  We study a non-convex low-rank promoting penalty function, the transformed
Schatten-1 (TS1), and its applications in matrix completion. The TS1 penalty,
as a matrix quasi-norm defined on its singular values, interpolates the rank
and the nuclear norm through a nonnegative parameter a. We consider the
unconstrained TS1 regularized low-rank matrix recovery problem and develop a
fixed point representation for its global minimizer. The TS1 thresholding
functions are in closed analytical form for all parameter values. The TS1
threshold values differ in subcritical (supercritical) parameter regime where
the TS1 threshold functions are continuous (discontinuous). We propose TS1
iterative thresholding algorithms and compare them with some state-of-the-art
algorithms on matrix completion test problems. For problems with known rank, a
fully adaptive TS1 iterative thresholding algorithm consistently performs the
best under different conditions with ground truth matrix being multivariate
Gaussian at varying covariance. For problems with unknown rank, TS1 algorithms
with an additional rank estimation procedure approach the level of IRucL-q
which is an iterative reweighted algorithm, non-convex in nature and best in
performance.
