Multivariate Regression with Gross Errors on Manifold-Valued Data.

Area of Science:

• Statistics
• Machine Learning
• Computational Geometry

Background:

• Multivariate regression typically assumes Euclidean output spaces.
• Manifold-valued data, common in fields like medical imaging, presents unique challenges due to its non-Euclidean geometry.
• Existing regression models struggle with the presence of outliers or gross errors in manifold-valued responses.

Purpose of the Study:

• To develop a robust multivariate regression model for manifold-valued outputs.
• To address the critical issue of grossly corrupted responses in practical applications.
• To introduce a novel optimization technique for non-convex and non-smooth problems on Riemannian manifolds.

Main Methods:

• A two-step approach involving geodesic curve-based correction of corrupted responses.
• Application of multivariate linear regression on the corrected manifold-valued data.
• Development and extension of proximal alternating linearized minimization techniques (PALMR) for Riemannian manifolds.

Main Results:

• The proposed PALMR method effectively corrects grossly corrupted manifold-valued responses.
• The model demonstrates convergence to a critical point under mild theoretical conditions.
• Empirical results show superior performance compared to existing multivariate regression models on synthetic and diffusion tensor imaging data.

Conclusions:

• The developed regression model offers a robust solution for manifold-valued data with outliers.
• PALMR is effective in identifying and correcting gross errors in practical scenarios.
• This work advances regression techniques for complex, non-Euclidean data structures.