Tractable and Scalable Schatten Quasi-Norm Approximations for Rank Minimization
This addresses scalability issues in rank minimization for applications like matrix completion and robust PCA, offering incremental improvements in algorithm speed and practicality.
The paper tackles the problem of slow and impractical algorithms for rank minimization using Schatten quasi-norms by introducing tractable approximations (bi-trace and tri-trace norms) that reduce computational complexity to smaller factor matrices, achieving efficiency and effectiveness verified in experiments compared to state-of-the-art methods.
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.