Sequential Low-Rank Change Detection

TL;DR

This paper introduces a new method for detecting low-rank changes in high-dimensional data using sketching techniques and eigenvalue statistics, enabling efficient online detection without full covariance matrix recovery.

Contribution

It proposes a sketching-based approach for low-rank change detection that reduces dimensionality while maintaining detection performance, with theoretical analysis and an online implementation.

Findings

High-probability equivalence of ranks between original and sketched covariance matrices.

Performance characterization of the eigenvalue statistic in terms of false alarms and detection delay.

Efficient online detection via subspace tracking.

Abstract

Detecting emergence of a low-rank signal from high-dimensional data is an important problem arising from many applications such as camera surveillance and swarm monitoring using sensors. We consider a procedure based on the largest eigenvalue of the sample covariance matrix over a sliding window to detect the change. To achieve dimensionality reduction, we present a sketching-based approach for rank change detection using the low-dimensional linear sketches of the original high-dimensional observations. The premise is that when the sketching matrix is a random Gaussian matrix, and the dimension of the sketching vector is sufficiently large, the rank of sample covariance matrix for these sketches equals the rank of the original sample covariance matrix with high probability. Hence, we may be able to detect the low-rank change using sample covariance matrices of the sketches without having to recover the original covariance matrix. We character the performance of the largest eigenvalue statistic in terms of the false-alarm-rate and the expected detection delay, and present an efficient online implementation via subspace tracking.