Tractable and Scalable Schatten Quasi-Norm Approximations for Rank Minimization

The Schatten quasi-norm was introduced tobridge the gap between the trace norm andrank function. However, existing algorithmsare too slow or even impractical for large-scale problems. Motivated by the equivalencerelation between the trace norm and its bilin-ear spectral penalty, we define two tractableSchatten norms, i.e. the bi-trace and tri-tracenorms, and prove that they are in essence theSchatten-1/2 and 1/3 quasi-norms, respec-tively. By applying the two defined Schat-ten quasi-norms to various rank minimiza-tion problems such as MC and RPCA, weonly need to solve much smaller factor matri-ces. We design two efficient linearized alter-nating minimization algorithms to solve ourproblems and establish that each boundedsequence generated by our algorithms con-verges to a critical point. We also provide therestricted strong convexity (RSC) based andMC error bounds for our algorithms. Our ex-perimental results verified both the efficiencyand effectiveness of our algorithms comparedwith the state-of-the-art methods.