  Numerous applications in data mining and machine learning require recovering
a matrix of minimal rank. Robust principal component analysis (RPCA) is a
general framework for handling this kind of problems. Nuclear norm based convex
surrogate of the rank function in RPCA is widely investigated. Under certain
assumptions, it can recover the underlying true low rank matrix with high
probability. However, those assumptions may not hold in real-world
applications. Since the nuclear norm approximates the rank by adding all
singular values together, which is essentially a $\ell_1$-norm of the singular
values, the resulting approximation error is not trivial and thus the resulting
matrix estimator can be significantly biased. To seek a closer approximation
and to alleviate the above-mentioned limitations of the nuclear norm, we
propose a nonconvex rank approximation. This approximation to the matrix rank
is tighter than the nuclear norm. To solve the associated nonconvex
minimization problem, we develop an efficient augmented Lagrange multiplier
based optimization algorithm. Experimental results demonstrate that our method
outperforms current state-of-the-art algorithms in both accuracy and
efficiency.
