Supervised Learning with Indefinite Topological Kernels

Topological Data Analysis (TDA) is a new branch of statistics devoted to the study of the ‘shape’ of the data. As TDA's tools are typically defined in complex spaces, kernel methods are often used to perform inferential task by implicitly mapping topological summaries, most noticeably the Persistence Diagram (PD), to vector spaces. For positive definite kernels defined on PDs, however, kernel embeddings do not fully retain the metric structure of the original space. We introduce a new exponential kernel, built on the geodesic space of PDs, and we show with simulated and real applications how it can be successfully used in regression and classification tasks, despite not being positive definite.