# Witness complex user manual#

Witness complex \(Wit(W,L)\) is a simplicial complex defined on two sets of points in \(\mathbb{R}^D\). The data structure is described in [5]. |
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## Definitions#

Witness complex is a simplicial complex defined on two sets of points in \(\mathbb{R}^D\):

\(W\) set of

**witnesses**and\(L\) set of

**landmarks**.

Even though often the set of landmarks \(L\) is a subset of the set of witnesses \(W\), it is not a requirement for the current implementation.

Landmarks are the vertices of the simplicial complex and witnesses help to decide on which simplices are inserted via a predicate “is witnessed”.

De Silva and Carlsson in their paper [15] differentiate **weak witnessing** and
**strong witnessing**:

*weak*: \(\sigma \subset L\) is witnessed by \(w \in W\) if \(\forall l \in \sigma,\ \forall l' \in \mathbf{L \setminus \sigma},\ d(w,l) \leq d(w,l')\)*strong*: \(\sigma \subset L\) is witnessed by \(w \in W\) if \(\forall l \in \sigma,\ \forall l' \in \mathbf{L},\ d(w,l) \leq d(w,l')\)

where \(d(.,.)\) is a distance function.

Both definitions can be relaxed by a real value \(\alpha\):

*weak*: \(\sigma \subset L\) is \(\alpha\)-witnessed by \(w \in W\) if \(\forall l \in \sigma,\ \forall l' \in \mathbf{L \setminus \sigma},\ d(w,l)^2 \leq d(w,l')^2 + \alpha^2\)*strong*: \(\sigma \subset L\) is \(\alpha\)-witnessed by \(w \in W\) if \(\forall l \in \sigma,\ \forall l' \in \mathbf{L},\ d(w,l)^2 \leq d(w,l')^2 + \alpha^2\)

which leads to definitions of **weak relaxed witness complex** (or just relaxed witness complex for short) and
**strong relaxed witness complex** respectively.

In particular case of 0-relaxation, weak complex corresponds to **witness complex** introduced in
[15], whereas 0-relaxed strong witness complex consists of just vertices and is not very
interesting. Hence for small relaxation weak version is preferable.
However, to capture the homotopy type (for example using `gudhi.SimplexTree.persistence()`

) it is
often necessary to work with higher filtration values. In this case strong relaxed witness complex is faster to compute
and offers similar results.

## Implementation#

The two complexes described above are implemented in the corresponding classes

The construction of the Euclidean versions of complexes follow the same scheme:

Construct a search tree on landmarks.

Construct lists of nearest landmarks for each witness.

Construct the witness complex for nearest landmark lists.

In the non-Euclidean classes, the lists of nearest landmarks are supposed to be given as input.

The constructors take on the steps 1 and 2, while the function `create_complex()`

executes the step 3.

## Constructing weak relaxed witness complex from an off file#

Let’s start with a simple example, which reads an off point file and computes a weak witness complex.

```
import gudhi
import argparse
parser = argparse.ArgumentParser(description='EuclideanWitnessComplex creation from '
'points read in a OFF file.',
epilog='Example: '
'example/witness_complex_diagram_persistence_from_off_file_example.py '
'-f ../data/points/tore3D_300.off -a 1.0 -n 20 -d 2'
'- Constructs a alpha complex with the '
'points from the given OFF file.')
parser.add_argument("-f", "--file", type=str, required=True)
parser.add_argument("-a", "--max_alpha_square", type=float, required=True)
parser.add_argument("-n", "--number_of_landmarks", type=int, required=True)
parser.add_argument("-d", "--limit_dimension", type=int, required=True)
args = parser.parse_args()
with open(args.file, 'r') as f:
first_line = f.readline()
if (first_line == 'OFF\n') or (first_line == 'nOFF\n'):
print("#####################################################################")
print("EuclideanWitnessComplex creation from points read in a OFF file")
witnesses = gudhi.read_points_from_off_file(off_file=args.file)
landmarks = gudhi.pick_n_random_points(points=witnesses, nb_points=args.number_of_landmarks)
message = "EuclideanWitnessComplex with max_edge_length=" + repr(args.max_alpha_square) + \
" - Number of landmarks=" + repr(args.number_of_landmarks)
print(message)
witness_complex = gudhi.EuclideanWitnessComplex(witnesses=witnesses, landmarks=landmarks)
simplex_tree = witness_complex.create_simplex_tree(max_alpha_square=args.max_alpha_square,
limit_dimension=args.limit_dimension)
message = "Number of simplices=" + repr(simplex_tree.num_simplices())
print(message)
else:
print(args.file, "is not a valid OFF file")
f.close()
```

## Example2: Computing persistence using strong relaxed witness complex#

Here is an example of constructing a strong witness complex filtration and computing persistence on it: