Robust PCA With Partial Subspace Knowledge
Abstract
In recent work, robust Principal Components Analysis (PCA) has been posed as a problem of recovering a lowrank matrix $\mathbf{L}$ and a sparse matrix $\mathbf{S}$ from their sum, $\mathbf{M}:= \mathbf{L} + \mathbf{S}$ and a provably exact convex optimization solution called PCP has been proposed. This work studies the following problem. Suppose that we have partial knowledge about the column space of the low rank matrix $\mathbf{L}$. Can we use this information to improve the PCP solution, i.e. allow recovery under weaker assumptions? We propose here a simple but useful modification of the PCP idea, called modifiedPCP, that allows us to use this knowledge. We derive its correctness result which shows that, when the available subspace knowledge is accurate, modifiedPCP indeed requires significantly weaker incoherence assumptions than PCP. Extensive simulations are also used to illustrate this. Comparisons with PCP and other existing work are shown for a stylized real application as well. Finally, we explain how this problem naturally occurs in many applications involving time series data, i.e. in what is called the online or recursive robust PCA problem. A corollary for this case is also given.
 Publication:

IEEE Transactions on Signal Processing
 Pub Date:
 July 2015
 DOI:
 10.1109/TSP.2015.2421485
 arXiv:
 arXiv:1403.1591
 Bibcode:
 2015ITSP...63.3332Z
 Keywords:

 Computer Science  Information Theory
 EPrint:
 19 pages, 9 figures, submitted to IEEE Transaction on Signal Processing