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To get rid of WARNING: document isn't included in any toctree - -Rips complex user manual -========================= -Definition ----------- - -==================================================================== ================================ ====================== -:Authors: Clément Maria, Pawel Dlotko, Vincent Rouvreau, Marc Glisse :Introduced in: GUDHI 2.0.0 :Copyright: GPL v3 -==================================================================== ================================ ====================== - -+-------------------------------------------+----------------------------------------------------------------------+ -| :doc:`rips_complex_user` | :doc:`rips_complex_ref` | -+-------------------------------------------+----------------------------------------------------------------------+ - -The `Rips complex <https://en.wikipedia.org/wiki/Vietoris%E2%80%93Rips_complex>`_ is a simplicial complex that -generalizes proximity (:math:`\varepsilon`-ball) graphs to higher dimensions. The vertices correspond to the input -points, and a simplex is present if and only if its diameter is smaller than some parameter α. Considering all -parameters α defines a filtered simplicial complex, where the filtration value of a simplex is its diameter. -The filtration can be restricted to values α smaller than some threshold, to reduce its size. - -The input discrete metric space can be provided as a point cloud plus a distance function, or as a distance matrix. - -When creating a simplicial complex from the graph, :doc:`RipsComplex <rips_complex_ref>` first builds the graph and -inserts it into the data structure. It then expands the simplicial complex (adds the simplices corresponding to cliques) -when required. The expansion can be stopped at dimension `max_dimension`, by default 1. - -A vertex name corresponds to the index of the point in the given range (aka. the point cloud). - -.. figure:: - ../../doc/Rips_complex/rips_complex_representation.png - :align: center - - Rips-complex one skeleton graph representation - -On this example, as edges (4,5), (4,6) and (5,6) are in the complex, simplex (4,5,6) is added with the filtration value -set with :math:`max(filtration(4,5), filtration(4,6), filtration(5,6))`. And so on for simplex (0,1,2,3). - -If the `RipsComplex` interfaces are not detailed enough for your need, please refer to rips_persistence_step_by_step.cpp -C++ example, where the graph construction over the Simplex_tree is more detailed. - -A Rips complex can easily become huge, even if we limit the length of the edges -and the dimension of the simplices. One easy trick, before building a Rips -complex on a point cloud, is to call `sparsify_point_set` which removes points -that are too close to each other. This does not change its persistence diagram -by more than the length used to define "too close". - -A more general technique is to use a sparse approximation of the Rips -introduced by Don Sheehy :cite:`sheehy13linear`. We are using the version -described in :cite:`buchet16efficient` (except that we multiply all filtration -values by 2, to match the usual Rips complex). :cite:`cavanna15geometric` proves -a :math:`\frac{1}{1-\varepsilon}`-interleaving, although in practice the -error is usually smaller. A more intuitive presentation of the idea is -available in :cite:`cavanna15geometric`, and in a video -:cite:`cavanna15visualizing`. Passing an extra argument `sparse=0.3` at the -construction of a `RipsComplex` object asks it to build a sparse Rips with -parameter :math:`\varepsilon=0.3`, while the default `sparse=None` builds the -regular Rips complex. - - -Point cloud ------------ - -Example from a point cloud -^^^^^^^^^^^^^^^^^^^^^^^^^^ - -This example builds the neighborhood graph from the given points, up to max_edge_length. -Then it creates a :doc:`Simplex_tree <simplex_tree_ref>` with it. - -Finally, it is asked to display information about the simplicial complex. - -.. testcode:: - - import gudhi - rips_complex = gudhi.RipsComplex(points=[[1, 1], [7, 0], [4, 6], [9, 6], [0, 14], [2, 19], [9, 17]], - max_edge_length=12.0) - - simplex_tree = rips_complex.create_simplex_tree(max_dimension=1) - result_str = 'Rips complex is of dimension ' + repr(simplex_tree.dimension()) + ' - ' + \ - repr(simplex_tree.num_simplices()) + ' simplices - ' + \ - repr(simplex_tree.num_vertices()) + ' vertices.' - print(result_str) - fmt = '%s -> %.2f' - for filtered_value in simplex_tree.get_filtration(): - print(fmt % tuple(filtered_value)) - -When launching (Rips maximal distance between 2 points is 12.0, is expanded -until dimension 1 - one skeleton graph in other words), the output is: - -.. testoutput:: - - Rips complex is of dimension 1 - 18 simplices - 7 vertices. - [0] -> 0.00 - [1] -> 0.00 - [2] -> 0.00 - [3] -> 0.00 - [4] -> 0.00 - [5] -> 0.00 - [6] -> 0.00 - [2, 3] -> 5.00 - [4, 5] -> 5.39 - [0, 2] -> 5.83 - [0, 1] -> 6.08 - [1, 3] -> 6.32 - [1, 2] -> 6.71 - [5, 6] -> 7.28 - [2, 4] -> 8.94 - [0, 3] -> 9.43 - [4, 6] -> 9.49 - [3, 6] -> 11.00 - -Notice that if we use - -.. code-block:: python - - rips_complex = gudhi.RipsComplex(points=[[1, 1], [7, 0], [4, 6], [9, 6], [0, 14], [2, 19], [9, 17]], - max_edge_length=12.0, sparse=2) - -asking for a very sparse version (theory only gives some guarantee on the meaning of the output if `sparse<1`), -2 to 5 edges disappear, depending on the random vertex used to start the sparsification. - -Example from OFF file -^^^^^^^^^^^^^^^^^^^^^ - -This example builds the :doc:`RipsComplex <rips_complex_ref>` from the given -points in an OFF file, and max_edge_length value. -Then it creates a :doc:`Simplex_tree <simplex_tree_ref>` with it. - -Finally, it is asked to display information about the Rips complex. - - -.. testcode:: - - import gudhi - point_cloud = gudhi.read_off(off_file=gudhi.__root_source_dir__ + '/data/points/alphacomplexdoc.off') - rips_complex = gudhi.RipsComplex(points=point_cloud, max_edge_length=12.0) - simplex_tree = rips_complex.create_simplex_tree(max_dimension=1) - result_str = 'Rips complex is of dimension ' + repr(simplex_tree.dimension()) + ' - ' + \ - repr(simplex_tree.num_simplices()) + ' simplices - ' + \ - repr(simplex_tree.num_vertices()) + ' vertices.' - print(result_str) - fmt = '%s -> %.2f' - for filtered_value in simplex_tree.get_filtration(): - print(fmt % tuple(filtered_value)) - -the program output is: - -.. testoutput:: - - Rips complex is of dimension 1 - 18 simplices - 7 vertices. - [0] -> 0.00 - [1] -> 0.00 - [2] -> 0.00 - [3] -> 0.00 - [4] -> 0.00 - [5] -> 0.00 - [6] -> 0.00 - [2, 3] -> 5.00 - [4, 5] -> 5.39 - [0, 2] -> 5.83 - [0, 1] -> 6.08 - [1, 3] -> 6.32 - [1, 2] -> 6.71 - [5, 6] -> 7.28 - [2, 4] -> 8.94 - [0, 3] -> 9.43 - [4, 6] -> 9.49 - [3, 6] -> 11.00 - -Distance matrix ---------------- - -Example from a distance matrix -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - -This example builds the one skeleton graph from the given distance matrix, and max_edge_length value. -Then it creates a :doc:`Simplex_tree <simplex_tree_ref>` with it. - -Finally, it is asked to display information about the simplicial complex. - -.. testcode:: - - import gudhi - rips_complex = gudhi.RipsComplex(distance_matrix=[[], - [6.0827625303], - [5.8309518948, 6.7082039325], - [9.4339811321, 6.3245553203, 5], - [13.0384048104, 15.6524758425, 8.94427191, 12.0415945788], - [18.0277563773, 19.6468827044, 13.152946438, 14.7648230602, 5.3851648071], - [17.88854382, 17.1172427686, 12.0830459736, 11, 9.4868329805, 7.2801098893]], - max_edge_length=12.0) - - simplex_tree = rips_complex.create_simplex_tree(max_dimension=1) - result_str = 'Rips complex is of dimension ' + repr(simplex_tree.dimension()) + ' - ' + \ - repr(simplex_tree.num_simplices()) + ' simplices - ' + \ - repr(simplex_tree.num_vertices()) + ' vertices.' - print(result_str) - fmt = '%s -> %.2f' - for filtered_value in simplex_tree.get_filtration(): - print(fmt % tuple(filtered_value)) - -When launching (Rips maximal distance between 2 points is 12.0, is expanded -until dimension 1 - one skeleton graph in other words), the output is: - -.. testoutput:: - - Rips complex is of dimension 1 - 18 simplices - 7 vertices. - [0] -> 0.00 - [1] -> 0.00 - [2] -> 0.00 - [3] -> 0.00 - [4] -> 0.00 - [5] -> 0.00 - [6] -> 0.00 - [2, 3] -> 5.00 - [4, 5] -> 5.39 - [0, 2] -> 5.83 - [0, 1] -> 6.08 - [1, 3] -> 6.32 - [1, 2] -> 6.71 - [5, 6] -> 7.28 - [2, 4] -> 8.94 - [0, 3] -> 9.43 - [4, 6] -> 9.49 - [3, 6] -> 11.00 - -Example from csv file -^^^^^^^^^^^^^^^^^^^^^ - -This example builds the :doc:`RipsComplex <rips_complex_ref>` from the given -distance matrix in a csv file, and max_edge_length value. -Then it creates a :doc:`Simplex_tree <simplex_tree_ref>` with it. - -Finally, it is asked to display information about the Rips complex. - - -.. testcode:: - - import gudhi - distance_matrix = gudhi.read_lower_triangular_matrix_from_csv_file(csv_file=gudhi.__root_source_dir__ + \ - '/data/distance_matrix/full_square_distance_matrix.csv') - rips_complex = gudhi.RipsComplex(distance_matrix=distance_matrix, max_edge_length=12.0) - simplex_tree = rips_complex.create_simplex_tree(max_dimension=1) - result_str = 'Rips complex is of dimension ' + repr(simplex_tree.dimension()) + ' - ' + \ - repr(simplex_tree.num_simplices()) + ' simplices - ' + \ - repr(simplex_tree.num_vertices()) + ' vertices.' - print(result_str) - fmt = '%s -> %.2f' - for filtered_value in simplex_tree.get_filtration(): - print(fmt % tuple(filtered_value)) - -the program output is: - -.. testoutput:: - - Rips complex is of dimension 1 - 18 simplices - 7 vertices. - [0] -> 0.00 - [1] -> 0.00 - [2] -> 0.00 - [3] -> 0.00 - [4] -> 0.00 - [5] -> 0.00 - [6] -> 0.00 - [2, 3] -> 5.00 - [4, 5] -> 5.39 - [0, 2] -> 5.83 - [0, 1] -> 6.08 - [1, 3] -> 6.32 - [1, 2] -> 6.71 - [5, 6] -> 7.28 - [2, 4] -> 8.94 - [0, 3] -> 9.43 - [4, 6] -> 9.49 - [3, 6] -> 11.00 - -Correlation matrix ------------------- - -Example from a correlation matrix -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - -Analogously to the case of distance matrix, Rips complexes can be also constructed based on correlation matrix. -Given a correlation matrix M, comportment-wise 1-M is a distance matrix. -This example builds the one skeleton graph from the given corelation matrix and threshold value. -Then it creates a :doc:`Simplex_tree <simplex_tree_ref>` with it. - -Finally, it is asked to display information about the simplicial complex. - -.. testcode:: - - import gudhi - import numpy as np - - # User defined correlation matrix is: - # |1 0.06 0.23 0.01 0.89| - # |0.06 1 0.74 0.01 0.61| - # |0.23 0.74 1 0.72 0.03| - # |0.01 0.01 0.72 1 0.7 | - # |0.89 0.61 0.03 0.7 1 | - correlation_matrix=np.array([[1., 0.06, 0.23, 0.01, 0.89], - [0.06, 1., 0.74, 0.01, 0.61], - [0.23, 0.74, 1., 0.72, 0.03], - [0.01, 0.01, 0.72, 1., 0.7], - [0.89, 0.61, 0.03, 0.7, 1.]], float) - - distance_matrix = np.ones((correlation_matrix.shape),float) - correlation_matrix - rips_complex = gudhi.RipsComplex(distance_matrix=distance_matrix, max_edge_length=1.0) - - simplex_tree = rips_complex.create_simplex_tree(max_dimension=1) - result_str = 'Rips complex is of dimension ' + repr(simplex_tree.dimension()) + ' - ' + \ - repr(simplex_tree.num_simplices()) + ' simplices - ' + \ - repr(simplex_tree.num_vertices()) + ' vertices.' - print(result_str) - fmt = '%s -> %.2f' - for filtered_value in simplex_tree.get_filtration(): - print(fmt % tuple(filtered_value)) - -When launching (Rips maximal distance between 2 points is 12.0, is expanded -until dimension 1 - one skeleton graph in other words), the output is: - -.. testoutput:: - - Rips complex is of dimension 1 - 15 simplices - 5 vertices. - [0] -> 0.00 - [1] -> 0.00 - [2] -> 0.00 - [3] -> 0.00 - [4] -> 0.00 - [0, 4] -> 0.11 - [1, 2] -> 0.26 - [2, 3] -> 0.28 - [3, 4] -> 0.30 - [1, 4] -> 0.39 - [0, 2] -> 0.77 - [0, 1] -> 0.94 - [2, 4] -> 0.97 - [0, 3] -> 0.99 - [1, 3] -> 0.99 - -.. note:: - As persistence diagrams points will be under the diagonal, - bottleneck distance and persistence graphical tool will not work properly, - this is a known issue. |