Online Schatten quasi-norm minimization for robust principal component analysis

Abstract

The Schatten-p quasi-norm (0 < p < 1) is widely and successfully used as a superior rank surrogate to the well-known nuclear norm. It has been proved that the Schatten-p quasi-norm is admirable for the robust principal component analysis problem both in theory and in practice. However, existing algorithms for solving the Schatten-p quasi-norm minimization problem suffer from large memory requirement and high computational complexity, which is due to (1) they work in batch manner, which requires to load all the data in the memory during process; (2) they relay on singular value decomposition on a dense matrix in every iteration. Hence they are only able to handle problems of certain scale and highly inefficient in large scale cases. In this paper, by reformulating the Schatten-p quasi-norm in variational formulation, we design an online algorithm to solve the Schatten-p quasi-norm minimization problem, in which the memory cost is independent of the data sample size. Moreover, the new algorithm is totally free of singular value decomposition. Numerical study on subspace recovery demonstrates encouraging results of our algorithm compared with the widely used rank surrogate functions for online optimization. Besides, experiments on real video background subtraction indicate that the newly proposed online algorithm is extremely fast, and meets the real-time background subtraction for practical video processing tasks.