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

2025-02-22

AI Summary

This paper addresses the challenge of jointly modeling multi-scale static topological structures and dynamic evolutionary patterns in time-varying data. We propose Quasi-Zigzag Persistent Homology (QZPH), a novel framework that introduces, for the first time, a theoretically stable and computationally feasible multiparameter Zigzag-type topological invariant. QZPH unifies the characterization of static topology across scales and temporal evolution patterns via zigzag filtration construction, index lattice reduction, and an efficient boundary matrix algorithm. Unlike conventional persistent homology, QZPH explicitly integrates the temporal dimension with multi-resolution geometric information. In sleep staging—a canonical time-series classification task—QZPH significantly improves both classification accuracy and generalization performance across diverse machine learning models. Empirical results demonstrate its superior capacity to capture intrinsic evolutionary dynamics of real-world time-series data.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

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-evolving datasets.

Problem

• Proposes Quasi Zigzag Persistent Homology for time-varying data analysis
• Introduces a stable topological invariant for static and dynamic features
• Demonstrates effectiveness in enhancing machine learning models

Innovation

• Integrates multiparameter and zigzag persistence
• Introduces stable topological invariant
• Efficient algorithm for invariant computation