An inexact proximal DC algorithm with sieving strategy for rank constrained least squares semidefinite programming

In this paper, the optimization problem of the supervised distance preserving projection (SDPP) for data dimension reduction (DR) is considered, which is equivalent to a rank constrained least squares semidefinite programming (RCLSSDP). In order to overcome the difficulties caused by rank constraint, the difference-of-convex (DC) regularization strategy was employed, then the RCLSSDP is transferred into a series of least squares semidefinite programming with DC regularization (DCLSSDP). An inexact proximal DC algorithm with sieving strategy (s-iPDCA) is proposed for solving the DCLSSDP, whose subproblems are solved by the accelerated block coordinate descent (ABCD) method. Convergence analysis shows that the generated sequence of s-iPDCA globally converges to stationary points of the corresponding DC problem. To show the efficiency of our proposed algorithm for solving the RCLSSDP, the s-iPDCA is compared with classical proximal DC algorithm (PDCA) and the PDCA with extrapolation (PDCAe) by performing DR experiment on the COIL-20 database, the results show that the s-iPDCA outperforms the PDCA and the PDCAe in solving efficiency. Moreover, DR experiments for face recognition on the ORL database and the YaleB database demonstrate that the rank constrained kernel SDPP (RCKSDPP) is effective and competitive by comparing the recognition accuracy with kernel semidefinite SDPP (KSSDPP) and kernal principal component analysis (KPCA).