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

Jonghyeok Lee, Yao Xie, Youngser Park, Jason Hindes, Ira Schwartz, Carey Priebe

TL;DR

The Multi-rank Subspace-CUSUM (MRS-C) procedure is proposed, which extends classical CUSUM by tracking projection energy onto an estimated signal subspace by tracking projection energy onto an estimated signal subspace.

Abstract

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.

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

TL;DR

The Multi-rank Subspace-CUSUM (MRS-C) procedure is proposed, which extends classical CUSUM by tracking projection energy onto an estimated signal subspace by tracking projection energy onto an estimated signal subspace.

Abstract

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.

Paper Structure

This paper contains 21 sections, 4 theorems, 61 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Problem setup
  4. Motivating application: robotic swarm monitoring
  5. Multi-rank Subspace-CUSUM
  6. Oracle benchmark: Exact CUSUM
  7. The MRS-C procedure
  8. Theoretical analysis
  9. Drift parameter
  10. Asymptotic EDD and oracle efficiency
  11. Parallel procedure for dimension selection
  12. Numerical experiments
  13. Simulation study: ARL calibration and baseline comparison
  14. Parallel procedure with unknown subspace dimension
  15. Robotic swarm monitoring
  16. ...and 6 more sections

Key Result

Lemma 4.3

Under Assumption assumption:covariance_structure, the expected increment under the post-change regime is given by, as $w \to \infty$,

Figures (4)

  • Figure 1: ARL versus EDD for three detectors under varying noise levels.
  • Figure 2: Snapshot of the synthetic milling swarm configuration at frame $t=6001$. Red dots denote agent positions, and blue arrows indicate instantaneous velocity vectors.
  • Figure 3: Detection statistics for the swarm dataset, zoomed in on the physical-time interval 780-820 s, as produced by different change-point detection methods. Both the MRS-C and iso-mirror methods identify the change-point at approximately time 800 s.
  • Figure 4: Illustration of swarm behavior before, during, and after the detected change-point from the UAVSwarm-13 sequence.

Theorems & Definitions (11)

  • Remark 2.1: Geometric interpretation via principal angles
  • Remark 4.2: Eigenvalue multiplicity
  • Lemma 4.3: Post-change increment
  • Remark 4.4
  • Proposition 4.5: First-order optimal EDD for arbitrary window length
  • Corollary 4.6: Optimal parameters
  • Theorem 4.7: First-order asymptotic optimality
  • Remark 4.8: Difference from rank-one case
  • proof : Proof of Lemma \ref{['lem:post_change_increment']}
  • proof : Proof of Proposition \ref{['thm:arl-edd']}
  • ...and 1 more