  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.
