Learning on topological and geometrical structures of data.

Topological data analysis (TDA) is a recent methodology for extracting topological and geometrical features from complex geometric data structures. Persistent homology, a new mathematical notion proposed by Edelsbrunner (2002), provides a multiscale descriptor for the topology of data, and has been recently applied to a variety of data analysis. In this talk I will introduce a machine learning framework of TDA by combining persistence homology and kernel methods. As an expression of persistent homology, persistence diagrams are widely used to express the lifetimes of generators of homology groups. While they serve as a compact representation of data, it is not straightforward to apply standard data analysis to persistence diagrams, since they consist of a set of points in 2D space expressing the lifetimes. We introduce a method of kernel embedding of the persistence diagrams to obtain their vector representation, which enables one to apply any kernel methods in topological data analysis, and propose a persistence weighted Gaussian kernel as a suitable kernel for vectorization of persistence diagrams. Some theoretical properties including Lipschitz continuity of the embedding are also discussed. I will also present applications to change point detection and time series analysis in the field of material sciences and biochemistry.