  Robust Principal Component Analysis (RPCA) and its associated non-convex
relaxation methods constitute a significant component of matrix completion
problems, wherein matrix factorization strategies effectively reduce
dimensionality and enhance computational speed. However, some non-convex
factorization forms lack theoretical guarantees. This paper proposes a novel
strategy in non-convex quasi-norm representation, introducing a method to
obtain weighted matrix quasi-norm factorization forms. Especially, explicit
bilinear factor matrix factorization formulations for the weighted logarithmic
norm and weighted Schatten-$q$ quasi norms with $q=1, 1/2, 2/3$ are provided,
along with the establishment of corresponding matrix completion models. An
Alternating Direction Method of Multipliers (ADMM) framework algorithm is
employed for solving, and convergence results of the algorithm are presented.
