Vietoris–Rips Complexes¶

Dionysus can compute Vietoris–Rips complexes. Given a point set $P$ , a Vietoris–Rips complex consists of all those simplices whose vertices are at pairwise distance no more than $r$ , $V R_{r} (P) = {σ \subseteq P ∣ \forall u, v \in σ, ‖ u - v ‖ \leq r}$ .

fill_rips() computes Vietoris–Rips filtrations (up to a specified skeleton dimension and distance $r$ ). It accepts points as NumPy arrays, following the standard convention that rows of a 2-dimensional array are interpreted as points in Euclidean space:

>>> import numpy as np
>>> points = np.random.random((100,2))
>>> f = d.fill_rips(points, 2, .3)
>>> print(f)
Filtration with 5974 simplices
>>> for s in f:
...     print(s)
<0> 0
...
<9,61,92> 0.299856
<9,72,92> 0.299856
<9,82,92> 0.299856

fill_rips() also accepts condensed distance matrices (linearized lower triangular part of a symmetric matrix):

>>> from scipy.spatial.distance import pdist
>>> dists = pdist(points)
>>> f = d.fill_rips(dists, 2, .3)
>>> print(f)
Filtration with 5974 simplices

SciPy provides a helper function squareform to convert between redundant square matrices ( $n \times n$ ) and condensed matrices (vectors with $(\binom{n}{2})$ elements).

>>> from scipy.spatial.distance import squareform
>>> sq_dist = squareform(dists)
>>> print(sq_dist.shape)
(100, 100)
>>> print(squareform(sq_dist).shape)
(4950,)