Supervised Learning with Indefinite Topological Kernels

TL;DR

This paper explores the use of topological data analysis in supervised learning by defining a novel topological exponential kernel that, despite not being positive semi-definite, effectively supports regression and classification tasks.

Contribution

It introduces a new topological exponential kernel for TDA summaries, expanding the toolkit for applying TDA in supervised learning.

Findings

The kernel can be used successfully in regression tasks.

The kernel supports classification tasks.

It is not positive semi-definite but still effective.

Abstract

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 be successfully used in regression and classification tasks.