Efficient convex PCA with applications to Wasserstein GPCA and ranked data

TL;DR

This paper advances convex PCA by providing new theoretical insights, developing a numerical implementation for polyhedral sets, and demonstrating applications to Wasserstein GPCA and financial data analysis.

Contribution

It introduces new theoretical results, a practical numerical method for convex PCA in finite dimensions, and applies these to Wasserstein GPCA and financial data.

Findings

Theoretical consistency and smoothness results for convex PCA.

A numerical algorithm for convex PCA with polyhedral convex sets.

Successful application to financial distribution data.

Abstract

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.