Robust PCA with partial subspace knowledge

In recent work, robust PCA has been posed as a problem of recovering a low-rank matrix L and a sparse matrix S from their sum, M := L + S and a provably exact convex optimization solution called PCP has been proposed. Suppose that we have a partial estimate of the column subspace of the low rank matrix L. Can we use this information to improve the PCP solution, i.e. allow recovery under weaker assumptions? We propose here a simple modification of the PCP idea, called modified-PCP, that allows us to use this knowledge. We derive its correctness result which shows that modified-PCP indeed requires significantly weaker incoherence assumptions than PCP, when the available subspace knowledge is accurate. Extensive simulations are also used to illustrate this. Finally, we explain how this problem naturally occurs in many applications involving time series data, e.g. in separating a video sequence into foreground and background layers, in which the subspace spanned by the background images is not fixed but changes over time and the changes are gradual. A corollary for this case is also given.