Compressive Online Robust Principal Component Analysis via $n$ - $\ell_1$ Minimization

This paper considers online robust principal component analysis (RPCA) in time-varying decomposition problems such as video foreground-background separation. We propose a compressive online RPCA algorithm that decomposes recursively a sequence of data vectors (e.g., frames) into sparse and low-rank components. Different from conventional batch RPCA, which processes all the data directly, our approach considers a small set of measurements taken per data vector (frame). Moreover, our algorithm can incorporate multiple prior information from previous decomposed vectors via proposing an <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula> minimization method. At each time instance, the algorithm recovers the sparse vector by solving the <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula> minimization problem—which promotes not only the sparsity of the vector but also its correlation with multiple previously recovered sparse vectors—and, subsequently, updates the low-rank component using incremental singular value decomposition. We also establish theoretical bounds on the number of measurements required to guarantee successful compressive separation under the assumptions of static or slowly changing low-rank components. We evaluate the proposed algorithm using numerical experiments and online video foreground-background separation experiments. The experimental results show that the proposed method outperforms the existing methods.