Tractable and Scalable Schatten Quasi-Norm Approximations for Rank Minimization

The Schatten quasi-norm was introduced to bridge the gap between the trace norm and rank function. However, existing algorithms are too slow or even impractical for large-scale problems. Motivated by the equivalence relation between the trace norm and its bilinear spectral penalty, we define two tractable Schatten norms, i.e.\ the bi-trace and tri-trace norms, and prove that they are in essence the Schatten-$1/2$ and $1/3$ quasi-norms, respectively. By applying the two defined Schatten quasi-norms to various rank minimization problems such as MC and RPCA, we only need to solve much smaller factor matrices. We design two efficient linearized alternating minimization algorithms to solve our problems and establish that each bounded sequence generated by our algorithms converges to a critical point. We also provide the restricted strong convexity (RSC) based and MC error bounds for our algorithms. Our experimental results verified both the efficiency and effectiveness of our algorithms compared with the state-of-the-art methods.