Abstract

This work studies the problem of sequentially recovering a sparse vector xt and a vector from a low-dimensional subspace `t from knowledge of their sum mt = xt + `t. If the primary goal is to recover the low-dimensional subspace where the `t’s lie, then the problem is one of online or recursive robust principal components analysis (PCA). To the best of our knowledge, this is the first correctness result for online robust PCA. We prove that if the `t’s obey certain denseness and slow subspace change assumptions, and the support of xt changes by at least a certain amount at least every so often, and some other mild assumptions hold, then with high probability, the support of xt will be recovered exactly, and the error made in estimating xt and `t will be small. An example of where such a problem might arise is in separating a sparse foreground and slowly changing dense background in a surveillance video. I.