# 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`