAGE:figures/fu…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

35 extracted references · 32 canonical work pages

[1]

Learning representations and generative models for 3d point clouds, 2018

Panos Achlioptas, Olga Diamanti, Ioannis Mitliagkas, and Leonidas Guibas. Learning representations and generative models for 3d point clouds, 2018

work page 2018
[2]

Akram Aldroubi, Shiying Li, and Gustavo K. Rohde. Partitioning signal classes using transport transforms for data analysis and machine learning. Sampl. Theory Signal Process. Data Anal., 19(6), 2021

work page 2021
[3]

A User’s Guide to Optimal Transport, pages 1–155

Luigi Ambrosio and Nicola Gigli. A User’s Guide to Optimal Transport, pages 1–155. Springer Berlin Heidelberg, Berlin, Heidelberg, 2013

work page 2013
[4]

Wasserstein generative adversarial networks

Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein generative adversarial networks. In Doina Precup and Yee Whye Teh, editors, Proceedings of Machine Learning Research, volume 70, pages 214–223. PMLR, 2017

work page 2017
[5]

Boyer, Yonatan Lipman, Elizabeth M

Doug M. Boyer, Yonatan Lipman, Elizabeth M. St Clair, Jorge Puente, Biren A. Patel, Thomas A. Funkhouser, and Jukka Jernvall. Algorithms to automatically quantify the geometric similarity of anatomical surfaces. Proceedings of the National Academy of Sciences, 108(45):18221–18226, 2011

work page 2011
[6]

Y . Brenier. Polar factorization and monotone rearrangement of vector-valued functions.Comm. Pure Appl. Math., 44(4):375–417, 1991

work page 1991
[7]

Approximating persistent homology for large datasets, 2022

Yueqi Cao and Anthea Monod. Approximating persistent homology for large datasets, 2022

work page 2022
[8]

Kung, Prabhjot Mundi, Joseph O

Yongxin Chen, Filemon Dela Cruz, Romeil Sandhu, Andrew L. Kung, Prabhjot Mundi, Joseph O. Deasy, and Allen Tannenbaum. Pediatric sarcoma data forms a unique cluster measured via the earth mover’s distance. Scientific Reports, 7(1):7035, 2017

work page 2017

Show all 35 references

[9]

Linearized wasserstein dimen- sionality reduction with approximation guarantees, 2023

Alexander Cloninger, Keaton Hamm, Varun Khurana, and Caroline Moosmüller. Linearized wasserstein dimen- sionality reduction with approximation guarantees, 2023

2023
[10]

Rates of estimation of optimal transport maps using plug-in estimators via barycentric projections

Nabarun Deb, Promit Ghosal, and Bodhisattva Sen. Rates of estimation of optimal transport maps using plug-in estimators via barycentric projections. Advances in Neural Information Processing Systems, 34:29736–29753, 2021

2021
[11]

The diffusion geometry of fibre bundles: Horizontal diffusion maps

Michael Gao. The diffusion geometry of fibre bundles: Horizontal diffusion maps. Applied and Computational Harmonic Analysis, 50:147–215, 2021. 21

2021
[12]

On Hölder continuity-in-time of the optimal transport map towards measures along a curve

Nicola Gigli. On Hölder continuity-in-time of the optimal transport map towards measures along a curve. Proceedings of the Edinburgh Mathematical Society, 54(2):401–409, 2011

2011
[13]

Supervised learning of sheared distributions using linearized optimal transport

Varun Khurana, Harish Kannan, Alexander Cloninger, and Caroline Moosmüller. Supervised learning of sheared distributions using linearized optimal transport. Sampling Theory, Signal Processing, and Data Analysis, 21(1), 2023

2023
[14]

Max Kuang and Esteban G. Tabak. Sample-based optimal transport and barycenter problems. Communications on Pure and Applied Mathematics, 72(8):1581–1630, 2019

2019
[15]

Kevrekidis, Joseph O

James Mathews, Maryam Pouryahya, Caroline Moosmüller, Ioannis G. Kevrekidis, Joseph O. Deasy, and Allen Tannenbaum. Molecular phenotyping using networks, diffusion, and topology: soft-tissue sarcoma. Scientific Reports, 9, 2019. Article number: 13982

2019
[16]

R. J. McCann. Polar factorization of maps on Riemannian manifolds. Geometric & Functional Analysis GAFA, 11(3):589–608, 2001

2001
[17]

Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space

Quentin Mérigot, Alex Delalande, and Frédéric Chazal. Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space. In Silvia Chiappa and Roberto Calandra, editors, Proceedings of the Twenty Third International Conference on Artificial Intellig...

2020
[18]

Linear Optimal Transport Embedding: Provable Wasserstein classification for certain rigid transformations and perturbations

Caroline Moosmüller and Alexander Cloninger. Linear Optimal Transport Embedding: Provable Wasserstein classification for certain rigid transformations and perturbations. To appear in: Information and Inference: A Journal of the IMA, 2022

2022
[19]

The geometry of dissipative evolution equations: The porus medium equation

Felix Otto. The geometry of dissipative evolution equations: The porus medium equation. Communications in Partial Differential Equations, 26(1-2):101–174, 2001

2001
[20]

Se Rim Park, Soheil Kolouri, Shinjini Kundu, and Gustavo K. Rohde. The cumulative distribution transform and linear pattern classification. Applied and Computational Harmonic Analysis, 45(3):616 – 641, 2018

2018
[21]

Computational optimal transport

Gabriel Peyré and Marco Cuturi. Computational optimal transport. Foundations and Trends in Machine Learning, 11(5-6):355–607, 2019

2019
[22]

Entropic estimation of optimal transport maps

Aram-Alexandre Pooladian and Jonathan Niles-Weed. Entropic estimation of optimal transport maps. arXiv:2109.12004, 2021

2021 arXiv
[23]

Qi, Hao Su, Kaichun Mo, and Leonidas J

Charles R. Qi, Hao Su, Kaichun Mo, and Leonidas J. Guibas. Pointnet: Deep learning on point sets for 3d classification and segmentation, 2017

2017
[24]

Qi, Li Yi, Hao Su, and Leonidas J

Charles R. Qi, Li Yi, Hao Su, and Leonidas J. Guibas. Pointnet++: Deep hierarchical feature learning on point sets in a metric space, 2017

2017
[25]

On a generalization of wasserstein distance and the beckmann problem to connection graphs, 2024

Sawyer Robertson, Dhruv Kohli, Gal Mishne, and Alexander Cloninger. On a generalization of wasserstein distance and the beckmann problem to connection graphs, 2024

2024
[26]

The earth mover’s distance as a metric for image retrieval

Yossi Rubner, Carlo Tomasi, and Leonidas J Guibas. The earth mover’s distance as a metric for image retrieval. International journal of computer vision, 40(2):99–121, 2000

2000
[27]

Wasserstein propagation for semi- supervised learning

Justin Solomon, Raif Rustamov, Leonidas Guibas, and Adrian Butscher. Wasserstein propagation for semi- supervised learning. In International Conference on Machine Learning, pages 306–314, 2014

2014
[28]

St Clair and Doug M

Elizabeth M. St Clair and Doug M. Boyer. Lower molar shape and size in prosimian and platyrrhine primates. American Journal of Physical Anthropology, 161(2):237–258, 2016

2016
[29]

Austin J. Stromme. Wasserstein barycenters: statistics and optimiztion. S.m. thesis, Massachusetts Institute of Technology, May 2020. Cataloged from the official PDF of thesis. Includes bibliographical references (pages 67-71)

2020
[30]

Optimal Transport: Old and New, volume 338

Cédric Villani. Optimal Transport: Old and New, volume 338. Springer Science & Business Media, 2008

2008
[31]

Ozolek, and Gustavo K

Wei Wang, Dejan Slepˇcev, Saurav Basu, John A. Ozolek, and Gustavo K. Rohde. A linear optimal transportation framework for quantifying and visualizing variations in sets of images. Int J Comput Vis, 101:254–269, 2013

2013
[32]

Sarma, Michael M

Yue Wang, Yongbin Sun, Ziwei Liu, Sanjay E. Sarma, Michael M. Bronstein, and Justin M. Solomon. Dynamic graph cnn for learning on point clouds, 2019

2019
[33]

Matthew Werenski, Brendan Mallery, Shuchin Aeron, and James M. Murphy. Linearized wasserstein barycenters: Synthesis, analysis, representational capacity, and applications, 2024

2024
[34]

Deep sets

Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabás Póczos, Ruslan Salakhutdinov, and Alexander Smola. Deep sets. In Advances in Neural Information Processing Systems (NeurIPS), volume 30, 2017. 22

2017
[35]

Understanding bag-of-words model: a statistical framework

Yin Zhang, Rong Jin, and Zhi-Hua Zhou. Understanding bag-of-words model: a statistical framework. Interna- tional Journal of Machine Learning and Cybernetics, 1(1-4):43–52, 2010. 23

2010

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Learning representations and generative models for 3d point clouds, 2018

Panos Achlioptas, Olga Diamanti, Ioannis Mitliagkas, and Leonidas Guibas. Learning representations and generative models for 3d point clouds, 2018

work page 2018

[2] [2]

Akram Aldroubi, Shiying Li, and Gustavo K. Rohde. Partitioning signal classes using transport transforms for data analysis and machine learning. Sampl. Theory Signal Process. Data Anal., 19(6), 2021

work page 2021

[3] [3]

A User’s Guide to Optimal Transport, pages 1–155

Luigi Ambrosio and Nicola Gigli. A User’s Guide to Optimal Transport, pages 1–155. Springer Berlin Heidelberg, Berlin, Heidelberg, 2013

work page 2013

[4] [4]

Wasserstein generative adversarial networks

Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein generative adversarial networks. In Doina Precup and Yee Whye Teh, editors, Proceedings of Machine Learning Research, volume 70, pages 214–223. PMLR, 2017

work page 2017

[5] [5]

Boyer, Yonatan Lipman, Elizabeth M

Doug M. Boyer, Yonatan Lipman, Elizabeth M. St Clair, Jorge Puente, Biren A. Patel, Thomas A. Funkhouser, and Jukka Jernvall. Algorithms to automatically quantify the geometric similarity of anatomical surfaces. Proceedings of the National Academy of Sciences, 108(45):18221–18226, 2011

work page 2011

[6] [6]

Y . Brenier. Polar factorization and monotone rearrangement of vector-valued functions.Comm. Pure Appl. Math., 44(4):375–417, 1991

work page 1991

[7] [7]

Approximating persistent homology for large datasets, 2022

Yueqi Cao and Anthea Monod. Approximating persistent homology for large datasets, 2022

work page 2022

[8] [8]

Kung, Prabhjot Mundi, Joseph O

Yongxin Chen, Filemon Dela Cruz, Romeil Sandhu, Andrew L. Kung, Prabhjot Mundi, Joseph O. Deasy, and Allen Tannenbaum. Pediatric sarcoma data forms a unique cluster measured via the earth mover’s distance. Scientific Reports, 7(1):7035, 2017

work page 2017