A Wasserstein-Type Distance for Gaussian Mixtures on Vector Bundles with Applications to Shape Analysis.

Area of Science:

• Computational geometry
• Geometric statistics
• Data analysis on manifolds

Background:

• Comparing populations on complex geometric spaces like Riemannian manifolds is challenging.
• Existing methods often struggle with the intrinsic geometry of these spaces.
• Representing populations as probability distributions is a common approach.

Purpose of the Study:

• To develop a novel framework for comparing populations residing on finite-dimensional parallelizable Riemannian manifolds and trivial vector bundles.
• To adapt statistical methods for analyzing population data with complex geometric structures.
• To enable robust analysis of shape data and time-series data in geometric contexts.

Main Methods:

• Representing populations as Gaussian mixtures on vector bundles, leveraging triviality.
• Employing a mode-based clustering algorithm for parameter estimation.
• Deriving a Wasserstein-type metric tailored to manifold geometry for comparing distributions.
• Establishing an identifiability result for Gaussian mixtures on manifold domains.
• Characterizing optimal couplings for Gaussian mixtures under the new metric.

Main Results:

• A novel Wasserstein-type metric is derived, accounting for manifold geometry.
• An identifiability result for Gaussian mixtures on manifold domains is proven.
• The framework is demonstrated on diverse geometric domains, including preshape spaces.
• The method successfully performs change-point detection on nanoparticle shape data from a manufacturing process.

Conclusions:

• The proposed framework provides a powerful and adaptable tool for comparing populations on geometric manifolds.
• The derived Wasserstein-type metric and associated methods enhance statistical analysis in geometric spaces.
• The application to nanoparticle manufacturing demonstrates practical utility in process monitoring and quality control.