Quasi Zigzag Persistence: A Topological Framework for Analyzing Time-Varying Data

TL;DR

This work introduces Quasi Zigzag Persistence (QZPH) to analyze time-varying data by blending multiparameter persistence with zigzag persistence over the quasi zigzag poset $\mathbb{ZZ}$. It defines Zz-Gril, a stable generalized-rank landscape computed via worm-shaped subposets, and provides an efficient algorithm that builds a quasi zigzag bi-filtration and evaluates Zz-Gril signatures along grid centers. The approach yields robust topological features that augment machine learning models, demonstrated by improved sleep-stage classification and multivariate time-series classification on real datasets. The combination of a stable invariant, boundary-based computation, and scalable filtering makes QZPH a practical tool for dynamic topological data analysis with broad applicability.

Abstract

In this paper, we propose Quasi Zigzag Persistent Homology (QZPH) as a framework for analyzing time-varying data by integrating multiparameter persistence and zigzag persistence. To this end, we introduce a stable topological invariant that captures both static and dynamic features at different scales. We present an algorithm to compute this invariant efficiently. We show that it enhances the machine learning models when applied to tasks such as sleep-stage detection, demonstrating its effectiveness in capturing the evolving patterns in time-varying datasets.

Figures (7)

• Figure 1: (left) A PCD (bottom row) and a resulting quasi zigzag bi-filtration; (middle) corresponding quasi zigzag persistence module and three intervals (light blue, yellow, green); (right) a rectangle (light yellow) is expanded for obtaining landscape width.
• Figure 2: The worm with blue boundary represents a worm centered at $\mathbf{p}$ with width $\delta=1$. The worm with red boundary represents an expanded worm, centered at $\mathbf{p}$ with width $\delta=2$.
• Figure 3: The top row shows a typical input to the Zz-Gril framework as a sequence of graphs. This sequence has 3 graphs. Observe that consecutive graphs cannot be linked by an inclusion in either direction because neither is a subgraph of the other. In order to circumvent this problem, we consider the unions of consecutive graphs as an intermediary step, as shown in the bottom row.
• Figure 4: A worm $I$ (shaded) on a quasi zigzag bi-filtration where the direction of the inclusion on horizontal arrows are shown at the bottom and on the vertical arrows on left. The worm centered at (3,3) has width 1. The part of the boundary colored green is $\partial_LI$ connecting the minima shown as green points and the part in purple is $\partial_UI$ connecting the maxima shown as purple points. The boundary cap is shown in orange which runs parallel to the boundary for a portion of it. Refer to Figure \ref{['fig:zz_bdry']} for the zigzag filtration along the boundary cap of the worm. A sequence of graphs with $T=3$ time steps is shown with the corresponding graph filtrations: $\mathcal{F}_{t_1}, \mathcal{F}_{t_2}, \mathcal{F}_{t_3}$ each with $L=5$ levels. The filtration of the unions are encircled by ovals. Zigzag filtration at the topmost level $\mathcal{Z}_L$ is shown in a rectangular box.
• Figure 5: This is the zigzag filtration $\mathcal{Z}_{bdry}$ along the boundary cap of the worm shown in Figure \ref{['fig:worm_qzz_bifil']}.

Theorems & Definitions (33)

• Definition 2.1: Persistence module
• Definition 2.2: Generalized rank
• Definition 3.1: Zigzag poset
• Definition 3.2: Quasi zigzag persistence module
• Definition 3.3: Worm
• Definition 3.4: Zz-Gril
• Definition 3.5
• Proposition 3.6
• Theorem 3.7
• Definition 4.1