Vietoris–Rips complex

노트

위키데이터

• ID : Q7928689

말뭉치

1. The Vietoris-Rips complex characterizes the topology of a point set.
2. The Vietoris–Rips complex of M 3 , for δ = 1, includes a simplex for every subset of points in M 3 , including a triangle for M 3 itself.
3. The Vietoris–Rips complex for δ = 1 contains an edge for every pair of points that are at unit distance or less in the given metric space.
4. As with unit disk graphs, the Vietoris–Rips complex has been applied in computer science to model the topology of ad hoc wireless communication networks.
5. Dionysus can compute Vietoris–Rips complexes.
6. We study Vietoris–Rips complexes of metric wedge sums and metric gluings.
7. We show that the Vietoris–Rips complex of a wedge sum, equipped with a natural metric, is homotopy equivalent to the wedge sum of the Vietoris–Rips complexes.
8. Given a metric space X and a distance threshold r > 0 , the Vietoris–Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r .
9. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris–Rips complex is homotopy equivalent to the original manifold.
10. Little is known about the behavior of Vietoris–Rips complexes for larger values of r , even though these complexes arise naturally in applications using persistent homology.
11. Now your first objection should be that computing a Vietoris-Rips complex still requires exponential time, because you have to scan all subsets for the possibility that they form a simplex.
12. It’s true, but one nice thing about the Vietoris-Rips complex is that it can be represented implicitly as a graph.
13. In a future post we’ll implement a method for speeding up the computation of the Vietoris-Rips complex, since this is the primary bottleneck for topological data analysis.
14. To establish these results, we describe the topology of the Vietoris-Rips filtrations arising from evolutionary histories indexed by galled trees.

메타데이터

위키데이터

• ID : Q7928689

Spacy 패턴 목록

• [{'LOWER': 'vietoris'}, {'OP': '*'}, {'LOWER': 'rips'}, {'LEMMA': 'complex'}]