Sparse principal component analysis via regularized rank-k matrix approximation

In calculating the sparse principal components (PCs), attaining k PCs simultaneously can reduce the accumulated error arising from the calculation process. We proposed the sparse principal component model via regularized rank-k matrix approximation and designed a block coordinate descent method (BCD-sPCA-rSVD) to solve this problem. Its main idea is to first divide variables into 2k blocks by coordinates, and then solve sub-problem with respect to each single coordinate block when keeping other 2k-1 variables fixed. By solving these sub-problems with explicit solutions recursively until the stopping criterion is satisfied, the BCD-sPCA-rSVD algorithm can be easily constructed. Its per-iteration complexity is linear in both sample size and variable dimensionality. The algorithm is convergent and easy to implement. Numerical simulation results show that the algorithm is feasible and effective when applied to real and synthetic data sets. The proposed method reduces the accumulated error and has lower computational complexity, which makes it well suited to handling large-scale problems of sparse principal component analysis.