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 L₀ 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.