Curvature Corrected Nonnegative Manifold Data Factorization

TL;DR

This paper introduces a curvature corrected nonnegative manifold data factorization method that leverages Riemannian geometry to analyze manifold-valued data, demonstrated on diffusion tensor MRI data.

Contribution

It proposes a novel geometry-aware factorization method for manifold data, extending nonnegative matrix factorization to Riemannian manifolds with an efficient algorithm.

Findings

Effective extraction of interpretable factors from manifold data

Demonstrated on real-world diffusion tensor MRI data

Provides a new tool for nonlinear geometric data analysis

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.