Shape of Data: Algebraic Topology and Shape Representation

This lecture introduces the concept of studying the 'topology of data sets' to efficiently represent and measure shape. It covers algebraic topology, Betti numbers, homology, and various methods for representing shape, such as Čech, Vietoris-Rips, and Deleaunay complexes. The instructor explains how homology provides signatures for different features and how complexes offer compressed representations of infinite shape information. The lecture also delves into measuring the shape of data through persistence vector spaces and module classification, with applications to natural image statistics and high-dimensional image analysis.