GUDHI Python modules documentation

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Data structures for cell complexes

Cubical complexes

Cubical complex representation

The cubical complex represents a grid as a cell complex with cells of all dimensions.

Author

Pawel Dlotko

Since

GUDHI 2.0.0

License

MIT

Simplicial complexes

Simplex tree

Simplex tree representation

The simplex tree is an efficient and flexible data structure for representing general (filtered) simplicial complexes.

The data structure is described in [5]

Author

Clément Maria

Since

GUDHI 2.0.0

License

MIT

Filtrations and reconstructions

Alpha complex

Alpha complex representation

Alpha complex is a simplicial complex constructed from the finite cells of a Delaunay Triangulation. It has the same persistent homology as the Čech complex and is significantly smaller.

Author

Vincent Rouvreau

Since

GUDHI 2.0.0

License

MIT (GPL v3)

Requires

Eigen \(\geq\) 3.1.0 and CGAL \(\geq\) 4.11.0

Rips complex

_images/rips_complex_representation.png

The Vietoris-Rips complex is a simplicial complex built as the clique-complex of a proximity graph.

We also provide sparse approximations, to speed-up the computation of persistent homology, and weighted versions, which are more robust to outliers.

Authors

Clément Maria, Pawel Dlotko, Vincent Rouvreau, Marc Glisse, Yuichi Ike

Since

GUDHI 2.0.0

License

MIT

Witness complex

Witness complex representation

Witness complex \(Wit(W,L)\) is a simplicial complex defined on two sets of points in \(\mathbb{R}^D\).

The data structure is described in [5].

Author

Siargey Kachanovich

Since

GUDHI 2.0.0

License

MIT (GPL v3 for Euclidean versions only)

Requires

Eigen \(\geq\) 3.1.0 and CGAL \(\geq\) 4.11.0 for Euclidean versions only

Cover complexes

Graph Induced Complex of a point cloud.

Nerves and Graph Induced Complexes are cover complexes, i.e. simplicial complexes that provably contain topological information about the input data. They can be computed with a cover of the data, that comes i.e. from the preimage of a family of intervals covering the image of a scalar-valued function defined on the data.

Author

Mathieu Carrière

Since

GUDHI 2.3.0

License

MIT (GPL v3)

Requires

CGAL \(\geq\) 4.11.0

Tangential complex

_images/tc_examples.png

A Tangential Delaunay complex is a simplicial complex designed to reconstruct a \(k\)-dimensional manifold embedded in \(d\)-dimensional Euclidean space. The input is a point sample coming from an unknown manifold. The running time depends only linearly on the extrinsic dimension \(d\) and exponentially on the intrinsic dimension \(k\).

Author

Clément Jamin

Since

GUDHI 2.0.0

License

MIT (GPL v3)

Requires

Eigen \(\geq\) 3.1.0 and CGAL \(\geq\) 4.11.0

Topological descriptors computation

Persistent cohomology

_images/3DTorus_poch.png

Rips Persistent Cohomology on a 3D Torus

The theory of homology consists in attaching to a topological space a sequence of (homology) groups, capturing global topological features like connected components, holes, cavities, etc. Persistent homology studies the evolution – birth, life and death – of these features when the topological space is changing.

Computation of persistent cohomology using the algorithm of [16] and [19] and the Compressed Annotation Matrix implementation of [3].

Author

Clément Maria

Since

GUDHI 2.0.0

License

MIT

Please refer to each data structure that contains persistence feature for reference:

Topological descriptors tools

Bottleneck distance

_images/perturb_pd.png

Bottleneck distance is the length of the longest edge

Bottleneck distance measures the similarity between two persistence diagrams. It’s the shortest distance b for which there exists a perfect matching between the points of the two diagrams (+ all the diagonal points) such that any couple of matched points are at distance at most b, where the distance between points is the sup norm in \(\mathbb{R}^2\).

Author

François Godi

Since

GUDHI 2.0.0

License

MIT (GPL v3)

Requires

CGAL \(\geq\) 4.11.0

Wasserstein distance

_images/perturb_pd.png

The q-Wasserstein distance measures the similarity between two persistence diagrams using the sum of all edges lengths (instead of the maximum). It allows to define sophisticated objects such as barycenters of a family of persistence diagrams.

Author

Theo Lacombe, Marc Glisse

Since

GUDHI 3.1.0

License

MIT, BSD-3-Clause

Persistence representations

_images/sklearn-tda.png

Vectorizations, distances and kernels that work on persistence diagrams, compatible with scikit-learn.

Author

Mathieu Carrière, Martin Royer

Since

GUDHI 3.1.0

License

MIT

Requires

Scikit-learn

Persistence graphical tools

_images/graphical_tools_representation.png

These graphical tools comes on top of persistence results and allows the user to display easily persistence barcode, diagram or density.

Note that these functions return the matplotlib axis, allowing for further modifications (title, aspect, etc.)

Author

Vincent Rouvreau, Theo Lacombe

Since

GUDHI 2.0.0

License

MIT

Requires

Matplotlib

Point cloud utilities

\((x_1, x_2, \ldots, x_d)\)
\((y_1, y_2, \ldots, y_d)\)

Utilities to process point clouds: read from file, subsample, find neighbors, embed time series in higher dimension, estimate a density, etc.

Authors

Vincent Rouvreau, Marc Glisse, Masatoshi Takenouchi

Since

GUDHI 2.0.0

License

MIT (GPL v3, BSD-3-Clause, Apache-2.0)

Clustering

_images/spiral-color.png

Clustering tools.

Author

Marc Glisse

Since

GUDHI 3.3.0

License

MIT

Datasets generators

_images/sphere_3d.png

Datasets generators (points).

Authors

Hind Montassif

Since

GUDHI 3.5.0

License

MIT (LGPL v3)

Requires

CGAL