  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.
