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To get rid of WARNING: document isn't included in any toctree - -Persistent cohomology user manual -================================= -Definition ----------- -===================================== ===================================== ===================================== -:Author: Clément Maria :Introduced in: GUDHI PYTHON 2.0.0 :Copyright: GPL v3 -===================================== ===================================== ===================================== - -+---------------------------------------------+----------------------------------------------------------------------+ -| :doc:`persistent_cohomology_user` | Please refer to each data structure that contains persistence | -| | feature for reference: | -| | | -| | * :doc:`simplex_tree_ref` | -+---------------------------------------------+----------------------------------------------------------------------+ - - -Computation of persistent cohomology using the algorithm of :cite:`DBLP:journals/dcg/SilvaMV11` and -:cite:`DBLP:journals/corr/abs-1208-5018` and the Compressed Annotation Matrix implementation of -:cite:`DBLP:conf/esa/BoissonnatDM13`. - -The theory of homology consists in attaching to a topological space a sequence of (homology) groups, capturing global -topological features like connected components, holes, cavities, etc. Persistent homology studies the evolution -- -birth, life and death -- of these features when the topological space is changing. Consequently, the theory is -essentially composed of three elements: - -* topological spaces -* their homology groups -* an evolution scheme. - -Topological Spaces ------------------- - -Topological spaces are represented by simplicial complexes. -Let :math:`V = \{1, \cdots ,|V|\}` be a set of *vertices*. -A *simplex* :math:`\sigma` is a subset of vertices :math:`\sigma \subseteq V`. -A *simplicial complex* :math:`\mathbf{K}` on :math:`V` is a collection of simplices :math:`\{\sigma\}`, -:math:`\sigma \subseteq V`, such that :math:`\tau \subseteq \sigma \in \mathbf{K} \Rightarrow \tau \in \mathbf{K}`. -The dimension :math:`n=|\sigma|-1` of :math:`\sigma` is its number of elements minus 1. -A *filtration* of a simplicial complex is a function :math:`f:\mathbf{K} \rightarrow \mathbb{R}` satisfying -:math:`f(\tau)\leq f(\sigma)` whenever :math:`\tau \subseteq \sigma`. - -Homology --------- - -For a ring :math:`\mathcal{R}`, the group of *n-chains*, denoted :math:`\mathbf{C}_n(\mathbf{K},\mathcal{R})`, of -:math:`\mathbf{K}` is the group of formal sums of n-simplices with :math:`\mathcal{R}` coefficients. The -*boundary operator* is a linear operator -:math:`\partial_n: \mathbf{C}_n(\mathbf{K},\mathcal{R}) \rightarrow \mathbf{C}_{n-1}(\mathbf{K},\mathcal{R})` -such that :math:`\partial_n \sigma = \partial_n [v_0, \cdots , v_n] = \sum_{i=0}^n (-1)^{i}[v_0,\cdots ,\widehat{v_i}, \cdots,v_n]`, -where :math:`\widehat{v_i}` means :math:`v_i` is omitted from the list. The chain groups form a sequence: - -.. math:: - - \cdots \ \ \mathbf{C}_n(\mathbf{K},\mathcal{R}) \xrightarrow{\ \partial_n\ } - \mathbf{C}_{n-1}(\mathbf{K},\mathcal{R}) \xrightarrow{\partial_{n-1}} \cdots \xrightarrow{\ \partial_2 \ } - \mathbf{C}_1(\mathbf{K},\mathcal{R}) \xrightarrow{\ \partial_1 \ } \mathbf{C}_0(\mathbf{K},\mathcal{R}) - -of finitely many groups :math:`\mathbf{C}_n(\mathbf{K},\mathcal{R})` and homomorphisms :math:`\partial_n`, indexed by -the dimension :math:`n \geq 0`. The boundary operators satisfy the property :math:`\partial_n \circ \partial_{n+1}=0` -for every :math:`n > 0` and we define the homology groups: - -.. math:: - - \mathbf{H}_n(\mathbf{K},\mathcal{R}) = \ker \partial_n / \mathrm{im} \ \partial_{n+1} - -We refer to :cite:`Munkres-elementsalgtop1984` for an introduction to homology -theory and to :cite:`DBLP:books/daglib/0025666` for an introduction to persistent homology. - -Indexing Scheme ---------------- - -"Changing" a simplicial complex consists in applying a simplicial map. An *indexing scheme* is a directed graph -together with a traversal order, such that two consecutive nodes in the graph are connected by an arrow (either forward -or backward). -The nodes represent simplicial complexes and the directed edges simplicial maps. - -From the computational point of view, there are two types of indexing schemes of interest in persistent homology: - -* linear ones - :math:`\bullet \longrightarrow \bullet \longrightarrow \cdots \longrightarrow \bullet \longrightarrow \bullet` - in persistent homology :cite:`DBLP:journals/dcg/ZomorodianC05`, -* zigzag ones - :math:`\bullet \longrightarrow \bullet \longleftarrow \cdots \longrightarrow \bullet \longleftarrow \bullet` - in zigzag persistent homology :cite:`DBLP:journals/focm/CarlssonS10`. - -These indexing schemes have a natural left-to-right traversal order, and we describe them with ranges and iterators. -In the current release of the Gudhi library, only the linear case is implemented. - -In the following, we consider the case where the indexing scheme is induced by a filtration. - -Ordering the simplices by increasing filtration values (breaking ties so as a simplex appears after its subsimplices of -same filtration value) provides an indexing scheme. - -Examples --------- - -We provide several example files: run these examples with -h for details on their use. - -.. only:: builder_html - - * :download:`alpha_complex_diagram_persistence_from_off_file_example.py <../example/alpha_complex_diagram_persistence_from_off_file_example.py>` - * :download:`periodic_cubical_complex_barcode_persistence_from_perseus_file_example.py <../example/periodic_cubical_complex_barcode_persistence_from_perseus_file_example.py>` - * :download:`rips_complex_diagram_persistence_from_off_file_example.py <../example/rips_complex_diagram_persistence_from_off_file_example.py>` - * :download:`rips_persistence_diagram.py <../example/rips_persistence_diagram.py>` - * :download:`rips_complex_diagram_persistence_from_distance_matrix_file_example.py <../example/rips_complex_diagram_persistence_from_distance_matrix_file_example.py>` - * :download:`random_cubical_complex_persistence_example.py <../example/random_cubical_complex_persistence_example.py>` - * :download:`tangential_complex_plain_homology_from_off_file_example.py <../example/tangential_complex_plain_homology_from_off_file_example.py>` - -Bibliography -============ - -.. bibliography:: ../../biblio/bibliography.bib - :filter: docnames - :style: unsrt |