diff options
author | ROUVREAU Vincent <vincent.rouvreau@inria.fr> | 2019-09-09 16:03:40 +0200 |
---|---|---|
committer | ROUVREAU Vincent <vincent.rouvreau@inria.fr> | 2019-09-09 16:03:40 +0200 |
commit | 68753b3c28321e28eedd5829c94234da84e25c8d (patch) | |
tree | 36003a30309b3203b41092ad4d7ee8fa78551452 /src/python/doc/rips_complex_user.rst | |
parent | dcbdaa0dc00eb069d1a13575f22c0a2f7d63dcc8 (diff) |
Code review: rename cython as python (make target and directory
Diffstat (limited to 'src/python/doc/rips_complex_user.rst')
-rw-r--r-- | src/python/doc/rips_complex_user.rst | 345 |
1 files changed, 345 insertions, 0 deletions
diff --git a/src/python/doc/rips_complex_user.rst b/src/python/doc/rips_complex_user.rst new file mode 100644 index 00000000..1d340dbe --- /dev/null +++ b/src/python/doc/rips_complex_user.rst @@ -0,0 +1,345 @@ +:orphan: + +.. 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. |