Topological data analysis

Summary

Topological data analysis studies the shape of data with tools from algebraic topology. It turns a finite set of points, usually in a metric space, into a family of simplicial complexes indexed by a scale. It then records how topological features such as components and loops appear and disappear as the scale grows. The main tool is persistent homology, whose output, a barcode or persistence diagram, is stable under small perturbations of the data. It is used to find clusters, loops and voids in data from biology, materials science and image analysis.