Introduction to Persistent Homology (Cech and Vietoris-Rips complex)

Motivation
Data is commonly represented as an unordered sequence of points in the Euclidean space $\mathbb{R}^n$ . The global `shape’ of the data may provide important information about the underlying phenomena of the data.

For data points in $\mathbb{R}^2$ , determining the global structure is not difficult, but for data in higher dimensions, a planar projection can be hard to decipher.
From point cloud data to simplicial complexes
To convert a collection of points $\{x_\alpha\}$ in a metric space into a global object, one can use the points as the vertices of a graph whose edges are determined by proximity (vertices within some chosen distance $\epsilon$ ). Then, one completes the graph to a simplicial complex. Two of the most natural methods for doing so are as follows:

Given a set of points $\{x_\alpha\}$ in Euclidean space $\mathbb{R}^n$ , the Cech complex (also known as the nerve), $\mathcal{C}_\epsilon$ , is the abstract simplicial complex where a set of $k+1$ vertices spans a $k$ -simplex whenever the $k+1$ corresponding closed $\epsilon/2$ -ball neighborhoods have nonempty intersection.

Given a set of points $\{x_\alpha\}$ in Euclidean space $\mathbb{R}^n$ , the Vietoris-Rips complex, $\mathcal{R}_\epsilon$ , is the abstract simplicial complex where a set $S$ of $k+1$ vertices spans a $k$ -simplex whenever the distance between any pair of points in $S$ is at most $\epsilon$ .

Top left: A fixed set of points. Top right: Closed balls of radius $\epsilon/2$ centered at the points. Bottom left: Cech complex has the homotopy type of the $\epsilon/2$ cover ( $S^1\vee S^1\vee S^1$ ) Bottom right: Vietoris-Rips complex has a different homotopy type ( $S^1\vee S^2$ ). Image from R. Ghrist, 2008, Barcodes: The Persistent Topology of Data.

Author: mathtuition88

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