Vietoris-Rips complex

The Vietoris-Rips complex is a mathematical way to analyze the shape of data points. Imagine connecting points that are close enough, forming lines, triangles, and higher-dimensional shapes. If all points in a triangle are pairwise connected, the triangle becomes a filled-in 2D shape. By doing this at various distance thresholds, we can uncover the underlying structure, like clusters or holes, in the data. It's commonly used in topological data analysis to help understand complex data by translating spatial relationships into a hierarchical network of shapes that reveal persistent patterns.