Manifold-valued Dirichlet Processes.

Area of Science:

• Statistics
• Data Science
• Computational Geometry

Background:

• Statistical models for manifold-valued data are crucial for analyzing data residing on curved spaces.
• Existing models often rely on local geodesic distances, limiting global parametric modeling on smooth manifolds.
• Current parametric models typically assume data are confined to small neighborhoods on the manifold.

Purpose of the Study:

• To address the 'locality' problem in manifold-valued data analysis.
• To propose a novel nonparametric model for global analysis of manifold-valued data.
• To unify multivariate general linear models (MGLMs) using multiple tangent spaces.

Main Methods:

• Developed a novel nonparametric framework unifying MGLMs via multiple tangent spaces.
• Proposed Dirichlet process mixtures of multivariate general linear models (DP-MGLM) on Riemannian manifolds.
• Utilized a mixture of local models to globally extend locally-defined parametric models.

Main Results:

• The proposed DP-MGLM framework generalizes existing Euclidean and non-Euclidean general linear models.
• The method enables grouping observations into sub-populations across multiple tangent spaces.
• Revealed hidden structures and discovered geodesic relationships between covariates (X) and manifold-valued responses (Y).

Conclusions:

• The DP-MGLM offers a powerful tool for global analysis of manifold-valued data.
• The framework provides insights into hidden geodesic relationships within complex datasets.
• Proof-of-concept experiments validate the model's efficacy in discovering data structure.