Regression Models on Riemannian Symmetric Spaces.

Area of Science:

• Statistics
• Differential Geometry
• Data Science

Background:

• Manifold-valued data, often found in medical imaging and computer vision, presents unique analytical challenges.
• Existing regression methods may not adequately capture the complex geometric structures inherent in Riemannian symmetric spaces (RSS).

Purpose of the Study:

• To develop a general, intrinsic regression framework for analyzing manifold-valued responses within Riemannian symmetric spaces (RSS).
• To establish methods for assessing the association between RSS-valued data and covariates in Euclidean space (e.g., age, gender).

Main Methods:

• Developed an intrinsic regression model based on conditional moment assumptions, avoiding parametric distribution specifications.
• Proposed novel link functions to connect Euclidean covariates with RSS responses.
• Implemented a two-stage estimation procedure and derived asymptotic distributions for parameter estimates.
• Constructed Wald and geodesic test statistics for hypothesis testing.

Main Results:

• The proposed regression framework provides a geometrically invariant approach to analyzing manifold-valued data.
• The methods were evaluated through simulation studies and a real-world data analysis, demonstrating good finite sample properties.
• Asymptotic distributions for parameter estimates and test statistics were derived.

Conclusions:

• The developed framework offers a powerful and flexible tool for regression analysis of manifold-valued data in RSS.
• The intrinsic approach ensures geometric invariance, crucial for reliable analysis in fields like medical imaging and computer vision.