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Online or recursive robust PCA can be posed as a problem of recovering asparse vector, $S_t$, and a dense vector, $L_t$, which lies in a slowlychanging low-dimensional subspace, from $M_t:= S_t + L_t$ on-the-fly as newdata comes in. For initialization, it is assumed that an accurate knowledge ofthe subspace in which $L_0$ lies is available. In recent works, Qiu et alproposed and analyzed a novel solution to this problem called recursiveprojected compressed sensing or ReProCS. In this work, we relax one limitingassumption of Qiu et al's result. Their work required that the $L_t$'s bemutually independent over time. However this is not a practical assumption,e.g., in the video application, $L_t$ is the background image sequence and onewould expect it to be correlated over time. In this work we relax this andallow the $L_t$'s to follow an autoregressive model. We are able to show thatunder mild assumptions and under a denseness assumption on the unestimated partof the changed subspace, with high probability (w.h.p.), ReProCS can exactlyrecover the support set of $S_t$ at all times; the reconstruction errors ofboth $S_t$ and $L_t$ are upper bounded by a time invariant and small value; andthe subspace recovery error decays to a small value within a finite delay of asubspace change time. Because the last assumption depends on an algorithmestimate, this result cannot be interpreted as a correctness result but only auseful step towards it.

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