Persistent Cohomology

## Classes | |

class | Gudhi::persistent_cohomology::Field_Zp |

Structure representing the coefficient field \(\mathbb{Z}/p\mathbb{Z}\). More... | |

class | Gudhi::persistent_cohomology::Multi_field |

Structure representing coefficients in a set of finite fields simultaneously using the chinese remainder theorem. More... | |

class | Gudhi::persistent_cohomology::Persistent_cohomology< FilteredComplex, CoefficientField > |

Computes the persistent cohomology of a filtered complex. More... | |

Computation of persistent cohomology using the algorithm of [18] and [19] and the Compressed Annotation Matrix implementation of [7]

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 and an evolution scheme.

Topological spaces are represented by simplicial complexes. Let \(V = \{1, \cdots ,|V|\}\) be a set of *vertices*. A *simplex* \(\sigma\) is a subset of vertices \(\sigma \subseteq V\). A *simplicial complex* \(\mathbf{K}\) on \(V\) is a collection of simplices \(\{\sigma\}\), \(\sigma \subseteq V\), such that \(\tau \subseteq \sigma \in \mathbf{K} \Rightarrow \tau \in \mathbf{K}\). The dimension \(n=|\sigma|-1\) of \(\sigma\) is its number of elements minus 1. A *filtration* of a simplicial complex is a function \(f:\mathbf{K} \rightarrow \mathbb{R}\) satisfying \(f(\tau)\leq f(\sigma)\) whenever \(\tau \subseteq \sigma\).

We define the concept FilteredComplex which enumerates the requirements for a class to represent a filtered complex from which persistent homology may be computed. We use the vocabulary of simplicial complexes, but the concept is valid for any type of cell complex. The main requirements are the definition of:

- type
`Indexing_tag`

, which is a model of the concept`IndexingTag`

, describing the nature of the indexing scheme, - type Simplex_handle to manipulate simplices,
- method
`int dimension(Simplex_handle)`

returning the dimension of a simplex, - type and method
`Boundary_simplex_range boundary_simplex_range(Simplex_handle)`

that returns a range giving access to the codimension 1 subsimplices of the input simplex, as-well-as the coefficients \((-1)^i\) in the definition of the operator \(\partial\). The iterators have value type`Simplex_handle`

, - type and method
`Filtration_simplex_range filtration_simplex_range ()`

that returns a range giving access to all the simplices of the complex read in the order assigned by the indexing scheme, - type and method
`Filtration_value filtration (Simplex_handle)`

that returns the value of the filtration on the simplex represented by the handle.

For a ring \(\mathcal{R}\), the group of *n-chains*, denoted \(\mathbf{C}_n(\mathbf{K},\mathcal{R})\), of \(\mathbf{K}\) is the group of formal sums of n-simplices with \(\mathcal{R}\) coefficients. The *boundary operator* is a linear operator \(\partial_n: \mathbf{C}_n(\mathbf{K},\mathcal{R}) \rightarrow \mathbf{C}_{n-1}(\mathbf{K},\mathcal{R})\) such that \(\partial_n \sigma = \partial_n [v_0, \cdots , v_n] = \sum_{i=0}^n (-1)^{i}[v_0,\cdots ,\widehat{v_i}, \cdots,v_n]\), where \(\widehat{v_i}\) means \(v_i\) is omitted from the list. The chain groups form a sequence:

\[\cdots \ \ \mathbf{C}_n(\mathbf{K},\mathcal{R}) \xrightarrow{\ \partial_n\ } \mathbf{C}_{n-1}(\mathbf{K},\mathcal{R}) \xrightarrow{\partial_{n-1}} \cdots \xrightarrow{\ \partial_2 \ } \mathbf{C}_1(\mathbf{K},\mathcal{R}) \xrightarrow{\ \partial_1 \ } \mathbf{C}_0(\mathbf{K},\mathcal{R}) \]

of finitely many groups \(\mathbf{C}_n(\mathbf{K},\mathcal{R})\) and homomorphisms \(\partial_n\), indexed by the dimension \(n \geq 0\). The boundary operators satisfy the property \(\partial_n \circ \partial_{n+1}=0\) for every \(n > 0\) and we define the homology groups:

\[\mathbf{H}_n(\mathbf{K},\mathcal{R}) = \ker \partial_n / \mathrm{im} \ \partial_{n+1}\]

We refer to [31] for an introduction to homology theory and to [22] for an introduction to persistent homology.

"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 \(\bullet \longrightarrow \bullet \longrightarrow \cdots \longrightarrow \bullet \longrightarrow \bullet\) in persistent homology [37] , and *zigzag* ones \(\bullet \longrightarrow \bullet \longleftarrow \cdots \longrightarrow \bullet \longleftarrow \bullet \) in zigzag persistent homology [11]. 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.

We provide several example files: run these examples with -h for details on their use, and read the README file.

- Rips_complex/rips_persistence.cpp computes the Rips complex of a point cloud and outputs its persistence diagram. $> ./rips_persistence ../../data/points/tore3D_1307.off -r 0.25 -m 0.5 -d 3 -p 3The complex contains 177838 simplicesand has dimension 33 0 0 inf3 1 0.0983494 inf3 1 0.104347 inf3 2 0.138335 inf

- Persistent_cohomology/rips_multifield_persistence.cpp computes the Rips complex of a point cloud and outputs its persistence diagram with a family of field coefficients.

- Rips_complex/rips_distance_matrix_persistence.cpp computes the Rips complex of a distance matrix and outputs its persistence diagram.

The file should contain square or lower triangular distance matrix with semicolons as separators. The code do not check if it is dealing with a distance matrix. It is the user responsibility to provide a valid input. Please refer to data/distance_matrix/lower_triangular_distance_matrix.csv for an example of a file.

- Rips_complex/rips_correlation_matrix_persistence.cpp computes the Rips complex of a correlation matrix and outputs its persistence diagram.

Note that no check is performed if the matrix given as the input is a correlation matrix. It is the user responsibility to ensure that this is the case. The input is to be given either as a square or a lower triangular matrix. Please refer to data/correlation_matrix/lower_triangular_correlation_matrix.csv for an example of a file.

- Alpha_complex/alpha_complex_3d_persistence.cpp computes the persistent homology with \(\mathbb{Z}/2\mathbb{Z}\) coefficients of the alpha complex on points sampling from an OFF file. $> ./alpha_complex_3d_persistence ../../data/points/tore3D_300.off -p 2 -m 0.45

- Alpha_complex/exact_alpha_complex_3d_persistence.cpp computes the persistent homology with \(\mathbb{Z}/2\mathbb{Z}\) coefficients of the alpha complex on points sampling from an OFF file. Here, as CGAL computes the exact values, it is slower, but it is necessary when points are on a grid for instance. $> ./exact_alpha_complex_3d_persistence ../../data/points/sphere3D_pts_on_grid.off -p 2 -m 0.1

- Alpha_complex/weighted_alpha_complex_3d_persistence.cpp computes the persistent homology with \(\mathbb{Z}/2\mathbb{Z}\) coefficients of the weighted alpha complex on points sampling from an OFF file and a weights file. $> ./weighted_alpha_complex_3d_persistence ../../data/points/tore3D_300.off../../data/points/tore3D_300.weights -p 2 -m 0.45Simplex_tree dim: 32 0 -1 inf2 1 -0.931784 0.0001033112 1 -0.906588 2.60165e-052 2 -0.43556 0.0393798

- Alpha_complex/alpha_complex_persistence.cpp computes the persistent homology with \(\mathbb{Z}/p\mathbb{Z}\) coefficients of the alpha complex on points sampling from an OFF file. $> ./alpha_complex_persistence -r 32 -p 2 -m 0.45 ../../data/points/tore3D_300.offAlpha complex is of dimension 3 - 9273 simplices - 300 vertices.Simplex_tree dim: 32 0 0 inf2 1 0.0682162 1.00012 1 0.0934117 1.000032 2 0.56444 1.03938

- Alpha_complex/periodic_alpha_complex_3d_persistence.cpp computes the persistent homology with \(\mathbb{Z}/2\mathbb{Z}\) coefficients of the periodic alpha complex on points sampling from an OFF file. The second parameter is a Iso-cuboid file with coordinates of the periodic cuboid. Note that the lengths of the sides of the periodic cuboid have to be the same. $> ./periodic_alpha_complex_3d_persistence ../../data/points/grid_10_10_10_in_0_1.off../../data/points/iso_cuboid_3_in_0_1.txt -p 3 -m 1.0Periodic Delaunay computed.Simplex_tree dim: 33 0 0 inf3 1 0.0025 inf3 1 0.0025 inf3 1 0.0025 inf3 2 0.005 inf3 2 0.005 inf3 2 0.005 inf3 3 0.0075 inf

- Persistent_cohomology/weighted_periodic_alpha_complex_3d_persistence.cpp computes the persistent homology with \(\mathbb{Z}/2\mathbb{Z}\) coefficients of the periodic alpha complex on weighted points from an OFF file. The additional parameters of this program are:

(a) The file with the weights of points. The file consist of a sequence of numbers (as many as points). Note that the weight of each single point have to be bounded by 1/64 times the square of the cuboid edge length.

(b) A Iso-cuboid file with coordinates of the periodic cuboid. Note that the lengths of the sides of the periodic cuboid have to be the same.

$> ./weighted_periodic_alpha_complex_3d_persistence ../../data/points/shifted_sphere.off../../data/points/shifted_sphere.weights ../../data/points/iso_cuboid_3_in_0_10.txt 3 1.0Weighted Periodic Delaunay computed.Simplex_tree dim: 33 0 -0.0001 inf3 1 16.0264 inf3 1 16.0273 inf3 1 16.0303 inf3 2 36.8635 inf3 2 36.8704 inf3 2 36.8838 inf3 3 58.6783 inf

- Persistent_cohomology/plain_homology.cpp computes the plain homology of a simple simplicial complex without filtration values.

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