Stability of the persistence transformation

TL;DR

This paper introduces the persistence transformation, a new stable method in Topological Data Analysis for analyzing time series data, which captures signal persistence and positional information, with a reduced version offering faster computation for specific applications.

Contribution

The paper presents the persistence transformation and its reduced form, demonstrating their stability, computational efficiency, and applicability to various domains, extending TDA techniques for time series analysis.

Findings

Persistence transformation is stable and outperforms traditional persistent diagrams.

Reduced persistence transformation offers faster computation with some accuracy trade-offs.

Applicable to domains like MALDI-Imaging where position matters more than signal height.

Abstract

In this paper, we introduce the persistence transformation, a novel methodology in Topological Data Analysis (TDA) for applications in time series data which can be obtained in various areas such as science, politics, economy, healthcare, engineering, and beyond. This approach captures the enduring presence or `persistence' of signal peaks in time series data arising from Morse functions while preserving their positional information. Through rigorous analysis, we demonstrate that the proposed persistence transformation exhibits stability and outperforms the persistent diagram of Morse functions (with respect to filtration, e.g., the upper levelset filtration). Moreover, we present a modified version of the persistence transformation, termed the reduced persistence transformation, which retains stability while enjoying dimensionality reduction in the data. Consequently, the reduced persistence transformation yields faster computational results for subsequent tasks, such as classification, albeit at the cost of reduced overall accuracy compared to the persistence transformation. However, the reduced persistence transformation finds relevance in specific domains, e.g., MALDI-Imaging, where positional information is of greater significance than the overall signal height. Finally, we provide a conceptual outline for extending the persistence diagram to accommodate higher-dimensional input while assessing its stability under these modifications.