  We consider an online version of the robust Principle Component Analysis
(PCA), which arises naturally in time-varying source separations such as video
foreground-background separation. This paper proposes a compressive online
robust PCA with prior information for recursively separating a sequences of
frames into sparse and low-rank components from a small set of measurements. In
contrast to conventional batch-based PCA, which processes all the frames
directly, the proposed method processes measurements taken from each frame.
Moreover, this method can efficiently incorporate multiple prior information,
namely previous reconstructed frames, to improve the separation and thereafter,
update the prior information for the next frame. We utilize multiple prior
information by solving $n\text{-}\ell_{1}$ minimization for incorporating the
previous sparse components and using incremental singular value decomposition
($\mathrm{SVD}$) for exploiting the previous low-rank components. We also
establish theoretical bounds on the number of measurements required to
guarantee successful separation under assumptions of static or slowly-changing
low-rank components. Using numerical experiments, we evaluate our bounds and
the performance of the proposed algorithm. In addition, we apply the proposed
algorithm to online video foreground and background separation from compressive
measurements. Experimental results show that the proposed method outperforms
the existing methods.
