Fast Topological Data Analysis for computationally efficient feature extraction in nonstationary time series
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 ($H_1$) from two-dimensional 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 ($\alpha$-complex) filtrations across noisy synthetic harmonic signals, experimental high-rate Dynamic Reproduction of Projectiles in Ballistic Environments for Advanced Research (DROPBEAR) 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, $\alpha$-complex persistence provides the lowest runtimes in the tested two-dimensional 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.