Anomaly-Driven Topological Signatures: A Universal Framework for Unexpected Feature Detection in Time Series

Building on Heo & Jung (2024), who tailor persistent homology (PH) analysis to specific domains by incorporating domain knowledge and influence vectors, this idea flips the perspective: instead of fine-tuning PH for known features, design a framework that learns "expected" topological signatures from large corpora of normal data, then automatically highlights significant deviations as potential anomalies. This approach introduces a meta-learning layer that abstracts away domain-specific features, focusing on the universality of "anomalous topology"—think of it as a persistent homology-based anomaly radar. Unlike PhoGAD (Yuan et al., 2024), which optimizes PH for graph-based anomaly detection in networks, this framework generalizes to arbitrary time series or sequential data, including cases where domain knowledge is sparse. The novelty is in developing tools to quantify and visualize the "unexpectedness" of topological features, potentially uncovering new classes of anomalies invisible to classical statistical or signal-based methods. This could be transformative for exploratory data analysis in fields ranging from finance to neuroscience.

References:

1. Persistent homology of featured time series data and its applications. Eunwoo Heo, Jae-Hun Jung (2024). AIMS Mathematics.
2. PhoGAD: Graph-based Anomaly Behavior Detection with Persistent Homology Optimization. Ziqi Yuan, Haoyi Zhou, Tianyu Chen, Jianxin Li (2024). Web Search and Data Mining.