Fast topological data analysis for computationally efficient feature extraction in nonstationary time series
TL;DR
Results position Fast TDA as a scalable geometry-guided surrogate for real-time feature extraction when VR-based persistence is computationally prohibitive, and noise-resilient topology-informed descriptors are desired.
Abstract
Topological data analysis (TDA) is a powerful tool for characterizing nonlinear and nonstationary dynamics, but the rapid computational growth of persistent homology filtrations, particularly Vietoris–Rips (VR) constructions, limits its use in high-rate and edge-computing applications. This paper introduces Fast topological data analysis (Fast TDA), a geometry-based framework designed to approximate one-dimensional persistent homology ( H1) from two-dimensional (2D) delay-coordinate embeddings with reduced computational cost. Fast TDA bypasses explicit filtration construction by utilizing a pipeline of least-squares ellipse fitting and k-dimensional tree (KD-tree)-based inscribed circle extraction to achieve efficient geometry-guided feature extraction. The method is formulated for both single-loop regimes and multi-loop point-cloud structures, reconstructing multiple geometric primitives to track evolving topologies. The performance of Fast TDA is evaluated against persistent homology computed with VR and alpha-complex ( α-complex) filtrations across noisy synthetic harmonic signals, experimental high-rate Dynamic Reproduction of Projectiles in Ballistic Environments for Advanced Research testbed data, and the chaotic Lorenz attractor. Results demonstrate that, for single-harmonic regimes, Fast TDA operates in the sub-millisecond range, reducing runtime relative to both persistence-based baselines evaluated in this study while exhibiting strong noise robustness. For multi-frequency and complex multi-loop signals, α-complex persistence provides the lowest runtimes in the tested 2D embeddings, but its extracted descriptors remain more sensitive to noise in the multi-loop cases considered. In contrast, Fast TDA remains substantially faster than VR-based persistence and maintains stronger noise resilience while preserving sub-second execution. Overall, these results position Fast TDA as a scalable geometry-guided surrogate for real-time feature extraction when VR-based persistence is computationally prohibitive, and noise-resilient topology-informed descriptors are desired.