Stable Topological Feature Vectors via Hermite Function Expansion on Persistence Curves

Abstract

Topological data analysis (TDA) is a rising field in the machine learning and has been proven useful in several scientific disciplines. Persistence diagrams are one of main tools in TDA. However, the space of persistence diagram lacks desirable structure for machine learning algorithms. Transforming the space of persistence diagrams into other space is therefore an active research area in TDA. The recently developed persistence framework transforms persistence diagrams into functions and has shown promising performance. In this article, we derive a grid-free representation of persistence curves that is efficient to compute and effective in machine learning tasks. Towards this end, we consider the coefficients of Hermite function expansion on persistence curves. The main contribution of this article is twofold. First, we find the explicit expression of the coefficients, derive their recursive relation for efficient computation, and prove the stability result. Second, we apply these coefficients on two classification problems in texture analysis, and sleep stages. We find that the performance is comparable with existing methods or in some cases outperforming the state-of-arts method.