  This work studies the recursive robust principal components analysis (PCA)
problem. If the outlier is the signal-of-interest, this problem can be
interpreted as one of recursively recovering a time sequence of sparse vectors,
$S_t$, in the presence of large but structured noise, $L_t$. The structure that
we assume on $L_t$ is that $L_t$ is dense and lies in a low dimensional
subspace that is either fixed or changes "slowly enough". A key application
where this problem occurs is in video surveillance where the goal is to
separate a slowly changing background ($L_t$) from moving foreground objects
($S_t$) on-the-fly. To solve the above problem, in recent work, we introduced a
novel solution called Recursive Projected CS (ReProCS). In this work we develop
a simple modification of the original ReProCS idea and analyze it. This
modification assumes knowledge of a subspace change model on the $L_t$'s. Under
mild assumptions and a denseness assumption on the unestimated part of the
subspace of $L_t$ at various times, we show that, with high probability
(w.h.p.), the proposed approach can exactly recover the support set of $S_t$ at
all times; and the reconstruction errors of both $S_t$ and $L_t$ are upper
bounded by a time-invariant and small value. In simulation experiments, we
observe that the last assumption holds as long as there is some support change
of $S_t$ every few frames.
