Parametrizing a point set using circle valued functions¶

The procedure described below is explained in detail in [dSVJ09].

One can use `examples/cohomology/rips-pairwise-cohomology.cpp` to compute persistent pairing of the Rips filtration using the persistent cohomology algorithm. It takes as input a file containing a point set in Euclidean space (one per line) as well as the following command-line flags:

`-p``, ``--prime`

The prime to use in the computation (defaults to 11).

`-m``, ``--max-distance`

Maximum cutoff parameter up to which to compute the complex.

`-s``, ``--skeleton-dimension`

Skeleton to compute; persistent pairs output will be this number minus 1 (defaults to 2).

`-b``, ``--boundary`

Filename where to output the boundary matrix.

`-c``, ``--cocycle`

Prefix of the filenames where to output the 1-dimensional cocycles.

`-v``, ``--vertices`

Filename where to output the simplex vertex mapping.

`-d``, ``--diagram`

Filename where to output the persistence diagram.

For example:

```rips-pairwise-cohomology points.txt -m 1 -b points.bdry -c points -v points.vrt -d points.dgm
```

Assuming that at the threshold value of 1 (`-m 1` above) Rips complex contains 1-dimensional cocycles, they will be output into filenames of the form `points-0.ccl`, `points-1.ccl`, etc.

Subsequently one can use `examples/cohomology/cocycle.py` to assign to each vertex of the input point set a circle-valued function. It takes the boundary matrix, cocycle, and simplex-vertex map as an input (all produced at the previous step):

```cocycle.py points.bdry points-0.ccl points.vrt
```

The above command outputs a file `points-0.val` which contains values assigned to the input points (the lines match the lines of the input file `points.txt`, but also contains the indices).

Plotting¶

Two auxilliary tools allow one to visualize the values assigned to the points (using Matplotlib): `tools/plot-values/plot.py` and `tools/plot-values/scatter.py`:

```plot.py points-0.val points.txt scatter.py points-0.val points-1.val
```

Dependency¶

The Python LSQR code (ported from the Stanford MATLAB implementation to Python by Jeffery Kline) included with Dionysus, and used in `examples/cohomology/cocycle.py`, requires CVXOPT.

Python cohomology computation¶

`examples/cohomology/rips-pairwise-cohomology.py` gives an example of the same computation performed in Python (but with the output in a different format).

After the simplicial complex is computed in a list simplices, and the list is sorted with respect to the Rips filtration order, the simplices are inserted into the `CohomologyPersistence` one by one:

```# list simplices is created

ch = CohomologyPersistence(prime)
complex = {}

for s in simplices:
i,d = ch.add([complex[sb] for sb in s.boundary], (s.dimension(), s.data))
complex[s] = i
if d:
dimension, birth = d
print dimension, birth, s.data
# else birth
```

Above dictionary complex maintains the map of simplices to indices returned by `CohomologyPersistence.add()`. The pair (dimension, data) is used as the birth value. Here data is the value associated with the simplex in the Rips filtration. The pair is returned back if a death occurs, and is printed on the standard output. After the for loop finishes, one may output infinite persistence classes with the following for loop:

```for ccl in ch:
dimension, birth = ccl.birth
if dimension >= skeleton: continue
print dimension, birth, 'inf'         # dimension, simplex data = birth
```

Naturally one may iterate over ccl which is of type `Cocycle` and extract more information. For example, this is necessary to get the coefficients that serve as the input for `examples/cohomology/cocycle.py`.