Topological Data Analysis (TDA) is a recent and growing branch of statistics devoted to the
study of the shape of the data. In this work we investigate the predictive power of TDA in the
context of supervised learning. Since topological summaries, most noticeably the
Persistence Diagram, are typically defined in complex spaces, we adopt a kernel approach
to translate them into more familiar vector spaces. We define a topological exponential
kernel, we characterize it, and we show that, despite not being positive semi-definite, it can …

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 …