Efficient Implementation of Truncated Reweighting Low-Rank Matrix Approximation

The weighted nuclear norm minimization and truncated nuclear norm minimization are two well-known low-rank constraint for visual applications. In this paper, by integrating their advantages into a unified formulation, we find a better weighting strategy, namely truncated reweighting norm minimization (TRNM), which provides better approximation to the target rank for some specific task. Albeit nonconvex and truncated, we prove that TRNM is equivalent to certain weighted quadratic programming problems, whose global optimum can be accessed by the newly presented reweighting singular value thresholding operator. More importantly, we design a computationally efficient optimization algorithm, namely momentum update and rank propagation (MURP), for the general TRNM regularized problems. The individual advantages of MURP include, first, reducing iterations through nonmonotonic search, and second, mitigating computational cost by reducing the size of target matrix. Furthermore, the descent property and convergence of MURP are proven. Finally, two practical models, i.e., Matrix Completion Problem via TRNM (MCTRNM) and Space Clustering Model via TRNM (SCTRNM), are presented for visual applications. Extensive experimental results show that our methods achieve better performance, both qualitatively and quantitatively, compared with several state-of-the-art algorithms.