Topological methods for data modelling

The rapidly developing field of topological data analysis represents data via graphs rather than as solutions to equations or as decompositions into clusters. This Review discusses the methods and provides examples from physics and other sciences. The analysis of large and complex data sets is one of the most important problems facing the scientific community, and physics in particular. One response to this challenge has been the development of topological data analysis (TDA), which models data by graphs or networks rather than by linear algebraic (matrix) methods or cluster analysis. TDA represents the shape of the data (suitably defined) in a combinatorial fashion. Methods for measuring shape have been developed within mathematics, providing a toolkit referred to as homology. In working with data, one can use this kind of modelling to obtain an understanding of the overall structure of the data set. There is a suite of methods for constructing vector representations of various kinds of unstructured data. In this Review, we sketch the basics of TDA and provide examples where this kind of analysis has been carried out.