A correctness result for online robust PCA

We study 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 this problem. We prove that if a good estimate of the initial subspace is available; the ℓ_t´s obey certain denseness and slow subspace change assumptions; and the support of x_t changes either at every frame or at least every so often, 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 where this problem occurs is in separating a sparse foreground and a slowly changing dense background from surveillance videos.