Persistence Paths and Signature Features in Topological Data Analysis

In this talk I will introduce the concept of the path signature and motivate its recent use in analysis of time-ordered data. I will then describe a new feature map for barcodes in persistent homology by first realizing each barcode as a path in a vector space, and then computing its signature which takes values in the tensor algebra over that vector space. The composition of these two operations — barcode to path, path to tensor series — results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness.