The Distributionally Robust Optimization Model of Sparse Principal Component Analysis

We consider sparse principal component analysis (PCA) under a stochastic setting where the underlying probability distribution of the random parameter is uncertain. This problem is formulated as a distributionally robust optimization (DRO) model based on a constructive approach to capturing uncertainty in the covariance matrix, which constitutes a nonsmooth constrained min-max optimization problem. We further prove that the inner maximization problem admits a closed-form solution, reformulating the original DRO model into an equivalent minimization problem on the Stiefel manifold. This transformation leads to a Riemannian optimization problem with intricate nonsmooth terms, a challenging formulation beyond the reach of existing algorithms. To address this issue, we devise an efficient smoothing manifold proximal gradient algorithm. We prove the Riemannian gradient consistency and global convergence of our algorithm to a stationary point of the nonsmooth minimization problem. Moreover, we establish the iteration complexity of our algorithm. Finally, numerical experiments are conducted to validate the effectiveness and scalability of our algorithm, as well as to highlight the necessity and rationality of adopting the DRO model for sparse PCA.