Geodesic Principal Component Analysis of Probability Measures Using Wasserstein Geometry
A new paper introduces Geodesic Principal Component Analysis (GPCA) for collections of probability distributions using the Otto-Wasserstein geometry, parameterizing geodesic curves with neural networks. The work addresses both Gaussian distributions—where computations are lifted to the space of invertible linear maps—and the more general case of absolutely continuous measures. The method offers a geometrically principled alternative to classical tangent PCA for analyzing variation in distributional datasets.
The paper, submitted to arXiv under machine learning and statistics, presents a framework for performing principal component analysis directly on spaces of probability measures equipped with the Wasserstein (optimal transport) metric. Unlike classical PCA, which operates in flat Euclidean space, GPCA seeks geodesic curves—the natural 'straight lines' in curved Wasserstein space—that best explain the variability in a dataset of distributions. For Gaussian distributions, the authors derive a tractable formulation by working in the space of invertible linear maps, exploiting known closed-form properties of Gaussian optimal transport. For general absolutely continuous measures, they introduce a neural network parameterization of Wasserstein geodesics, enabling flexible and scalable computation. The approach is validated through synthetic examples and real-world datasets, with comparisons to tangent PCA—a common approximation that linearizes the space around a reference point—demonstrating cases where the geodesic approach captures nonlinear structure that tangent methods miss.
What's missing
The paper is a preprint and has not yet undergone formal peer review. Key open questions include computational scalability to high-dimensional or large-sample distributional datasets, theoretical guarantees on the neural network parameterization (e.g., consistency or convergence rates), sensitivity to hyperparameter choices in the network architecture, and whether the geodesic uniqueness assumptions required by the method hold broadly in practice.
What different sources said
- arXiv cs.LGCenter
On the Wasserstein Geodesic Principal Component Analysis of probability measures
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