Streaming Persistent Homology for Real-Time High-Velocity Data

While Liza et al. (2025) address computational efficiency via the Lazy Witness Complex for large static datasets, the challenge of streaming high-dimensional data remains open. This research proposes incremental and distributed algorithms for updating persistence diagrams as new data points arrive, possibly leveraging sketching and reservoir sampling. The novelty lies in maintaining topological summaries in real time, with provable bounds on approximation and memory use. This would be transformative for applications like financial tick data, sensor networks, or social media streams, where the topology of the data evolves rapidly and timely detection of topological anomalies or regime shifts is critical.

References:

1. Exploring the Lazy Witness Complex for Efficient Persistent Homology in Large-Scale Data. Mst Zinia Afroz Liza, Md. Al-Imran, M. Shiraj, Tozam Hossain, Masum Murshed, Nasima Akhter (2025). Tensor: Pure and Applied Mathematics Journal.