  This work studies the problem of sequentially recovering a sparse vector
$x_t$ and a vector from a low-dimensional subspace $l_t$ from knowledge of
their sum $m_t = x_t + l_t$. If the primary goal is to recover the
low-dimensional subspace where the $l_t$'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 $l_t$'s obey certain denseness and slow subspace change
assumptions, and the support of $x_t$ 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 $x_t$ will be recovered exactly, and the error made
in estimating $x_t$ and $l_t$ 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.
