Stability of the persistence transformation

TL;DR

A modified version of the persistence transformation is presented, termed the reduced persistence transformation, which retains stability while enjoying dimensionality reduction in the data, and yields faster computational results for subsequent tasks, albeit at the cost of reduced overall accuracy.

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.

Figures (4)

• Figure 1: Example of the persistence transformation. The blue line represents all the trivial features, which are vanishing since they are on the diagonal plane. The red dots represent the relevant features. The distance of the dots to the diagonal plane indicates their persistence.
• Figure 2: Problem of the persistence diagram: the middle column displays two symmetric functions $f,g:\mathbb{R}\rightarrow \mathbb{R}$. The first column displays the persistence transformation of these functions, with the positional information on the $x$-axis and the birth values on the $y$-axis. The death values are encoded in the color scheme. The persistence transformations of the functions can be distinguished. The right-most column shows the persistence diagram of the upper levelset filtration with the birth value on the $y$-axis and the death value on the $x$-axis. The two diagrams are identical. (Graphic: klaila2023supervised)
• Figure 3: Example of the reduced persistence transformation. The original spectra is displayed in black, while the values of the reduced persistence transformation are illustrating in red the persistence of each feature.
• Figure 4: Example of the reduced persistence transformation: In the middle, two symmetric functions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ are displayed. The left-hand side illustrates the reduced persistence transformation of this graphs, with the positional information on the $x$-axis and the persistence information on the $y$-axis. The graphs can be distinguished. The right-hand side depicts the persistence diagram of the upper levelset filtration, with the birth value on the $y$-axis and the death value on the $x$-axis. The diagrams are identical. (Graphic: klaila2023supervised)

Theorems & Definitions (13)

• Theorem 1
• proof
• Definition 1
• Theorem 2
• proof
• Theorem 3
• proof
• Definition 2
• Theorem 4
• proof