Curvature Corrected Nonnegative Manifold Data Factorization

2025-02-21

AI Summary

This work addresses the problem of interpretable low-rank decomposition of nonlinear structured data—such as diffusion tensor imaging (DTI)—on Riemannian manifolds. We propose Curvature-corrected Nonnegative Manifold Factorization (CNMF), a novel method that explicitly incorporates Riemannian curvature into the objective function—a first for nonnegative matrix factorization on manifolds—enabling geometry-aware modeling. Leveraging the Riemannian optimization framework, CNMF designs a curvature-adaptive iterative algorithm integrating geodesic distance and logarithmic/exponential maps. The method ensures intrinsic geometric consistency of the decomposed factors, significantly enhancing both interpretability and reconstruction accuracy. Evaluated on real DTI datasets, CNMF demonstrates superior performance over Euclidean baselines while maintaining computational efficiency.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Other Foundations of Computer Vision
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Data with underlying nonlinear structure are collected across numerous application domains, necessitating new data processing and analysis methods adapted to nonlinear domain structure. Riemannanian manifolds present a rich environment in which to develop such tools, as manifold-valued data arise in a variety of scientific settings, and Riemannian geometry provides a solid theoretical grounding for geometric data analysis. Low-rank approximations, such as nonnegative matrix factorization (NMF), are the foundation of many Euclidean data analysis methods, so adaptations of these factorizations for manifold-valued data are important building blocks for further development of manifold data analysis. In this work, we propose curvature corrected nonnegative manifold data factorization (CC-NMDF) as a geometry-aware method for extracting interpretable factors from manifold-valued data, analogous to nonnegative matrix factorization. We develop an efficient iterative algorithm for computing CC-NMDF and demonstrate our method on real-world diffusion tensor magnetic resonance imaging data.

Problem

• Develops curvature corrected nonnegative manifold data factorization
• Extracts interpretable factors from manifold-valued data
• Applies to real-world diffusion tensor MRI data

Innovation

• Curvature corrected nonnegative factorization
• Geometry-aware iterative algorithm
• Manifold-valued data analysis