MANTRA: Manifold Triangulations Assembly

Getting the Dataset

The raw MANTRA dataset consisting of the $2$ and $3$ manifolds with up to $10$ vertices is provided here. For machine learning applications and research, we provide a custom PyTorch Geometric dataset in the form of a python package.

For installations via pip, run

The raw datasets, consisting of the 2 and 3 manifolds with up to 10 vertices, can be manually downloaded here. A pytorch geometric wrapper for the dataset is installable via the following command.

pip install mantra-dataset

After installation the dataset can be used with the follwing snippet.

from mantra.datasets import ManifoldTriangulations

dataset = ManifoldTriangulations(root="./data", manifold="2", version="latest")

Folder Structure

Data Format

This section is mostly information-oriented and provides a brief overview of the data format, followed by a short example.

Each dataset consists of a list of triangulations, with each triangulation having the following attributes:

• id (required, str): This attribute refers to the original ID of the triangulation as used by the creator of the dataset (see below). This facilitates comparisons to the original dataset if necessary.

• triangulation (required, list of list of int): A doubly-nested list of the top-level simplices of the triangulation.

• n_vertices (required, int): The number of vertices in the triangulation. This is not the number of simplices.

• name (required, str): A canonical name of the triangulation, such as S^2 for the two-dimensional sphere. If no canonical name exists, we store an empty string.

• betti_numbers (required, list of int): A list of the Betti numbers of the triangulation, computed using $Z$ coefficients. This implies that torsion coefficients are stored in another attribute.

• torsion_coefficients (required, list of str): A list of the torsion coefficients of the triangulation. An empty string "" indicates that no torsion coefficients are available in that dimension. Otherwise, the original spelling of torsion coefficients is retained, so a valid entry might be "Z_2".

• genus (optional, int): For 2-manifolds, contains the genus of the triangulation.

• orientable (optional, bool): Specifies whether the triangulation is orientable or not.

Example

[
  {
    "id": "manifold_2_4_1",
    "triangulation": [
      [1,2,3],
      [1,2,4],
      [1,3,4],
      [2,3,4]
    ],
    "dimension": 2,
    "n_vertices": 4,
    "betti_numbers": [
      1,
      0,
      1
    ],
    "torsion_coefficients": [
      "",
      "",
      ""
    ],
    "name": "S^2",
    "genus": 0,
    "orientable": true
  },
  {
    "id": "manifold_2_5_1",
    "triangulation": [
      [1,2,3],
      [1,2,4],
      [1,3,5],
      [1,4,5],
      [2,3,4],
      [3,4,5]
    ],
    "dimension": 2,
    "n_vertices": 5,
    "betti_numbers": [
      1,
      0,
      1
    ],
    "torsion_coefficients": [
      "",
      "",
      ""
    ],
    "name": "S^2",
    "genus": 0,
    "orientable": true
  }
]

Design Decisions

This section is understanding-oriented and provides additional justifications for our data format.

The datasets are converted from their original (mixed) lexicographical format. A triangulation in lexicographical format could look like this:

manifold_lex_d2_n6_#1=[[1,2,3],[1,2,4],[1,3,4],[2,3,5],[2,4,5],[3,4,6],
  [3,5,6],[4,5,6]]

A triangulation in mixed lexicographical format could look like this:

manifold_2_6_1=[[1,2,3],[1,2,4],[1,3,5],[1,4,6],
  [1,5,6],[2,3,4],[3,4,5],[4,5,6]]

This format is hard to parse. Moreover, any additional information about the triangulations, including information about homology groups or orientability, for instance, requires additional files.

We thus decided to use a format that permits us to keep everything in one place, including any additional attributes for a specific triangulation. A desirable data format needs to satisfy the following properties:

1. It should be easy to parse and modify, ideally in a number of programming languages.

2. It should be human-readable and diff-able in order to permit simplified comparisons.

3. It should scale reasonably well to larger triangulations.

After some considerations, we decided to opt for gzip-compressed JSON files. JSON is well-specified and supported in virtually all major programming languages out of the box. While the compressed file is not human-readable on its own, the uncompressed version can easily be used for additional data analysis tasks. This also greatly simplifies maintenance operations on the dataset. While it can be argued that there are formats that scale even better, they are not well-applicable to our use case since each triangulation typically consists of different numbers of top-level simplices. This rules out column-based formats like Parquet.

We are open to revisiting this decision in the future.

As for the storage of the data as such, we decided to keep only the top-level simplices (as is done in the original format) since this substantially saves disk space. The drawback is that the client has to supply the remainder of the triangulation. Given that the triangulations in our dataset are not too large, we deem this to be an acceptable compromise. Moreover, data structures such as simplex trees can be used to further improve scalability if necessary.

The decision to keep only top-level simplices is final.

Finally, our data format includes, whenever possible and available, additional information about a triangulation, including the Betti numbers and a name, i.e., a canonical description, of the topological space described by the triangulation. We opted to minimize any inconvenience that would arise from having to perform additional parsing operations.

Please use the following citation for our work:

@misc{ballester2024mantramanifoldtriangulationsassemblage,
      title={ {MANTRA}: {T}he {M}anifold {T}riangulations {A}ssemblage}, 
      author={Rub{\'e}n Ballester and Ernst R{\"o}ell and Daniel Bin Schmid and Mathieu Alain and Sergio Escalera and Carles Casacuberta and Bastian Rieck},
      year={2024},
      eprint={2410.02392},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/2410.02392}, 
}

Acknowledgments

This work is dedicated to Frank H. Lutz, who passed away unexpectedly on November 10, 2023. May his memory be a blessing.