A Correctness Result for Online Robust PCA

TL;DR

This paper presents the first correctness proof for online robust PCA, demonstrating that under certain conditions, the method accurately recovers sparse and low-dimensional components in real-time applications.

Contribution

It provides the first theoretical guarantees for online robust PCA, showing exact support recovery and small estimation errors under specific assumptions.

Findings

Support of sparse vector is recovered exactly with high probability

Estimation errors for sparse and low-dimensional vectors are small

Applicable to separating foreground and background in surveillance videos

Abstract

This work studies the problem of sequentially recovering a sparse vector $x_t$ and a vector from a low-dimensional subspace $l_t$ from knowledge of their sum $m_t = x_t + l_t$. If the primary goal is to recover the low-dimensional subspace where the $l_t$'s lie, then the problem is one of online or recursive robust principal components analysis (PCA). To the best of our knowledge, this is the first correctness result for online robust PCA. We prove that if the $l_t$'s obey certain denseness and slow subspace change assumptions, and the support of $x_t$ changes by at least a certain amount at least every so often, and some other mild assumptions hold, then with high probability, the support of $x_t$ will be recovered exactly, and the error made in estimating $x_t$ and $l_t$ will be small. An example of where such a problem might arise is in separating a sparse foreground and slowly changing dense background in a surveillance video.