Efficient Convex PCA with Applications to Wasserstein GPCA and Ranked Data
本文提出了凸PCA的新理论结果和数值实现方法，并将其应用于股票收益分布和资本分布曲线的金融分析，对研究排序数据和分布数据的学者有参考价值。
Convex PCA, which was introduced by Bigot et al. modifies Euclidean PCA by restricting the data and the principal components to lie in a given convex subset of a Hilbert space. This setting arises naturally in many applications, including distributional data in the Wasserstein space of an interval, and ranked compositional data under the Aitchison geometry. Our contribution in this article is 3-fold. First, we present several new theoretical results including consistency as well as continuity and differentiability of the objective function in the finite dimensional case. Second, we develop a numerical implementation of finite dimensional convex PCA when the convex set is polyhedral, and show that this provides a natural approximation of Wasserstein GPCA. Third, we illustrate our results with two financial applications, namely distributions of stock returns ranked by size and the capital distribution curve, both of which are of independent interest in stochastic portfolio theory. Supplementary materials for this article are available online.