Supervised learning with indefinite topological Kernels

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