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To get rid of WARNING: document isn't included in any toctree - -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 Gudhi::persistent_cohomology::Persistent_cohomology) 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 '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_off(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>` - -Bibliography -============ - -.. bibliography:: ../../biblio/bibliography.bib - :filter: docnames - :style: unsrt |