Scalable Algorithms for Tractable Schatten Quasi-Norm Minimization
This work addresses scalability issues in matrix completion for applications like recommendation systems, though it is incremental as it builds on existing quasi-norm approaches.
The paper tackled the inefficiency of existing Schatten-p quasi-norm minimization algorithms for large-scale matrix completion by defining tractable Schatten quasi-norms and designing efficient algorithms that update smaller factor matrices, resulting in algorithms that are more accurate and orders of magnitude faster than state-of-the-art methods.
The Schatten-p quasi-norm $(0<p<1)$ is usually used to replace the standard nuclear norm in order to approximate the rank function more accurately. However, existing Schatten-p quasi-norm minimization algorithms involve singular value decomposition (SVD) or eigenvalue decomposition (EVD) in each iteration, and thus may become very slow and impractical for large-scale problems. In this paper, we first define two tractable Schatten quasi-norms, i.e., the Frobenius/nuclear hybrid and bi-nuclear quasi-norms, and then prove that they are in essence the Schatten-2/3 and 1/2 quasi-norms, respectively, which lead to the design of very efficient algorithms that only need to update two much smaller factor matrices. We also design two efficient proximal alternating linearized minimization algorithms for solving representative matrix completion problems. Finally, we provide the global convergence and performance guarantees for our algorithms, which have better convergence properties than existing algorithms. Experimental results on synthetic and real-world data show that our algorithms are more accurate than the state-of-the-art methods, and are orders of magnitude faster.