Curvature Corrected Nonnegative Manifold Data Factorization

TL;DR

The paper addresses the challenge of performing interpretable, low-rank factorization on manifold-valued data by introducing curvature corrected nonnegative manifold data factorization (CC-NMDF) for symmetric Riemannian manifolds. It recasts a Euclidean semi-NMF into a tangent-space framework, then applies curvature corrections to better approximate exact manifold reconstructions while remaining computationally tractable. An alternating, multiplicative-update algorithm is developed, with an initialization scheme via tangent-space K-means and corrections to mitigate factor cancellations; the method is validated on diffusion tensor MRI data, showing improved reconstruction and interpretability over tangent-space NMDF and competitive performance against curvature-corrected SVD variants. The work demonstrates how geometry-aware, low-rank decompositions can yield meaningful, identifiable factors that respect the underlying manifold structure, with practical implications for analyzing non-Euclidean scientific data such as DT-MRI.

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.

Figures (4)

• Figure 1: Visualization of the DTI data set (left) and two "slices" from the data (right).
• Figure 2: The error incurred by CC-NMDF using approximate zero ($\mathbf{q}$) and the data barycenter ($\mathbf{r}$) as the base point of linearization, plotted against approximation rank ($K$).
• Figure 3: The factors obtained from rank-20 CC-NMDF using (a) $\mathbf{q}$, approximately zero, and (b) $\mathbf{r}$, the data barycenter, as the base point of linearization. (b) shows a clear deterioration in the interpretability of the factors when compared to (a) and the original data set.
• Figure 4: (a) Approximation error incurred by CC-NMDF, T-NMDF, and CC-SVD for approximation ranks 2-35. (b) Manifold-valued factors obtained by CC-NMDF. (c) Manifold-valued factors obtained by T-NMDF. (d) Manifold-valued factors obtained by CC-SVD.