Čech complexes are one of the main tools for building a simplicial complex from data and are used regularly for Persistent Homology. This post provides an interactive demonstration of the Čech complex for a 2-dimensional data set. It is meant to help newcomers gain an intuition about the complex.

Visualization

The resulting simplicial complex is overlaid on top of the covers. Here we only calculate the nerve up to order 2, i.e. vertices, edges, and triangles.

• the slider lets you change the radius of the balls and
• the data points can be rearranged by dragging them.

Mathematical Background

The Čech complex of a data set \(X\) with radius \(r\) is defined as the simplicial complex

$$ Č(X) = \big\{ \sigma_I \mid \bigcup_{i \in I} B_r(x_i) \neq \emptyset \big\} $$

All this really says is we add a simplex \(\sigma_I\) on vertices \( \{ x_i\}_{i\in I} \) when balls with radius \(r\) around each vertex all have a nonempty intersection.