Scalable Nuclear-norm Minimization by Subspace Pursuit Proximal Riemannian Gradient
This work addresses scalability issues in low-rank matrix problems for machine learning practitioners, though it appears incremental as it builds on existing regularization methods.
The authors tackled the computational expense of nuclear-norm minimization in tasks like low-rank matrix recovery by proposing a proximal Riemannian gradient scheme and a subspace pursuit paradigm, which avoid large-rank SVDs and demonstrate superiority in empirical studies on matrix completion and subspace clustering.
Nuclear-norm regularization plays a vital role in many learning tasks, such as low-rank matrix recovery (MR), and low-rank representation (LRR). Solving this problem directly can be computationally expensive due to the unknown rank of variables or large-rank singular value decompositions (SVDs). To address this, we propose a proximal Riemannian gradient (PRG) scheme which can efficiently solve trace-norm regularized problems defined on real-algebraic variety $\mMLr$ of real matrices of rank at most $r$. Based on PRG, we further present a simple and novel subspace pursuit (SP) paradigm for general trace-norm regularized problems without the explicit rank constraint $\mMLr$. The proposed paradigm is very scalable by avoiding large-rank SVDs. Empirical studies on several tasks, such as matrix completion and LRR based subspace clustering, demonstrate the superiority of the proposed paradigms over existing methods.