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Convex PCA, which was introduced by Bigot et al., is a dimension reductionmethodology for data with values in a convex subset of a Hilbert space. Thissetting arises naturally in many applications, including distributional data inthe Wasserstein space of an interval, and ranked compositional data under theAitchison geometry. Our contribution in this paper is threefold. First, wepresent several new theoretical results including consistency as well ascontinuity and differentiability of the objective function in the finitedimensional case. Second, we develop a numerical implementation of finitedimensional convex PCA when the convex set is polyhedral, and show that thisprovides a natural approximation of Wasserstein geodesic PCA. Third, weillustrate our results with two financial applications, namely distributions ofstock returns ranked by size and the capital distribution curve, both of whichare of independent interest in stochastic portfolio theory.

Efficient Convex PCA with applications to Wasserstein geodesic PCA and ranked data