Multi-Rank Subspace Change-Point Detection for Monitoring Robotic Swarms

We study real-time detection of low-rank changes in the covariance structure of high-dimensional streaming data, motivated by robotic swarm monitoring. Building on the spiked covariance model, we propose the Multi-rank Subspace-CUSUM (MRS-C) procedure, which extends classical CUSUM by tracking projection energy onto an estimated signal subspace. We analyze performance by characterizing the expected detection delay (EDD) under a prescribed average run length (ARL), deriving closed-form asymptotically optimal choices of the window size and drift. We further prove that MRS-C is first-order asymptotically optimal relative to the oracle Exact CUSUM, with an explicit efficiency constant that depends on heterogeneity in spike strengths. When the signal rank is unknown, we use a parallel procedure. Simulations and robotic swarm-behavior data illustrate robustness and effectiveness.