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 $p=100$s or hours for $p=1,000$s 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.
翻译:主要成分分析(PCA)是获取主要组成部分的流行的维度减少技术,这些主要组成部分是原始特征中一小部分的线性组合。现有方法无法用超过1美元=100美元的变量提供可证实的最佳主要组成部分。通过将稀有的五氯苯甲醚改制为混凝土混凝土半脱硫优化问题,我们设计了一种切割机方法,以解决在从p=300变量中选择 k=5 共变数的尺度上可证实的最佳性的问题,并在更大的尺度上提供小的捆绑差距。我们还提议了一个convex 放松和贪婪四舍五入方案,在实际操作中以分钟内提供1-2美元=100美元或1 000美元的小时的捆绑差距,因此是精确比例法的一种可行的替代方法。我们使用真实世界的金融和医疗数据集,说明我们的方法在规模上可以获取可解释的主要组成部分的能力。