Fast PCA in 1-D Wasserstein Spaces via B-Splines Representation and Metric Projection

Abstract

We address the problem of performing Principal Component Analysis over a family of probability measures on the real line, using the Wasserstein geometry. We present a novel representation of the 2-Wasserstein space, based on a well known isometric bijection and a B-spline expansion. Thanks to this representation, we are able to reinterpret previous work and derive more efficient optimization routines for existing approaches. As shown in our simulations, the solution of these optimization problems can be costly in practice and thus pose a limit to their usage. We propose a novel definition of Principal Component Analysis in the Wasserstein space that, when used in combination with the B-spline representation, yields a straightforward optimization problem that is extremely fast to compute. Through extensive simulation studies, we show how our PCA performs similarly to the ones already proposed in the literature while retaining a much smaller computational cost. We apply our method to a real dataset of mortality rates due to Covid-19 in the US, concluding that our analyses are consistent with the current scientific consensus on the disease.