A Decomposition Algorithm for the Sparse Generalized Eigenvalue Problem

The sparse generalized eigenvalue problem arises in a number of standard and modern statistical learning models, including sparse principal component analysis, sparse Fisher discriminant analysis, and sparse canonical correlation analysis. However, this problem is difficult to solve since it is NP-hard. In this paper, we consider a new effective decomposition method to tackle this problem. Specifically, we use random or/and swapping strategies to find a working set and perform global combinatorial search over the small subset of variables. We consider a bisection search method and a coordinate descent method for solving the quadratic fractional programming subproblem. In addition, we provide some theoretical analysis for the proposed method. Our experiments on synthetic data and real-world data have shown that our method significantly and consistently outperforms existing solutions in term of accuracy.