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 $p=100s$ covariates. 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=10s covariates from p=300 variables, and provides small bound gaps at a larger scale. We also propose two convex relaxations and randomized rounding schemes that provide certifiably near-exact solutions within minutes for p=100s or hours for p=1,000s. Using real-world financial and medical datasets, we illustrate our approach's ability to derive interpretable principal components tractably at scale.

通过将稀疏主成分分析重新制定为凸混合整数半定规划问题，并设计一个切平面方法，该方法可以以确切的最优性选择5个协变量从300个变量并能在更大范围内提供小的界限间隙。我们还提出了一种凸松弛和贪心舍入方案，可在几分钟内提供1-2％的绑定间隙，使其成为规模上的可行替代方法。使用真实的金融和医疗数据集，我们展示了我们的方法在可解释性和易于计算的情况下，在不同范围内可行性的能力。

解决大规模稀疏PCA至可证明（近似）最优