Abstract

Linear regression is a parametric model which is ubiquitous in scientifific analysis. The classical setup where the observations and responses, i.e., (xi , yi) 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 classi- fifiers, principal components analysis, and dictionary learning on certain types of manifolds. For parametric regression specififically, results within the last year provide mechanisms to regress one real-valued parameter, xi ∈ R, against a manifold-valued variable, yi ∈ 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 effificiently 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.