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Witness complex user manual
===========================

.. include:: witness_complex_sum.inc

Definitions
-----------

Witness complex is a simplicial complex defined on two sets of points in :math:`\mathbb{R}^D`:

- :math:`W` set of **witnesses** and
- :math:`L` set of **landmarks**.

Even though often the set of landmarks :math:`L` is a subset of the set of witnesses :math:`W`, it is not a requirement
for the current implementation.

Landmarks are the vertices of the simplicial complex and witnesses help to decide on which simplices are inserted via a
predicate "is witnessed".

De Silva and Carlsson in their paper :cite:`de2004topological` differentiate **weak witnessing** and
**strong witnessing**:

- *weak*:   :math:`\sigma \subset L` is witnessed by :math:`w \in W` if :math:`\forall l \in \sigma,\ \forall l' \in \mathbf{L \setminus \sigma},\ d(w,l) \leq d(w,l')`
- *strong*: :math:`\sigma \subset L` is witnessed by :math:`w \in W` if :math:`\forall l \in \sigma,\ \forall l' \in \mathbf{L},\ d(w,l) \leq d(w,l')`

where :math:`d(.,.)` is a distance function.

Both definitions can be relaxed by a real value :math:`\alpha`:

- *weak*:   :math:`\sigma \subset L` is :math:`\alpha`-witnessed by :math:`w \in W` if :math:`\forall l \in \sigma,\ \forall l' \in \mathbf{L \setminus \sigma},\ d(w,l)^2 \leq d(w,l')^2 + \alpha^2`
- *strong*: :math:`\sigma \subset L` is :math:`\alpha`-witnessed by :math:`w \in W` if :math:`\forall l \in \sigma,\ \forall l' \in \mathbf{L},\ d(w,l)^2 \leq d(w,l')^2 + \alpha^2`

which leads to definitions of **weak relaxed witness complex** (or just relaxed witness complex for short) and
**strong relaxed witness complex** respectively.

.. figure:: ../../doc/Witness_complex/swit.svg
    :alt: Strongly witnessed simplex
    :figclass: align-center

    Strongly witnessed simplex


In particular case of 0-relaxation, weak complex corresponds to **witness complex** introduced in
:cite:`de2004topological`, whereas 0-relaxed strong witness complex consists of just vertices and is not very
interesting. Hence for small relaxation weak version is preferable.
However, to capture the homotopy type (for example using :func:`gudhi.SimplexTree.persistence`) it is
often necessary to work with higher filtration values. In this case strong relaxed witness complex is faster to compute
and offers similar results.

Implementation
--------------

The two complexes described above are implemented in the corresponding classes

- :doc:`witness_complex_ref`
- :doc:`strong_witness_complex_ref`
- :doc:`euclidean_witness_complex_ref`
- :doc:`euclidean_strong_witness_complex_ref`

The construction of the Euclidean versions of complexes follow the same scheme:

1. Construct a search tree on landmarks.
2. Construct lists of nearest landmarks for each witness.
3. Construct the witness complex for nearest landmark lists.

In the non-Euclidean classes, the lists of nearest landmarks are supposed to be given as input.

The constructors take on the steps 1 and 2, while the function :func:`!create_complex` executes the step 3.

Constructing weak relaxed witness complex from an off file
----------------------------------------------------------

Let's start with a simple example, which reads an off point file and computes a weak witness complex.

.. code-block:: python

    import gudhi
    import argparse

    parser = argparse.ArgumentParser(description='EuclideanWitnessComplex creation from '
                                     'points read in a OFF file.',
                                     epilog='Example: '
                                     'example/witness_complex_diagram_persistence_from_off_file_example.py '
                                     '-f ../data/points/tore3D_300.off -a 1.0 -n 20 -d 2'
                                     '- Constructs a alpha complex with the '
                                     'points from the given OFF file.')
    parser.add_argument("-f", "--file", type=str, required=True)
    parser.add_argument("-a", "--max_alpha_square", type=float, required=True)
    parser.add_argument("-n", "--number_of_landmarks", type=int, required=True)
    parser.add_argument("-d", "--limit_dimension", type=int, required=True)

    args = parser.parse_args()

    with open(args.file, 'r') as f:
        first_line = f.readline()
        if (first_line == 'OFF\n') or (first_line == 'nOFF\n'):
            print("#####################################################################")
            print("EuclideanWitnessComplex creation from points read in a OFF file")

            witnesses = gudhi.read_points_from_off_file(off_file=args.file)
            landmarks = gudhi.pick_n_random_points(points=witnesses, nb_points=args.number_of_landmarks)

            message = "EuclideanWitnessComplex with max_edge_length=" + repr(args.max_alpha_square) + \
                " - Number of landmarks=" + repr(args.number_of_landmarks)
            print(message)

            witness_complex = gudhi.EuclideanWitnessComplex(witnesses=witnesses, landmarks=landmarks)
            simplex_tree = witness_complex.create_simplex_tree(max_alpha_square=args.max_alpha_square,
                limit_dimension=args.limit_dimension)

            message = "Number of simplices=" + repr(simplex_tree.num_simplices())
            print(message)
        else:
            print(args.file, "is not a valid OFF file")

        f.close()


Example2: Computing persistence using strong relaxed witness complex
--------------------------------------------------------------------

Here is an example of constructing a strong witness complex filtration and computing persistence on it:

* :download:`euclidean_strong_witness_complex_diagram_persistence_from_off_file_example.py <../example/euclidean_strong_witness_complex_diagram_persistence_from_off_file_example.py>`
