Performance Guarantees for ReProCS -- Correlated Low-Rank Matrix Entries Case

TL;DR

This paper extends the ReProCS method for online robust PCA to handle correlated low-rank components modeled as autoregressive processes, providing theoretical guarantees for support recovery and error bounds.

Contribution

It relaxes the independence assumption on low-rank components, allowing for correlated data, and proves support recovery and error bounds under this new model.

Findings

Support set of sparse vector recovered exactly with high probability

Reconstruction errors are bounded and small over time

Subspace recovery error diminishes after finite delay

Abstract

Online or recursive robust PCA can be posed as a problem of recovering a sparse vector, $S_t$, and a dense vector, $L_t$, which lies in a slowly changing low-dimensional subspace, from $M_t:= S_t + L_t$ on-the-fly as new data comes in. For initialization, it is assumed that an accurate knowledge of the subspace in which $L_0$ lies is available. In recent works, Qiu et al proposed and analyzed a novel solution to this problem called recursive projected compressed sensing or ReProCS. In this work, we relax one limiting assumption of Qiu et al's result. Their work required that the $L_t$'s be mutually independent over time. However this is not a practical assumption, e.g., in the video application, $L_t$ is the background image sequence and one would expect it to be correlated over time. In this work we relax this and allow the $L_t$'s to follow an autoregressive model. We are able to show that under mild assumptions and under a denseness assumption on the unestimated part of the changed subspace, with high probability (w.h.p.), ReProCS can exactly recover the support set of $S_t$ at all times; the reconstruction errors of both $S_t$ and $L_t$ are upper bounded by a time invariant and small value; and the subspace recovery error decays to a small value within a finite delay of a subspace change time. Because the last assumption depends on an algorithm estimate, this result cannot be interpreted as a correctness result but only a useful step towards it.