Geodesic Principal Component Analysis of Probability Measures Using Wasserstein Geometry
Researchers have developed a new method for performing Principal Component Analysis (PCA) on collections of probability distributions using Wasserstein geometry, a mathematical framework for measuring distances between distributions. The approach handles both Gaussian distributions through lifted computations and general probability measures using neural network parameterization of geodesics. This work extends classical dimensionality reduction techniques to probability spaces, with potential applications in analyzing complex distributional data.
The paper introduces Geodesic Principal Component Analysis (GPCA) applied to probability measures under the Otto-Wasserstein geometry framework. The authors first develop solutions for Gaussian distributions by lifting computations to the space of invertible linear maps, then extend to more general absolutely continuous probability measures using a novel neural network approach for parameterizing geodesics in Wasserstein space. The methodology is compared against classical tangent PCA approaches across multiple examples and real-world datasets. This work bridges differential geometry and machine learning by enabling dimensionality reduction directly in the space of probability distributions rather than in Euclidean spaces. The contribution addresses a gap in existing methods for capturing modes of variation in distributional data.
What's missing
The paper does not discuss computational complexity or scalability limitations of the proposed neural network approach for high-dimensional probability spaces, nor does it address convergence guarantees or theoretical error bounds for the geodesic parameterization method.
What different sources said
- arXiv cs.LGCenter
On the Wasserstein Geodesic Principal Component Analysis of probability measures
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