PETRELS: Parallel Subspace Estimation and Tracking by Recursive Least Squares from Partial Observations

TL;DR

PETRELS is an online algorithm for low-rank subspace estimation and tracking from partial observations, suitable for high-dimensional data streams with applications in video denoising, network monitoring, and anomaly detection.

Contribution

It introduces PETRELS, a recursive least squares based method for parallel subspace estimation and tracking in streaming data with missing entries, improving efficiency over batch methods.

Findings

PETRELS effectively tracks changing subspaces in real-time.

It outperforms batch algorithms in speed and accuracy.

Demonstrated on direction-of-arrival estimation and matrix completion tasks.

Abstract

Many real world data sets exhibit an embedding of low-dimensional structure in a high-dimensional manifold. Examples include images, videos and internet traffic data. It is of great significance to reduce the storage requirements and computational complexity when the data dimension is high. Therefore we consider the problem of reconstructing a data stream from a small subset of its entries, where the data is assumed to lie in a low-dimensional linear subspace, possibly corrupted by noise. We further consider tracking the change of the underlying subspace, which can be applied to applications such as video denoising, network monitoring and anomaly detection. Our problem can be viewed as a sequential low-rank matrix completion problem in which the subspace is learned in an on-line fashion. The proposed algorithm, dubbed Parallel Estimation and Tracking by REcursive Least Squares (PETRELS), first identifies the underlying low-dimensional subspace via a recursive procedure for each row of the subspace matrix in parallel with discounting for previous observations, and then reconstructs the missing entries via least-squares estimation if required. Numerical examples are provided for direction-of-arrival estimation and matrix completion, comparing PETRELS with state of the art batch algorithms.