Solving Large-Scale Sparse PCA to Certifiable (Near) Optimality

Sparse principal component analysis (PCA) is a popular dimensionality reduction technique for obtaining principal components which are linear combinations of a small subset of the original features. Existing approaches cannot supply certifiably optimal principal components with more than p=100s of variables. By reformulating sparse PCA as a convex mixed-integer semidefinite optimization problem, we design a cutting-plane method which solves the problem to certifiable optimality at the scale of selecting k=5 covariates from p=300 variables, and provides small bound gaps at a larger scale. We also propose a convex relaxation and greedy rounding scheme that provides bound gaps of 1-2% in practice within minutes for p=100s or hours for p=1,000s and is therefore a viable alternative to the exact method at scale. Using real-world financial and medical datasets, we illustrate our approach's ability to derive interpretable principal components tractably at scale.