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

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.