Recursive Robust PCA or Recursive Sparse Recovery in Large but Structured Noise

This paper studies the recursive robust principal components analysis 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 paper, 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, 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.