Barcodes: The persistent topology of data

This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets — persistent homology — and a novel representation of this algebraic characterization — barcodes. We sketch an application of these techniques to the classification of natural images. 1. The shape of data When a topologist is asked, “How do you visualize a four-dimensional object?” the appropriate response is a Socratic rejoinder: “How do you visualize a threedimensional object?” We do not see in three spatial dimensions directly, but rather via sequences of planar projections integrated in a manner that is sensed if not comprehended. We spend a significant portion of our first year of life learning how to infer three-dimensional spatial data from paired planar projections. Years of practice have tuned a remarkable ability to extract global structure from representations in a strictly lower dimension. The inference of global structure occurs on much finer scales as well, with regards to converting discrete data into continuous images. Dot-matrix printers, scrolling LED tickers, televisions, and computer displays all communicate images via arrays of discrete points which are integrated into coherent, global objects. This also is a skill we have practiced from childhood. No adult does a dot-to-dot puzzle with anything approaching anticipation. 1.1. Topological data analysis. Problems of data analysis share many features with these two fundamental integration tasks: (1) how does one infer high dimensional structure from low dimensional representations; and (2) how does one assemble discrete points into global structure. The principal themes of this survey of the work of Carlsson, de Silva, Edelsbrunner, Harer, Zomorodian, and others are the following: (1) It is beneficial to replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter. This converts the data set into global topological objects. (2) It is beneficial to view these topological complexes through the lens of algebraic topology — specifically, via a novel theory of persistent homology adapted to parameterized families. (3) It is beneficial to encode the persistent homology of a data set in the form of a parameterized version of a Betti number: a barcode. The author gratefully acknowledges the support of DARPA # HR0011-07-1-0002. The work reviewed in this article is funded by the DARPA program TDA: Topological Data Analysis.