Fast Sparse PCA via Positive Semidefinite Projection for Unsupervised Feature Selection

In the field of unsupervised feature selection, sparse principal component analysis (SPCA) methods have attracted more and more attention recently. Compared to spectral-based methods, SPCA methods don’t rely on the construction of similarity matrix and show better feature selection ability on real-world data. The original SPCA formulates a nonconvex optimization problem. And existing convex SPCA methods reformulate SPCA as a convex model by regarding reconstruction matrix as optimization variable. However, they are lack of constraints equivalent to the orthogonality restriction in SPCA, leading to larger solution space. In this paper,  it’s proved that the optimal solution to a convex SPCA model falls onto the PSD cone. A new convex SPCA model with PSD constraint is then proposed. Further, a two-step fast optimization algorithm is presented to solve the proposed model. Firstly, adopt the derivative of objective function to obtain the unconstrained solution. Secondly, project the solution onto the PSD cone.The convergence of the  proposed optimization algorithm is studied. With PSD projection, the proposed method takes less number of iterations and running  time than other convex SPCA methods and has stronger feature selection ability. Experiments on both synthetic and real-world datasets demonstrate the effectiveness and efficiency of the proposed method.