Orthogonal Sparse PCA and Covariance Estimation via Procrustes Reformulation
The problem of estimating sparse eigenvectors of a symmetric matrix has attracted a lot of attention in many applications, especially those with a high dimensional dataset. While classical eigenvectors can be obtained as the solution of a maximization problem, existing approaches formulate this problem by adding a penalty term into the objective function that encourages a sparse solution. However, the vast majority of the resulting methods achieve sparsity at the expense of sacrificing the orthogonality property. In this paper, we develop a new method to estimate dominant sparse eigenvectors without trading off their orthogonality. The problem is highly nonconvex and hard to handle. We apply the minorization-maximization framework, wherein we iteratively maximize a tight lower bound (surrogate function) of the objective function over the Stiefel manifold. The inner maximization problem turns out to be a rectangular Procrustes problem, which has a closed-form solution. In addition, we propose a method to improve the covariance estimation problem when its underlying eigenvectors are known to be sparse. We use the eigenvalue decomposition of the covariance matrix to formulate an optimization problem wherein we impose sparsity on the corresponding eigenvectors. Numerical experiments show that the proposed eigenvector extraction algorithm outperforms existing algorithms in terms of support recovery and explained variance, whereas the covariance estimation algorithms improve the sample covariance estimator significantly.