Low-rank approximation models have been widely developed in computer vision, image analysis, signal processing, web data analysis, bioinformatics, etc. Generally, we assume that the intrinsic data lies in a low-dimensional subspace, and we need to extract the low-rank representation given observations. There are many well-known works such as Principal Component Analysis (PCA), factor analysis, least squares, etc. However, their performance may be affected when dealing with outliers. Robust PCA (RPCA) plays an important role in such cases, but RPCA based methods suffer from expensive computation costs. In this thesis, we discussed how to improve the performance of RPCA in terms of both speed and accuracy. The comparison between convex and non-convex models is also discussed. Notably, we propose a theory about matrix decomposition with unknown rank. A nonlinear RPCA approach is also proposed, given the assumption that data lie on a manifold. Then, we take examples from seismic event detection and 2D image denoising. The numerical experiments show the robustness of our techniques and present speedup and higher recovery accuracy compared with existing approaches. It is usually common in practice that observed data has missing values. So, we need to make a low-rank approximation based on incomplete data. Also, it may take a long time for offline matrix completion since we need to collect all data first. The online version can offer up-to-date results based on a continuous data stream. Online matrix completion has applications in computer vision and web data analysis, especially in video image transmission and recommendation systems. To be better applied on color images with three channels, we introduced online quaternion matrix completion. We can get an updated result for every new observed entry using stochastic gradient descent on the quaternion matrix.
Read
