Online (Recursive) Robust Principal Components Analysis

This work studies the problem of sequentially recovering a sparse vector St and a vector from a low-dimensional subspace Lt from knowledge of their sum Mt := Lt + St. If the primary goal is to recover the low-dimensional subspace in which the Lt’s lie, then the problem is one of online or recursive robust principal components analysis (PCA). An example of where such a problem might arise is in separating a sparse foreground and a slowly changing dense background in a surveillance video. In this chapter, we describe our recently proposed algorithm, called Recursive Projected Compressed Sensing (ReProCS), to solve this problem and demonstrate its significant advantage over other robust PCA based methods for this video layering problem. We also summarize the performance guarantees for ReProCS. Lastly, we briefly describe our work on modified-PCP, which is a piecewise batch approach that removes a key limitation of ReProCS, however it retains a key limitation of existing work.