Multivariate General Linear Models (MGLM) on Riemannian Manifolds with Applications to Statistical Analysis of Diffusion Weighted Images

Linear regression is a parametric model which is ubiquitous in scientific analysis. The classical setup where the observations and responses, i.e., (x_i, y_i) pairs, are Euclidean is well studied. The setting where yi is manifold valued is a topic of much interest, motivated by applications in shape analysis, topic modeling, and medical imaging. Recent work gives strategies for max-margin classifiers, principal components analysis, and dictionary learning on certain types of manifolds. For parametric regression specifically, results within the last year provide mechanisms to regress one real-valued parameter, x_i ∈ R, against a manifold-valued variable, y_i ∈ M. We seek to substantially extend the operating range of such methods by deriving schemes for multivariate multiple linear regression -- a manifold-valued dependent variable against multiple independent variables, i.e., f: ℝⁿ → M. Our variational algorithm efficiently solves for multiple geodesic bases on the manifold concurrently via gradient updates. This allows us to answer questions such as: what is the relationship of the measurement at voxel y to disease when conditioned on age and gender. We show applications to statistical analysis of diffusion weighted images, which give rise to regression tasks on the manifold GL(n)/O(n) for diffusion tensor images (DTI) and the Hilbert unit sphere for orientation distribution functions (ODF) from high angular resolution acquisition. The companion open-source code is available on nitrc.org/projects/riem_mglm.