Proximal Riemannian Pursuit for Large-Scale Trace-Norm Minimization

Trace-norm regularization plays an important role in many areas such as computer vision and machine learning. When solving general large-scale trace-norm regularized problems, existing methods may be computationally expensive due to many high-dimensional truncated singular value decompositions (SVDs) or the unawareness of matrix ranks. In this paper, we propose a proximal Riemannian pursuit (PRP) paradigm which addresses a sequence of trace-norm regularized subproblems defined on nonlinear matrix varieties. To address the subproblem, we extend the proximal gradient method on vector space to nonlinear matrix varieties, in which the SVDs of intermediate solutions are maintained by cheap low-rank QR decompositions, therefore making the proposed method more scalable. Empirical studies on several tasks, such as matrix completion and low-rank representation based subspace clustering, demonstrate the competitive performance of the proposed paradigms over existing methods.