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Sparse principal component analysis (PCA) is a popular dimensionalityreduction technique for obtaining principal components which are linearcombinations of a small subset of the original features. Existing approachescannot supply certifiably optimal principal components with more than $p=100s$of variables. By reformulating sparse PCA as a convex mixed-integersemidefinite optimization problem, we design a cutting-plane method whichsolves the problem to certifiable optimality at the scale of selecting k=5covariates from p=300 variables, and provides small bound gaps at a largerscale. We also propose a convex relaxation and greedy rounding scheme thatprovides bound gaps of $1-2\%$ in practice within minutes for $p=100$s or hoursfor $p=1,000$s and is therefore a viable alternative to the exact method atscale. Using real-world financial and medical datasets, we illustrate ourapproach's ability to derive interpretable principal components tractably atscale.

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