:orphan:

.. _tutorials:

Tutorials
==================

Each of the tutorials below can be downloaded as a Jupyter notebook (scroll to the bottom of each page for a download link).
Here are some topics we cover, or plan to cover soon:



- **Persistent homology (barcodes, persistence diagrams)**

    - Vietoris-Rips complexes

- **Optimal cycle representatives**

    - Vietoris-Rips complexes
    - Dowker complexes


- **Vietoris-Rips complexes**

    - Construction
    - Simplices
    - Boundary matrices

      - Oracles
      - Scipy sparse formats
      - Visualization

    - Computing (co)boundaries
    - Computing (co)bounding chains
    - Enclosing radius

- **Umatch decomposition**

  - basic construction
  - generalized matching matrix
  - differential COBM
  - R = DV factorization
  - RU = D factorization
  - cycle representatives
  - bounding chains
  - cocycle representatives
  - cobounded chains

- **Plotting**


- **Point clouds**








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Dowker complexes and Hypergraphs
--------------------------------------




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    <div class="sphx-glr-thumbcontainer" tooltip="In this example we compute the homology of a Dowker complex.">

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  .. image:: /auto_examples/dowker/images/thumb/sphx_glr_plot_dowker_homology_thumb.png
    :alt:

  :ref:`sphx_glr_auto_examples_dowker_plot_dowker_homology.py`

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      <div class="sphx-glr-thumbnail-title">Homology</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="In this example we will">

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  .. image:: /auto_examples/dowker/images/thumb/sphx_glr_plot_optimize_thumb.png
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  :ref:`sphx_glr_auto_examples_dowker_plot_optimize.py`

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      <div class="sphx-glr-thumbnail-title">Optimal cycles</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Homology calculations are only available in OAT for one type of hypergraph: a list of sorted-lists of integers. However we provide tools to translate back and forth between this and other data fromats, e.g. a dictionary of lists of strings.  ">

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  .. image:: /auto_examples/dowker/images/thumb/sphx_glr_plot_relabel_reduce_thumb.png
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  :ref:`sphx_glr_auto_examples_dowker_plot_relabel_reduce.py`

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      <div class="sphx-glr-thumbnail-title">Reduce & Relabel</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="The Restricted Barycentric Subdivision Homology of a hypergraph is the homology of the clique complex of the simple graph G such that">

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  .. image:: /auto_examples/dowker/images/thumb/sphx_glr_plot_rbs_homology_thumb.png
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  :ref:`sphx_glr_auto_examples_dowker_plot_rbs_homology.py`

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      <div class="sphx-glr-thumbnail-title">Restricted Barycentric Subdivision (Vietoris-Rips)</div>
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.. toctree::
   :hidden:

   /auto_examples/dowker/plot_dowker_homology
   /auto_examples/dowker/plot_optimize
   /auto_examples/dowker/plot_relabel_reduce
   /auto_examples/dowker/plot_rbs_homology

Plotting
------------------




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    <div class="sphx-glr-thumbcontainer" tooltip="It&#x27;s often useful to plot collections of objects (points, edges, triangles, tetrahedra, etc.) on a colorscale. In some cases Plotly allows this natively">

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  .. image:: /auto_examples/plotting/images/thumb/sphx_glr_plot_color_mapping_thumb.png
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  :ref:`sphx_glr_auto_examples_plotting_plot_color_mapping.py`

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      <div class="sphx-glr-thumbnail-title">Color Mapping</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Edges in 2D">

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  .. image:: /auto_examples/plotting/images/thumb/sphx_glr_plot_edges_2d_thumb.png
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  :ref:`sphx_glr_auto_examples_plotting_plot_edges_2d.py`

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      <div class="sphx-glr-thumbnail-title">Edges in 2D</div>
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  .. image:: /auto_examples/plotting/images/thumb/sphx_glr_plot_edges_3d_thumb.png
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  :ref:`sphx_glr_auto_examples_plotting_plot_edges_3d.py`

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      <div class="sphx-glr-thumbnail-title">Edges in 3D</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="This gallery contains fun miscellaneous plots.">

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  .. image:: /auto_examples/plotting/images/thumb/sphx_glr_plot_misc_surfaces_thumb.png
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  :ref:`sphx_glr_auto_examples_plotting_plot_misc_surfaces.py`

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      <div class="sphx-glr-thumbnail-title">Miscellaneous Surfaces</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="This tutorial contains tips that we found helpful when plotting cycle representatives.">

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  .. image:: /auto_examples/plotting/images/thumb/sphx_glr_plot_cycle_representatives_thumb.png
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  :ref:`sphx_glr_auto_examples_plotting_plot_cycle_representatives.py`

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      <div class="sphx-glr-thumbnail-title">Strategies for Plotting Cycle Representatives</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="This tutorial shows how to adjust the layout of 3D plots in Plotly. For more advanced usage, see the Plotly documentation: https://plotly.com/python/reference/layout/.">

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  .. image:: /auto_examples/plotting/images/thumb/sphx_glr_plot_style_3d_thumb.png
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  :ref:`sphx_glr_auto_examples_plotting_plot_style_3d.py`

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      <div class="sphx-glr-thumbnail-title">Styling in 3D</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Triangles in 2D">

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  .. image:: /auto_examples/plotting/images/thumb/sphx_glr_plot_triangles_2d_thumb.png
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  :ref:`sphx_glr_auto_examples_plotting_plot_triangles_2d.py`

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      <div class="sphx-glr-thumbnail-title">Triangles in 2D</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="This tutorial covers commonly asked questions about ploting triangles in 3D using Plotly.">

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  .. image:: /auto_examples/plotting/images/thumb/sphx_glr_plot_triangles_3d_thumb.png
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  :ref:`sphx_glr_auto_examples_plotting_plot_triangles_3d.py`

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      <div class="sphx-glr-thumbnail-title">Triangles in 3D</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="This tutorial covers vertex embedding to visualize simplices and simplicial complexes.">

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  .. image:: /auto_examples/plotting/images/thumb/sphx_glr_plot_vertex_embedding_thumb.png
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  :ref:`sphx_glr_auto_examples_plotting_plot_vertex_embedding.py`

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      <div class="sphx-glr-thumbnail-title">Vertex Embedding</div>
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.. toctree::
   :hidden:

   /auto_examples/plotting/plot_color_mapping
   /auto_examples/plotting/plot_edges_2d
   /auto_examples/plotting/plot_edges_3d
   /auto_examples/plotting/plot_misc_surfaces
   /auto_examples/plotting/plot_cycle_representatives
   /auto_examples/plotting/plot_style_3d
   /auto_examples/plotting/plot_triangles_2d
   /auto_examples/plotting/plot_triangles_3d
   /auto_examples/plotting/plot_vertex_embedding

Point cloud generators
----------------------




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    <div class="sphx-glr-thumbcontainer" tooltip="This gallery shows how to generate several different types of point clouds.">

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  .. image:: /auto_examples/point_cloud_generators/images/thumb/sphx_glr_plot_generate_point_clouds_thumb.png
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  :ref:`sphx_glr_auto_examples_point_cloud_generators_plot_generate_point_clouds.py`

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      <div class="sphx-glr-thumbnail-title">Point Cloud Generators</div>
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.. toctree::
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   /auto_examples/point_cloud_generators/plot_generate_point_clouds

Vietoris Rips complexes
-------------------------




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    <div class="sphx-glr-thumbcontainer" tooltip="The Laplacian of a simplicial complex K with boundary matrix D is the matrix L = D^T D + D D^T.">

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  .. image:: /auto_examples/vietoris_rips/images/thumb/sphx_glr_plot_laplacian_thumb.png
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  :ref:`sphx_glr_auto_examples_vietoris_rips_plot_laplacian.py`

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      <div class="sphx-glr-thumbnail-title">Laplacian</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="This tutorial covers persistent homology of point clouds, and filtered Vietoris-Rips complexes more generally.">

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  .. image:: /auto_examples/vietoris_rips/images/thumb/sphx_glr_plot_persistent_homology_dragon_thumb.png
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  :ref:`sphx_glr_auto_examples_vietoris_rips_plot_persistent_homology_dragon.py`

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      <div class="sphx-glr-thumbnail-title">Persistent Homology: Stanford Dragon</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="R = DV factorization is a standard tool to compute persistent homology. OAT obtains R=DV factorizations by way of Umatch decompositions. This tutorial shows how to obtain">

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  .. image:: /auto_examples/vietoris_rips/images/thumb/sphx_glr_plot_rdv_thumb.png
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  :ref:`sphx_glr_auto_examples_vietoris_rips_plot_rdv.py`

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      <div class="sphx-glr-thumbnail-title">R=DV and other matrix decompositions</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Sparse matrices in applied topology present several challenges.">

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  .. image:: /auto_examples/vietoris_rips/images/thumb/sphx_glr_plot_matrices_thumb.png
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  :ref:`sphx_glr_auto_examples_vietoris_rips_plot_matrices.py`

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      <div class="sphx-glr-thumbnail-title">Sparse Matrices</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="A common pain point in computational homological algebra is passing between matrix and integer indices for boundary and other matrices. OAT offers a oat_python.core.vietoris_rips.SubmatrixIndexTool to help wth this.">

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  .. image:: /auto_examples/vietoris_rips/images/thumb/sphx_glr_plot_submatrix_index_tool_thumb.png
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  :ref:`sphx_glr_auto_examples_vietoris_rips_plot_submatrix_index_tool.py`

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      <div class="sphx-glr-thumbnail-title">Submatrix Index Tool</div>
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  .. image:: /auto_examples/vietoris_rips/images/thumb/sphx_glr_plot_vector_index_tool_thumb.png
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  :ref:`sphx_glr_auto_examples_vietoris_rips_plot_vector_index_tool.py`

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      <div class="sphx-glr-thumbnail-title">Vector Index Tool</div>
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.. toctree::
   :hidden:

   /auto_examples/vietoris_rips/plot_laplacian
   /auto_examples/vietoris_rips/plot_persistent_homology_dragon
   /auto_examples/vietoris_rips/plot_rdv
   /auto_examples/vietoris_rips/plot_matrices
   /auto_examples/vietoris_rips/plot_submatrix_index_tool
   /auto_examples/vietoris_rips/plot_vector_index_tool


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      :download:`Download all examples in Python source code: auto_examples_python.zip </auto_examples/auto_examples_python.zip>`

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download all examples in Jupyter notebooks: auto_examples_jupyter.zip </auto_examples/auto_examples_jupyter.zip>`


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    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_