Online (and Offline) Robust PCA: Novel Algorithms and Performance Guarantees

TL;DR

This paper introduces a novel online RPCA algorithm based on ReProCS, providing the first comprehensive correctness guarantees under mild assumptions, advancing real-time data separation in video analytics.

Contribution

It develops an improved online RPCA algorithm with complete performance guarantees, overcoming key limitations of previous batch and online methods.

Findings

Algorithm achieves accurate separation of low-rank and sparse components in real-time.

Provides the first full correctness guarantees for online RPCA.

Outperforms previous methods in accuracy and robustness under mild assumptions.

Abstract

In this work, we study the online robust principal components' analysis (RPCA) problem. In recent work, RPCA has been defined as a problem of separating a low-rank matrix (true data), $L$, and a sparse matrix (outliers), $S$, from their sum, $M:=L + S$. A more general version of this problem is to recover $L$ and $S$ from $M:=L + S + W$ where $W$ is the matrix of unstructured small noise/corruptions. An important application where this problem occurs is in video analytics in trying to separate sparse foregrounds (e.g., moving objects) from slowly changing backgrounds. While there has been a large amount of recent work on solutions and guarantees for the batch RPCA problem, the online problem is largely open."Online" RPCA is the problem of doing the above on-the-fly with the extra assumptions that the initial subspace is accurately known and that the subspace from which $l_t$ is generated changes slowly over time. We develop and study a novel "online" RPCA algorithm based on the recently introduced Recursive Projected Compressive Sensing (ReProCS) framework. Our algorithm improves upon the original ReProCS algorithm and it also returns even more accurate offline estimates. The key contribution of this work is a correctness result (complete performance guarantee) for this algorithm under reasonably mild assumptions. By using extra assumptions -- accurate initial subspace knowledge, slow subspace change, and clustered eigenvalues -- we are able to remove one important limitation of batch RPCA results and two key limitations of a recent result for ReProCS for online RPCA. To our knowledge, this work is among the first few correctness results for online RPCA. Most earlier results were only partial results, i.e., they required an assumption on intermediate algorithm estimates.