An augmented Lagrangian approach for sparse principal component analysis

Principal component analysis (PCA) is a widely used technique for data analysis and dimension reduction with numerous applications in science and engineering. However, the standard PCA suffers from the fact that the principal components (PCs) are usually linear combinations of all the original variables, and it is thus often difficult to interpret the PCs. To alleviate this drawback, various sparse PCA approaches were proposed in the literature (Cadima and Jolliffe in J Appl Stat 22:203-214, 1995; d'Aspremont et al. in J Mach Learn Res 9:1269-1294, 2008; d'Aspremont et al. SIAM Rev 49:434-448, 2007; Jolliffe in J Appl Stat 22:29-35, 1995; Journ,e et al. in J Mach Learn Res 11:517-553, 2010; Jolliffe et al. in J Comput Graph Stat 12:531-547, 2003; Moghaddam et al. in Advances in neural information processing systems 18:915-922, MIT Press, Cambridge, 2006; Shen and Huang in J Multivar Anal 99(6):1015-1034, 2008; Zou et al. in J Comput Graph Stat 15(2):265-286, 2006). Despite success in achieving sparsity, some important properties enjoyed by the standard PCA are lost in these methods such as uncorrelation of PCs and orthogonality of loading vectors. Also, the total explained variance that they attempt to maximize can be too optimistic. In this paper we propose a new formulation for sparse PCA, aiming at finding sparse and nearly uncorrelated PCs with orthogonal loading vectors while explaining as much of the total variance as possible. We also develop a novel augmented Lagrangian method for solving a class of nonsmooth constrained optimization problems, which is well suited for our formulation of sparse PCA. We show that it converges to a feasible point, and moreover under some regularity assumptions, it converges to a stationary point. Additionally, we propose two nonmonotone gradient methods for solving the augmented Lagrangian subproblems, and establish their global and local convergence. Finally, we compare our sparse PCA approach with several existing methods on synthetic (Zou et al. in J Comput Graph Stat 15(2):265-286, 2006), Pitprops (Jeffers in Appl Stat 16:225-236, 1967), and gene expression data (Chin et al in Cancer Cell 10:529C-541C, 2006), respectively. The computational results demonstrate that the sparse PCs produced by our approach substantially outperform those by other methods in terms of total explained variance, correlation of PCs, and orthogonality of loading vectors. Moreover, the experiments on random data show that our method is capable of solving large-scale problems within a reasonable amount of time.