Algorithms Based On Robust Principal Component Analysis And Its Applications

The robust principal component analysis (RPCA) has become one of the international hot issues in the field of signal processing recently. As a classic method for data analysis, RPCA is widely used in science and engineering, such as video surveillance, face modeling, subspace clustering, etc. There are main two kinds of methods at present, including low-rank matrix analysis and Bayesian.Besides, one problem closely related to RPCA is the low rank matrix factorization(LRMF).The characters of the low-rank component and the sparse component are analysed in this paper firstly. Then we combine the Bayesian model, Ising model and signal model to propose a mixture model framework namely Bayesian-Ising-Signal (BIS). The equivalence of our model with the RPCA model is then proved theoretically at the same time. Then the pros and cons between the model and truncated SVD are performed further analysis.Secondly, this paper proposes a novel algorithm based on sparse Bayesian learning principles and Markov random fields. Generally, the model assumes s-parse noise and characterizes the error term by the l1-norm. However, the sparse noise has clustering effect in practice so that using a certain lp-norm simply is not appropriate for modeling. The low-rank matrix and outlier support are estimated simultaneously in BIS, by enforcing the low-rank constraint in a matrix factoriza-tion formulation and incorporating the contiguity prior as a sparsity constraint.Besides, this paper employs independent sparsity priors on the individual factors with a common sparsity profile which favors low-rank solutions. Subsequently, we use the variational inference method to infer the posterior. As the prior knowledge on the spatial distribution of outliers has been considered, this paper incorporates such contiguity prior to estimate the outlier support matrix using MRFs further.Simulation data and real data finally are used to analyse and verify the BIS,which is proved to be very effective for low-rank matrix recovery and contiguous outliers detection. Compared with other state-of-the-art methods, BIS not only improves the accuracy and the speed, but also it can effectively deal with more complex information, which means better robustness.