Informed Non-convex Robust Principal Component Analysis with Features

Many machine learning and artificial intelligence tasks involve the separation of a data matrix into a low-rank structure and a sparse part capturing different information. Robust principal component analysis (RPCA) Candes et al. [2011], Chandrasekaran et al. [2011] is a popular framework that logically characterizes this matrix separation problem. Nevertheless, prior side information, oftentimes in the form of features, may also be present in practice. For instance, features are available for the following tasks: - Collaborative filtering: apart from ratings of an item by other users, the profile of the user and the description of the item can also be exploited in making recommendations Chiang et al. [2015]; - Relationship prediction: user behaviours and message exchanges can assist in finding missing links on social media networks Xu et al. [2013]; - Person-specific facial deformable models: an orthonormal subspace learnt from manually annotated data captured in-the-wild, when fed into an im-1 age congealing procedure, can help produce more correct fittings Sagonas et al. [2014]. It is thus reasonable to investigate how propitious it is for RPCA to exploit the available features. Indeed, recent results Liu et al. [2017] indicate that features are not redundant at all. In the setting of multiple subspaces, RPCA degrades as the number of subspaces grows because of the increased row-coherence. On the other hand, the use of feature dictionaries allows accurate low-rank recovery by removing the dependency on row-coherence.