Topological Machine Learning with Persistence Indicator Functions

Abstract: Techniques from computational topology, in particular persistent homology, are becoming increasingly relevant for data analysis. Their stable metrics permit the use of many distance-based data analysis methods, such as multidimensional scaling, while providing a firm theoretical ground. Many modern machine learning algorithms, however, are based on kernels. This paper presents persistence indicator functions (PIFs), which summarize persistence diagrams, i.e., feature descriptors in topological data analysis. PIFs can be calculated and compared in linear time and have many beneficial properties, such as the availability of a kernel-based similarity measure. We demonstrate their usage in common data analysis scenarios, such as confidence set estimation and classification of complex structured data.