论文标题
有效的凸PCA，并应用于Wasserstein GPCA并排名
Efficient convex PCA with applications to Wasserstein GPCA and ranked data
论文作者
论文摘要
凸PCA，是在Bigot等人中引入的。 （2017年），通过限制数据和主要组件位于希尔伯特空间的给定凸子集中来修改欧几里得PCA。这种设置自然出现在许多应用中，包括间隔的Wasserstein空间中的分布数据，并在Aitchison几何形状下排名的组成数据。我们在本文中的贡献是三倍。首先，我们提出了几个新的理论结果，包括在有限维度的情况下的目标函数的一致性以及连续性和可不同的性能。其次，当凸组为多面体时，我们会开发出有限尺寸凸PCA的数值实现，并表明这提供了Wasserstein GPCA的自然近似值。第三，我们通过两种财务应用来说明结果，即按规模排名的股票收益和资本分配曲线的分配，这两者都具有随机投资组合理论的独立感兴趣。本文的补充材料可在线获得。
Convex PCA, which was introduced in Bigot et al. (2017), 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 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 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.
