  Matrix completion focuses on recovering a matrix from a small subset of its
observed elements, and has already gained cumulative attention in computer
vision. Many previous approaches formulate this issue as a low-rank matrix
approximation problem. Recently, a truncated nuclear norm has been presented as
a surrogate of traditional nuclear norm, for better estimation to the rank of a
matrix. The truncated nuclear norm regularization (TNNR) method is applicable
in real-world scenarios. However, it is sensitive to the selection of the
number of truncated singular values and requires numerous iterations to
converge. Hereby, this paper proposes a revised approach called the double
weighted truncated nuclear norm regularization (DW-TNNR), which assigns
different weights to the rows and columns of a matrix separately, to accelerate
the convergence with acceptable performance. The DW-TNNR is more robust to the
number of truncated singular values than the TNNR. Instead of the iterative
updating scheme in the second step of TNNR, this paper devises an efficient
strategy that uses a gradient descent manner in a concise form, with a
theoretical guarantee in optimization. Sufficient experiments conducted on real
visual data prove that DW-TNNR has promising performance and holds the
superiority in both speed and accuracy for matrix completion.
