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

arXiv: https://arxiv.org/abs/1708.08143 · HTML: https://mlsanity.com/paper/1708.08143/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Aug 2017.

## Built on by (most cited first, 5 of 5)

- [Nearly Tight Convergence Bounds for Semi-discrete Entropic Optimal Transport](https://mlsanity.com/paper/2110.12678/) (2021)
- [Tangential Wasserstein Projections](https://mlsanity.com/paper/2207.14727/) (2022)
- [Low dimensional representation of multi-patient flow cytometry datasets using optimal transport for minimal residual disease detection in leukemia](https://mlsanity.com/paper/2407.17329/) (2024)
- [Wasserstein multivariate auto-regressive models for modeling distributional time series](https://mlsanity.com/paper/2207.05442/) (2022)
- [Fused Gromov-Wasserstein Variance Decomposition with Linear Optimal Transport](https://mlsanity.com/paper/2411.10204/) (2024)

## Abstract (arXiv)

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.