Log-PCA versus Geodesic PCA of histograms in the Wasserstein space

TL;DR

This paper compares log-PCA and geodesic PCA methods for analyzing histogram data in Wasserstein space, proposing a new algorithm for geodesic PCA and evaluating both methods on 1D and 2D data sets.

Contribution

It introduces a novel forward-backward algorithm for approximate geodesic PCA and provides a detailed comparison with log-PCA for histogram data.

Findings

Geodesic PCA offers advantages in capturing data geometry.

Log-PCA is computationally simpler but less accurate.

The methods perform differently depending on data dimensionality.

Abstract

This paper is concerned by the statistical analysis of data sets whose elements are random histograms. For the purpose of learning principal modes of variation from such data, we consider the issue of computing the PCA of histograms with respect to the 2-Wasserstein distance between probability measures. To this end, we propose to compare the methods of log-PCA and geodesic PCA in the Wasserstein space as introduced by Bigot et al. (2015) and Seguy and Cuturi (2015). Geodesic PCA involves solving a non-convex optimization problem. To solve it approximately, we propose a novel forward-backward algorithm. This allows a detailed comparison between log-PCA and geodesic PCA of one-dimensional histograms, which we carry out using various data sets, and stress the benefits and drawbacks of each method. We extend these results for two-dimensional data and compare both methods in that setting.