Convex PCA, which was introduced by Bigot et al., is a dimension reduction methodology for data with values in a 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 paper is threefold. 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 geodesic PCA. 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.

翻译：凸PCA由Bigot等人提出，是一种针对希尔伯特空间中凸子集内数值数据的降维方法。该框架自然适用于诸多应用场景，包括区间Wasserstein空间中的分布数据，以及在Aitchison几何下排序的成分数据。本文贡献如下：首先，提出若干新理论结果，包括一致收敛性以及有限维情形下目标函数的连续性与可微性；其次，针对凸集为多面体的情形，开发了有限维凸PCA的数值实现方法，并证明其可自然近似Wasserstein测地线PCA；最后，通过两个金融应用——即按规模排序的股票收益分布与资本分布曲线——验证了所提方法，这两个应用在随机投资组合理论中具有独立研究价值。