Time-optimal persistent homology representatives for univariate time series

TL;DR

The paper addresses interpretability in persistent homology for time-varying data by introducing time-optimal PH cycle representatives for univariate time series. It defines two notions—vertex-based and simplex-based time-optimal cycles—via a time-dispersion objective and a linear programming formulation using an $L_1$-norm, with a relaxation parameter $\varepsilon$ controlling the minimum persistence. The approach relies on sliding-window embeddings and Vietoris–Rips filtrations to produce cycle representatives that align with temporal proximity, demonstrated on quasi-periodic signals and ENSO models where representatives span roughly one period. Overall, the results indicate comparable performance between simplex- and vertex-based methods, with vertex-based optimization often performing better, and the work opens avenues for higher-degree PH representatives and extension to time-varying networks; the authors provide code at GitHub.

Abstract

Persistent homology (PH) is one of the main methods used in Topological Data Analysis. An active area of research in the field is the study of appropriate notions of PH representatives, which allow to interpret the meaning of the information provided by PH, making it an important problem in the application of PH, and in the study of its interpretability. Computing optimal PH representatives is a problem that is known to be NP-hard, and one is therefore interested in developing context-specific optimality notions that are computable in practice. Here we introduce time-optimal PH representatives for time-varying data, allowing one to extract representatives that are close in time in an appropriate sense. We illustrate our methods on quasi-periodic synthetic time series, as well as time series arising from climate models, and we show that our methods provide optimal PH representatives that are better suited for these types of problems than existing optimality notions, such as length-optimal PH representatives.

Figures (2)

• Figure 1: Pipeline for computing time-optimal PH cycle representatives for univariate time series. A) One starts by embedding a univariate time series into Euclidean space. For quasi-periodic times series, one can obtain an embedding together with a lower bound ($\alpha$) on persistence that allows to distinguish between components, holes, voids, that one may consider as "significant", and features that may be due to noise. B) Given the significant PH features, we attempt to find representatives that are optimal with respect to their cohesion in time. Given representatives of the same PH feature (e.g., green and orange cycles on the left), the ones that correspond to a continuous trajectory (green) in the original signal are preferred over the ones that are discontinuous (orange).
• Figure 2:

Theorems & Definitions (7)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Example 3.6
• Example 3.7