Rips complex

rips_persistence

This program computes the persistent homology with coefficient field Z/pZ of a Rips complex defined on a set of input points, using Euclidean distance. The output diagram contains one bar per line, written with the convention:

p dim birth death

where dim is the dimension of the homological feature, birth and death are respectively the birth and death of the feature, and p is the characteristic of the field Z/pZ used for homology coefficients (p must be a prime number).

Usage

rips_persistence [options] <OFF input file>

Allowed options

  • -h [ --help ] Produce help message
  • -o [ --output-file ] Name of file in which the persistence diagram is written. Default print in standard output.
  • -r [ --max-edge-length ] (default = inf) Maximal length of an edge for the Rips complex construction.
  • -d [ --cpx-dimension ] (default = 1) Maximal dimension of the Rips complex we want to compute.
  • -p [ --field-charac ] (default = 11) Characteristic p of the coefficient field Z/pZ for computing homology.
  • -m [ --min-persistence ] (default = 0) Minimal lifetime of homology feature to be recorded. Enter a negative value to see zero length intervals.

Beware: this program may use a lot of RAM and take a lot of time if max-edge-length is set to a large value.

Example 1 with Z/2Z coefficients

rips_persistence ../../data/points/tore3D_1307.off -r 0.25 -m 0.5 -d 3 -p 2

Example 2 with Z/3Z coefficients

rips_persistence ../../data/points/tore3D_1307.off -r 0.25 -m 0.5 -d 3 -p 3

rips_distance_matrix_persistence

Same as rips_persistence but taking a distance matrix as input.

Usage

rips_distance_matrix_persistence [options] <CSV input file>

where <CSV input file> is the path to the file containing a distance matrix. Can be square or lower triangular matrix. Separator is ‘;’. 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.

Example

rips_distance_matrix_persistence data/distance_matrix/full_square_distance_matrix.csv -r 15 -d 3 -p 3 -m 0

rips_correlation_matrix_persistence

Same as rips_distance_matrix_persistence but taking a correlation matrix as input.

Usage

rips_correlation_matrix_persistence [options] <CSV input file>

where <CSV input file> is the path to the file containing a correlation matrix. Can be square or lower triangular matrix. Separator is ‘;’. 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. Please refer to data/correlation_matrix/lower_triangular_correlation_matrix.csv for an example of a file.

Example

rips_correlation_matrix_persistence data/distance_matrix/full_square_distance_matrix.csv -r 15 -d 3 -p 3 -m 0

Warning

As persistence diagrams points will be under the diagonal, bottleneck distance and persistence graphical tool will not work properly, this is a known issue.

sparse_rips_persistence

This program computes the persistent homology with coefficient field Z/pZ of a sparse (1+epsilon)-approximation of the Rips complex defined on a set of input Euclidean points. The output diagram contains one bar per line, written with the convention:

p dim birth death

where dim is the dimension of the homological feature, birth and death are respectively the birth and death of the feature, and p is the characteristic of the field Z/pZ used for homology coefficients (p must be a prime number).

Usage

sparse_rips_persistence [options] <OFF input file>

Allowed options

  • -h [ --help ] Produce help message
  • -o [ --output-file ] Name of file in which the persistence diagram is written. Default print in standard output.
  • -e [ --approximation ] (default = .5) Epsilon, where the sparse Rips complex is a (1+epsilon)-approximation of the Rips complex.
  • -d [ --cpx-dimension ] (default = 1) Maximal dimension of the Rips complex we want to compute.
  • -p [ --field-charac ] (default = 11) Characteristic p of the coefficient field Z/pZ for computing homology.
  • -m [ --min-persistence ] (default = 0) Minimal lifetime of homology feature to be recorded. Enter a negative value to see zero length intervals.

Example with Z/2Z coefficients

sparse_rips_persistence ../../data/points/tore3D_1307.off -e .5 -m .2 -d 3 -p 2