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doc/python/cohomology-persistence.rst

author | Dmitriy Morozov <dmitriy@mrzv.org> |

Sun, 24 Nov 2019 13:35:45 -0800 | |

changeset 298 | 02e680f6d987 |

parent 267 | 2f02384a4d9b |

permissions | -rw-r--r-- |

Add .build.yml

:class:`CohomologyPersistence` class ==================================== The :ref:`rips-pairwise-cohomology` illustrates the use of :class:`CohomologyPersistence`. .. class:: CohomologyPersistence .. method:: __init__(prime = 11) Initializes :class:`CohomologyPersistence` with the given `prime`; from this point on all the computation will be performed with coefficients in :math:`\mathbb{Z}/prime \mathbb{Z}`. .. method:: add(boundary, birth, [store = True], [image = True], [coefficients = []]) Adds a simplex with the given `boundary` to the complex, i.e. :math:`K_{i+1} = K_i \cup \sigma` and `boundary` = :math:`\partial \sigma`. If a new class is born as a result of the addition, `birth` is stored with it for future reference. If `store` is ``False`` and a class is born, it will not be stored in :class:`CohomologyPersistence`. This avoids wasting space on the classes of the dimension equal to the maximum-dimensional simplices of the complex since such classes will never die. The `image` parameter allows one to work with a case of a space :math:`L \subseteq K` where the filtration of :math:`K` induces a filtration of :math:`L`. In this case, one may want to compute **image persistence** (i.e. the persistence of the sequences of the images given by the inclusion of :math:`L` in :math:`K`). `image` indicates whether the simplex added belongs to :math:`L` or not. If given, `coefficients` is a list parallel to `boundary` that provides coefficients for the corresponding boundary elements. If empty, it is assumed to be :math:`(-1)^i`. :returns: a triple (`i`, `d`, `ccl`). The first element is the index `i`. It is the internal representation of the newly added simplex, and should be used later when constructing the boundaries of its cofaces. In other words, `boundary` must consist of these indices. The second element `d` is the death element. It is `None` if a birth occurred, otherwise it contains the value passed as `birth` to :meth:`~CohomologyPersistence.add` when the class that just died was born. The third element `ccl` returns the dying cocycle (iterable over instances of :class:`CHSNode`), in case of a death. It's empty if a birth occurs. .. method:: __iter__() Iterator over the live cocycles stored in :class:`CohomologyPersistence`. The returned elements are of the type :class:`Cocycle` below. .. class:: Cocycle .. attribute:: birth The birth value associated with the cocycle. It is passed to :class:`CohomologyPersistence` in method :meth:`~CohomologyPersistence.add`. .. method:: __iter__() Iterator over the individual nodes (simplices) of the cocycle, each of type :class:`CHSNode`. .. class:: CHSNode .. attribute:: si The index of the simplex, of type :class:`CHSimplexIndex`. .. attribute:: coefficient Coefficient in :math:`\mathbb{Z}/prime \mathbb{Z}` associated with the simplex. .. class:: CHSimplexIndex .. attribute:: order The count associated with the simplex when it is inserted into :class:`CohomologyPersistence`. Adaptor ------- :class:`StaticCohomologyPersistence` provides a wrapper around class :class:`CohomologyPersistence` that's compatible with :class:`StaticPersistence`. See the documentation of the latter class for details. .. class:: StaticCohomologyPersistence .. method:: __init__(filtration, prime = 2) Initializes :class:`StaticCohomologyPersistence` with the given :class:`Filtration`. `prime` is passed straight to the wrapped :class:`CohomologyPersistence` class. .. method:: pair_simplices() Pairs simplices using :class:`CohomologyPersistence` class. .. method:: __call__(i) Given a node in the internal representation, the method returns its integer offset from the beginning of the filtration. .. method:: make_simplex_map(filtration) Creates an auxilliary map from the nodes to the simplices:: smap = persistence.make_simplex_map(filtration) for i in persistence: if i.unpaired(): print smap[i] .. method:: __iter__() Iterator over the nodes (representing individual simplices). See :class:`APNode`. .. method:: __len__() Returns the number of nodes (i.e. the number of simplices). .. class:: APNode The following methods behave the same way as they do in :class:`SPNode`. .. method:: sign() .. method:: pair() .. method:: unpaired() The only crucial distinction in the behavior comes with the attribute :attr:`cocycle`. .. attribute:: cocycle If the simplex is positive, this attribute stores a cocycle it created (recorded at the time of its death). The 1-dimensional cocycles can be used with the :func:`circular.smooth` function to turn them into circle-valued maps. :: for i in persistence: if i.sign(): print i.cocycle .. class:: ImagePersistence This class is another wrapper around :class:`CohomologyPersistence` that can compute image persistence induced by inclusion of a subcomplex. Its interface is the same as :class:`StaticCohomologyPersistence` above, except for the constructor: .. method:: __init__(filtration, subcomplex) `subcomplex` is a function called with every simplex. It should return ``True`` if the simplex belong to the subcomplex; ``False`` otherwise. Circular coordinates -------------------- .. function:: circular.smooth(filtration, cocycle) Returns a map from the vertices of the simplicial complex `filtration` to a circle :math:`[-.5, .5]`, where the opposite ends of the interval are identified.