Vietoris-Rips Complex

Consider a discrete subset $\Omega \subset \mathbb{R}^n$ of Eucledian space. If one receives $\Omega$ as a collection of data points — a point cloud — sampled from a submanifold, it may be desirable to reconstruct the underlying submanifold from the point cloud using a simplicial complex.

For chosen constraint $\varepsilon > 0$, the Vietoris-Rips complex of scale $\varepsilon$ on point cloud $\Omega$, written $\mathrm{VR}_\varepsilon(\Omega)$, is the simplicial complex whose simplices are all those finite collections of points in $\Omega$ of pairwise-distance $d \leq \varepsilon$.

By sampling while varying pairwise-distance $\varepsilon_1 \leq \varepsilon_2 \leq \cdots \leq \varepsilon_k$, we obtain a grading:

$$\mathrm{VR}_{\varepsilon_1} \subseteq \mathrm{VR}_{\varepsilon_2} \subseteq \cdots \subseteq \mathrm{VR}_{\varepsilon_k}$$

which induces a grading in chain complexes:

$$\mathcal{C}_\bullet\big(\mathrm{VR}_{\varepsilon_1}\big) \hookrightarrow \mathcal{C}_\bullet\big(\mathrm{VR}_{\varepsilon_2}\big) \hookrightarrow \cdots \hookrightarrow \mathcal{C}_\bullet\big(\mathrm{VR}_{\varepsilon_k}\big)$$


Topics in Topological Data Analysis

Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License