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REVIEW 5 major objections 5 minor 86 references

EuLearn: A 3D database for learning Euler characteristics

T0 review · 5 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read EuLearn, a new dataset of 3,300 watertight surfaces with uniformly distributed genus from 0 to 10, shows that feeding adjacency information to PointNet and attention layers lifts genus-classification accuracy from at most 0.63 to about…

desk verdict A genuinely useful uniform-genus dataset, but the construction as written cannot produce genus 0 and the labels are never checked; fixable but not ready to cite. read the letter →

arxiv 2505.13539 v1 pith:YURX4M2N submitted 2025-05-18 cs.CG cs.CVcs.LGmath.DGmath.GT

Rodrigo Fritz , Pablo Suárez-Serrato , Victor Mijangos , Anayanzi D. Martinez-Hernandez , Eduardo Ivan Velazquez Richards This is my paper · ORCID
classification cs.CGcs.CVcs.LGmath.DGmath.GT MSC 57M2568T0765D18
keywords Eulercharacteristicgenusclassificationsingularknots3DsurfacedatasetgraphsamplingPointNettransformerattentionMarchingCubes
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces EuLearn, a dataset of 3,300 watertight, randomly embedded surfaces in 3D space with exactly 300 examples for each genus from 0 to 10, so that topological type is uniformly distributed for the first time in a surface dataset. The surfaces are built by thickening self-intersecting Fourier knots (Lissajous and Fibonacci families) into tubular neighborhoods and discretizing them with Marching Cubes; the number of self-intersections determines the genus. The authors report that standard 3D neural architectures—PointNet, PointNet++, FNO, DGCNN, and a transformer—classify genus poorly on this data, reaching at most 0.63 accuracy, whereas their adjacency-preserving graph-sampling method combined with adjacency-informed PointNet and attention layers reaches about 0.79–0.81 accuracy. The point is to expose and help close a blind spot: current deep learning tools do not reliably recognize topological structure, and a controlled, topologically balanced dataset makes that measurable.

What carries the argument

The load-bearing construction is a pipeline: parameterize a singular knot (Lissajous or Fibonacci), detect its self-intersections by minimizing pair distances with differential evolution, thicken the curve into a tubular neighborhood whose radius is kept below the surface reach so distinct regions do not merge, discretize the signed-distance scalar field on a $100^3$ grid, and extract the surface with Marching Cubes. The identity that carries the genus control is the Euler characteristic $\chi = V - E + F = 2 - 2g$, with $n$ self-intersections producing genus $n+1$. For learning, the central device is graph sampling (Algorithm 2): starting from the mesh triangulation, it iteratively removes a random subset of neighbors of each traversed node and reconnects the retained node to the removed nodes' neighbors, so a 3,000-point sample inherits the adjacency structure of the original mesh. The adapted layers, GS PointNet and GS attention, replace Euclidean-neighbor aggregation with aggregation over these adjacency-defined neighbors, and the attention variant drops the learned linear projections, using a masked softmax over raw point dot products.

What would settle it

Compute $V - E + F$ on the 3,000-point sampled graphs of EuLearn and compare with the mesh genus; any mismatch falsifies the coarsening claim. Also check whether 300 distinct genus-0 meshes exist in the released files and whether their construction is documented, since the 'n self-intersections → genus n+1' rule allows no genus-0 case.

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Extended reading notes

Core claim

EuLearn is claimed to be the first 3D surface dataset whose topological types are both well-defined and uniformly distributed: by construction, every instance is a single watertight, compact, orientable triangulated surface, and the genus is uniform across the dataset (Theorem 3). Each surface is the boundary of a tubular neighborhood of a closed, self-intersecting curve, with each self-intersection adding one handle; the genus is then computed from the Euler characteristic $\chi = V - E + F = 2 - 2g$. The paper further claims that when the point cloud is sampled by a graph-coarsening procedure that preserves adjacency (Algorithm 2), and when PointNet and attention layers aggregate over mesh neighbors rather than over a Euclidean ball, genus classification accuracy rises from 0.10–0.63 for vanilla baselines to 0.79–0.81, with attention reaching 0.81. This is presented as evidence that topological information in the adjacency structure is what current architectures lack.

Load-bearing premise

The pipeline preserves the genus of every instance: thickening the curve, discretizing it on a 100³ grid, and graph-sampling it down to 3,000 points never changes the Euler characteristic, and the genus-0 class is actually produced by the stated construction.

Editorial extensions

If this is right

  • If the construction works as claimed, researchers gain a benchmark where genus is a controlled label, so failures in topology recognition become attributable to the model rather than to dataset imbalance.
  • The adjacency-preserving sampling method is a standalone contribution: any point-cloud network can consume EuLearn's 3,000-point graphs, and the reported gains suggest that retaining mesh edges matters more than the specific architecture.
  • The dataset's scalar-field files (100³ grids per surface) enable testing of FNO-style and other field-based models on the same topology classification task.
  • The reported accuracy of about 0.8 on 11 balanced classes is a concrete target: any future model can be compared against it as a baseline on EuLearn.

Reading between the lines

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

  • If genus is truly preserved by the coarsening, the same pipeline could generate larger datasets with arbitrary genus ranges (including $g \geq 11$) by adding more Fourier-knot parameter families; the paper's data-generation methodology is designed to be reused.
  • The strong role of adjacency suggests a natural experiment the paper does not run: feeding the same 3,000-point samples to a model that uses only geometric features (e.g., persistent-homology descriptors) should change accuracy in a way that isolates what topological information adjacency actually encodes.
  • The stated rule 'n self-intersections result in a surface with genus n+1' implies a minimum genus of 1; if genus-0 surfaces come from a separate construction (e.g., non-singular knots), the paper's description leaves that unspecified, and a reader should check the released dataset metadata for how the 300 genus-0 instances were generated.
  • The sensitivity analysis finding that random seed is the most influential hyperparameter suggests the reported 0.79–0.81 accuracies may have a few-point spread across seeds; re-running with a fixed seed list would give a tighter performance estimate.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper introduces EuLearn, a synthetic dataset of 3,300 embedded triangulated surfaces, with 300 instances claimed for each genus from 0 to 10. Each surface is built by thickening a closed, possibly self-intersecting curve, discretizing the resulting scalar field on a 100^3 grid, and extracting a mesh via Marching Cubes; each instance is shipped as a mesh, a smoothed mesh, and a scalar field. The authors also propose a graph-based point sampling scheme and two adjacency-informed architectures (GS PointNet and GS Attention), and report that these reach about 0.79-0.81 accuracy on an 11-class genus classification task, compared with 0.10-0.63 for vanilla baselines. The central theoretical claim is Theorem 3, that the genus distribution in EuLearn is uniform, with a proof stated as 'by construction.'

Significance. If the uniform-genus guarantee were actually verified, EuLearn would fill a real gap: existing 3D datasets are topologically unbalanced or have ill-defined genus, and a controlled benchmark for topological classification could be useful for geometric deep learning. The paper also contributes a public dataset, a reproducible generation workflow, and a non-Euclidean sampling idea that demonstrably changes model behavior. However, the significance is currently conditional. The proof of the main theorem is not a proof, and the construction as stated cannot produce genus-0 surfaces. The experimental comparison is also presented without repeated-run statistics, and the adjacency matrix input makes the classification task partly a graph-topology task rather than a raw point-cloud task. These issues are fixable within the manuscript's scope, but they need to be addressed before the dataset guarantee can be accepted.

major comments (5)
  1. [§2.1, Appendix B] The stated construction rule, 'n self-intersections result in a surface with genus n+1' (Section 2.1), is inconsistent with the presence of 300 genus-0 instances. A thickened simple closed curve has a torus boundary, hence genus 1, so the family of thickened closed curves described in the paper cannot produce genus-0 surfaces. Appendix B still lists 15 singular knot types per genus, including genus 0, but no construction for the sphere-like instances is given. Please describe explicitly how the genus-0 instances are generated, or revise the rule linking self-intersections to genus; this is load-bearing for the uniform-distribution claim.
  2. [Appendix A.3, Appendix J] Theorem 3 is not proved. The proof says only 'By construction' and states that the Euler characteristic 'we may verify,' but no actual Euler characteristic counts, per-instance audit, or verification script are reported anywhere in the manuscript. The datasheet (Appendix J, item 9) asserts 'None that we are aware of' without an audit, and item 17 admits that some parameterizations were discarded because the genus was altered when the radius exceeded the reach. Please provide a complete per-instance audit of all 3,300 meshes, reporting chi = 2 - 2g, and release the verification code; otherwise the uniform genus labels are asserted rather than established.
  3. [§3.1, Algorithm 2] The graph sampling procedure is claimed to 'preserve the topological structure of the surface' and is described as 'a fine grained coarsening of the graph itself,' but no check is reported that the 3,000-vertex sampled graph has the same Euler characteristic as the original mesh. Since all model inputs are sampled graphs, any topology-changing sample would invalidate the classification labels. Please report Euler characteristic or genus before and after sampling across the dataset, and add a safeguard that rejects samples that change the genus.
  4. [§3.3, Table 1, Appendix I.2] All reported accuracies, precisions, recalls, and F1 scores come from a single run, with no standard deviations or confidence intervals. The sensitivity analysis in Appendix I.2 reports that the random seed is the most influential hyperparameter (relative importance 0.45), so the observed gaps between models could be sensitive to initialization. Please report repeated-seed means and standard deviations for all methods. In addition, the per-class tables (Tables 6 and 7) show F1 scores as low as 0.24 (genus 8, GS Attention) and recall as low as 0.30 (genus 5, GS PointNet), so the average accuracy claim overstates the degree to which the models 'recognize' all genera.
  5. [§3.2, §D.4, §D.6] The comparison between vanilla models and the GS models conflates two different inputs: the GS models receive the mesh adjacency matrix, which directly encodes the combinatorial structure of the triangulation (edge count, and with coordinates, much of the face structure). Since the target is the topological genus, a large part of the performance gain may come from the input containing the answer in combinatorial form rather than from the models learning geometry. This does not invalidate the experiments, but the paper should reframe the task as a graph-topology-informed classification benchmark and should include a control, for example k-NN graphs or permuted adjacency matrices, to quantify how much of the gain is due to genuine mesh connectivity rather than the raw embedding coordinates.
minor comments (5)
  1. [Appendix B.4 vs Appendix J item 17] The text says the 20 sinusoidal variations 'by construction, have the same isotopy type,' while the datasheet says some variations were discarded 'because the genus was altered due to the contact of the surface with itself when the radius exceeded the reach.' Please reconcile these two statements.
  2. [Appendix A.1, A.2] The proofs of Theorems 1 and 2 are described as 'By computation,' but the computations, the version of Trimesh, and the exact counting procedures are not fully specified; please include a reproducibility script or more detail, since these theorems are used to motivate the dataset.
  3. [Throughout] There are numerous typos and minor errors, including 'hi+i' and 'comutative' in the FNO and edge-convolution definitions, 'PoitNet++' in Appendix D.3, 'PointNett++' in Section 3.1, and 'presernts' in Appendix I.2; a careful proofreading pass is needed.
  4. [Appendix D, Definitions] The definitions are numbered as Definition 3.1 through 3.7 even though they appear across Sections 2 and 3 and in the appendices; please renumber them consistently with the section in which they appear.
  5. [Appendix J, item 33] The datasheet says the dataset 'will be made available upon submission of the paper,' while the abstract and datasheet item 27 give a HuggingFace repository URL; please clarify the current release status.

Circularity Check

1 steps flagged · score 2.0 of 10

Only the dataset-uniformity theorem is self-definitional; the learning experiments are not circular.

  1. self definitional [Section A.3 (Theorem 3); Appendix J items 16-17]
    "Theorem 3. The distribution of topological types of surfaces in the EuLearn dataset is uniform. Proof. By construction. ... we may verify that, by the detailed construction explained in this paper, the Euler characteristic is uniformly distributed. ... Once we knew the number of self-intersections, then we manually add the corresponding parameters to a list ordered by genus."

    The uniform distribution asserted in Theorem 3 is the intended output of the curation process, not a consequence derived from the random-knot construction. Appendix J states that parameters were manually added to a list ordered by genus and that the final subset was manually selected, so the uniform class counts (300 per genus) are imposed by the authors' balancing decision. The 'proof by construction' therefore reduces to a definitional property of how the dataset was assembled; no independent Euler-characteristic computation is reported to verify the labels.

full rationale

Apart from the self-definitional Theorem 3, the paper's derivation chain is self-contained. The EuLearn meshes are generated independently of the trained models; the train/test split is fixed and no test-set fitting is described, so the accuracy gains are not statistically forced predictions. The GS PointNet/GS Attention models receive the adjacency matrix as an explicit input, and the paper openly attributes the improvement to that structural information; although the adjacency matrix of a closed triangulated surface contains combinatorial data from which the Euler characteristic could be computed, the models do not compute V-E+F directly and the observed 0.79-0.81 accuracy is an empirical result rather than a tautology. There are no load-bearing self-citations or imported uniqueness theorems. The stated rule 'n self-intersections result in a surface with genus n+1' would exclude genus 0 (requiring n=-1), so the 300 genus-0 instances lack a described construction, and the datasheet's 'None that we are aware of' error claim is unverified; these are correctness and verification gaps that could invalidate the labels, but they are not circularity.

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

The central dataset claim rests on standard surface classification (chi = 2 - 2g), a stated but unproved relation between curve self-intersections and the genus of a thickened tube, and an unverified assumption that Marching Cubes and graph sampling preserve topology. No constants are fitted to the classification labels; the free parameters listed are construction choices (thickening radius, grid resolution, sample size, augmentation frequencies). No invented entities are introduced.

free parameters (4)
  • thickening radius delta
    Per-surface parameter chosen so the tubular neighborhood stays below the reach; if too large, surface regions merge and the genus changes. Selection rule is described qualitatively, not fitted to any training label.
  • voxel grid resolution = 100^3
    Fixed discretization for the scalar field; coarse enough that thin tubes could be lost, so mesh face sizes are constrained. A construction choice, not fitted.
  • graph sampling size = 3000
    Number of sampled vertices per mesh, chosen to fit GPU memory and batching; not fitted to classification accuracy.
  • sinusoidal variation frequencies = 1 to 20
    Data augmentation by varying tube radius; the 20 frequencies per seed knot are chosen by hand, not optimized.
assumptions (4)
  • domain assumption Thickening a singular closed curve with n transverse self-intersections produces a surface of genus n+1.
    Used in Section 2 to derive the genus label from the knot's self-intersection count; stated without proof.
  • standard math A connected orientable triangulated surface is classified by its Euler characteristic chi = 2 - 2g, so chi uniquely determines the genus.
    Invoked in Theorem 3 and Appendix A to compute genus from mesh counts.
  • domain assumption Marching Cubes on a signed distance field produces a watertight, single-component, manifold mesh with the topology of the zero level set.
    Central to Theorem 3 and to the claim that each EuLearn object is a single well-defined mesh; discretization artifacts are controlled by face-size constraints but not formally proven.
  • domain assumption The listed FAUST, ShapeNet, Thingi10K, ABC, and SimJEB files used in the computations are processed correctly by Trimesh for Euler characteristic and watertightness checks.
    Theorems 1 and 2 are 'by computation' and depend on the mesh-processing library returning correct values.

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Cite this review

Pith. "Pith review of EuLearn: A 3D database for learning Euler characteristics." pith.science (2026). https://pith.science/paper/YURX4M2N

@misc{pith2026250513539,
  author       = {Pith},
  title        = {Pith review of: EuLearn: A 3D database for learning Euler characteristics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YURX4M2N}},
  note         = {Machine review of arXiv:2505.13539}
}
read the original abstract

We present EuLearn, the first surface datasets equitably representing a diversity of topological types. We designed our embedded surfaces of uniformly varying genera relying on random knots, thus allowing our surfaces to knot with themselves. EuLearn contributes new topological datasets of meshes, point clouds, and scalar fields in 3D. We aim to facilitate the training of machine learning systems that can discern topological features. We experimented with specific emblematic 3D neural network architectures, finding that their vanilla implementations perform poorly on genus classification. To enhance performance, we developed a novel, non-Euclidean, statistical sampling method adapted to graph and manifold data. We also introduce adjacency-informed adaptations of PointNet and Transformer architectures that rely on our non-Euclidean sampling strategy. Our results demonstrate that incorporating topological information into deep learning workflows significantly improves performance on these otherwise challenging EuLearn datasets.

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    For what purpose was the dataset created?To be able to train machine learning and deep learning systems to understand topological variation in a uniformly controlled way

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    Each object has a mesh (.stl), a scalar field (.txt), and a smoothed mesh (.stl)

    What do the instances that comprise the dataset represent (for example, documents, photos, people, countries)?The instances are embedded and possibly knotted 2-dimensional surfaces in 3-space. Each object has a mesh (.stl), a scalar field (.txt), and a smoothed mesh (.stl)

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    The total number of files is 9 900, as each surface has 3 associated files

    How many instances are there in total (of each type, if appropriate)?We have a total of 3 300 instances, with 300 surfaces for each genus, from 0 to 10. The total number of files is 9 900, as each surface has 3 associated files

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    Does the dataset contain all possible instances or is it a sample (not necessarily random) of instances from a larger set?The dataset is a random sample, whose construction builds upon parameterizations of random knots. EULEARN: A 3D DATABASE FOR LEARNING EULER CHARACTERISTICS 33

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    What data does each instance consist of?Three files, two in .stl format and another in .txt format

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    Is there a label or target associated with each instance?The target associated with each instance is its topological genus. The name labeling indicates the surface genus, knot type, its six associated parameters (frequencies and phases), frequency, constant and amplitude for t...

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    Are relationships between individual instances made explicit (for example, users’ movie ratings, social network links)?Yes, multiple instances are generated by a single random knot type, thus these are related by belonging to the same isotopy type of embedding

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    This method randomly selects part of the data to train and to evaluate a model, saving these sets in a local directory, so the model can call the necessary batches one at a time

    Are there recommended data splits (for example, training, development/validation, testing)? Due to local memory restrictions on our GPU units, for our reported experiments we adapted a data split method that saves training and testing subsets to our local hard disk for batch p...

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    Please do not hesitate to contact us if any appear to be there

    Are there any errors, sources of noise, or redundancies in the dataset?None that we are aware of. Please do not hesitate to contact us if any appear to be there

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    Is the dataset self-contained, or does it link to or otherwise rely on external resources (for example, websites, tweets, other datasets)?The EuLearn dataset is self-contained

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    12.Does the dataset identify any subpopulations (for example, by age, gender)?There are no population identifications

    Does the dataset contain data that might be considered confidential (for example, data that is protected by legal privilege or by doctor–patient confidentiality, data that includes the content of individuals’ non-public communications)?None of the data is confidential. 12.Does...

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    Is it possible to identify individuals (that is, one or more natural persons), either directly or indirectly (that is, in combination with other data) from the dataset?This dataset does not include individual data

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    Does the dataset contain data that might be considered sensitive in any way (for example, data that reveals race or ethnic origins, sexual orientations, religious beliefs, political opinions or union memberships, or locations; financial or health data; biometric or genetic dat...

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    How was the data associated with each instance acquired? Was the data directly observable (for example, raw text, movie ratings), reported by subjects (for example, survey responses), or indirectly inferred/ derived from other data (for example, part-of-speech tags, model-base...

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    For this purpose, a code script was designed to compute the number of self-intersections for each triple of frequencies with one of those phases

    What mechanisms or procedures were used to collect the data (for example, hardware apparatuses or sensors, manual human curation, software programs, software APIs)? The parameterizations needed to generate each of the instances were drawn from a pool limited to frequencies up ...

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    If the dataset is a sample from a larger set, what was the sampling strategy (for example, deterministic, probabilistic with specific sampling probabilities)? The sampling strategy was deterministic, since we manually selected the most complete subset of instances from a large...

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    Who was involved in the data collection process (for example, students, crowdworkers, contractors) and how were they compensated (for example, how much were crowdworkers paid)?Faculty, postdocs and graduate students, receiving a salary or a fellowship

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    The data generation procedure for the final version spanned two months, November-December 2024

    Over what time frame was the data collected?The research into the data design process started in fall 2021. The data generation procedure for the final version spanned two months, November-December 2024

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    We relied on HPC supercomputing resources, obtained through a national grant application

    Did you collect the data from the individuals in question directly, or obtain it via third parties or other sources (for example, websites)?No individuals were involved. We relied on HPC supercomputing resources, obtained through a national grant application

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    Were any ethical review processes conducted (for example, by an institutional review board)? None were needed, none were conducted

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    Has an analysis of the potential impact of the dataset and its use on data subjects (for example, a data protection impact analysis) been conducted?It has not, as this is a synthetic dataset and no data protection analysis is required

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    Was any preprocessing/cleaning/labeling of the data done (for example, discretization or bucketing, tokenization, part-of-speech tagging, SIFT feature extraction, removal of instances, processing of missing values)?Yes, a scalar field was created in order to construct the fina...

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    The available data may be considered the raw data itself

    Was the “raw” data saved in addition to the preprocessed/cleaned/ labeled data (for example, to support unanticipated future uses)?While we did not save the curve data that the database relies on as a 3D object, we do have a list of the parameters used to create the associated...

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    Namely, an implementation of Marching Cubes, Blender for smoothing, and CUDA for computing the scalar fields

    Is the software that was used to preprocess/clean/label the data available?Yes. Namely, an implementation of Marching Cubes, Blender for smoothing, and CUDA for computing the scalar fields. 26.Has the dataset been used for any tasks already?Not yet

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    Is there a repository that links to any or all papers or systems that use the dataset? https://huggingface.co/datasets/appliedgeometry/EuLearn

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    We mentioned specific engineering and scientific applications where these techniques are relevant in the main body

    What (other) tasks could the dataset be used for?It could be of crucial assitance in model pretraining, so that a proposed architecture is guaranteed to have the ability to distinguish varying topological types in 3D data. We mentioned specific engineering and scientific appli...

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    As long as these remain in use, there should be no foreseeable impact on future users

    Is there anything about the composition of the dataset or the way it was collected and preprocessed/cleaned/labeled that might impact future uses?The files are in .stl and .txt EULEARN: A 3D DATABASE FOR LEARNING EULER CHARACTERISTICS 35 formats. As long as these remain in use...

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    Will the dataset be distributed to third parties outside of the entity (for example, company, institution, organization) on behalf of which the dataset was created?It will be made openly and publicly available

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    How will the dataset be distributed (for example, tarball on website, API, GitHub)?Through HuggingFace and in an institutional repository

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    When will the dataset be distributed?The dataset will be made available upon submission of the paper

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    Will the dataset be distributed under a copyright or other intellectual property (IP) license, and/or under applicable terms of use (ToU)?With a MIT license

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    Have any third parties imposed IP-based or other restrictions on the data associated with the instances?None

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    37.Who will be supporting/hosting/maintaining the dataset?The authors

    Do any export controls or other regulatory restrictions apply to the dataset or to individual instances?No. 37.Who will be supporting/hosting/maintaining the dataset?The authors

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    39.Is there an erratum?Not yet

    How can the owner/curator/ manager of the dataset be contacted?Via email, found in the associated publication. 39.Is there an erratum?Not yet

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    Will the dataset be updated (for example, to correct labeling errors, add new instances, delete instances)?Perhaps after the submission revision process

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    For those interested in augmenting the available instance in the genus range{0,

    If others want to extend/augment/build on/contribute to the dataset, is there a mechanism for them to do so?This includes two possibilities. For those interested in augmenting the available instance in the genus range{0, . . . ,10}, they would have to construct curves that use...

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    Association for Computational Linguistics

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.