Multivariate General Linear Models (MGLM) on Riemannian Manifolds with Applications to Statistical Analysis of Diffusion Weighted Images
Hyunwoo J. Kim, Nagesh Adluru, Maxwell D. Collins, Moo K. Chung, Barbara B. Bendlin, Sterling C. Johnson, Richard J. Davidson, Vikas Singh,
Multivariate General Linear Models (MGLM) on Riemannian Manifolds with Applications to Statistical Analysis of Diffusion Weighted Images
, In IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR), June 2014.
Abstract
Linear regression is a parametric model which is ubiquitous in scientific 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 maxmargin 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 realvalued parameter, xi in R, against a manifoldvalued variable, yi in M. We seek to substantially extend the operating range of such methods by deriving schemes for multivariate multiple linear regression  a manifoldvalued 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 ten sor images (DTI) and the Hilbert unit sphere for orientation distribution functions (ODF) from high angular resolution acquisition. The companion opensource code is available on nitrc.org/projects/riem_mglm.
Acknowledgments
This work was supported in part by NIH R01 AG040396; R01 AG037639; R01 AG027161; R01 AG021155; NSF CAREER award 1252725; NSF RI 1116584; Wisconsin Partnership Fund; UW ADRC P50 AG033514; UW ICTR 1UL1RR025011; a VA Merit Review Grant I01CX000165; NCCAM P01 AT00495204 and the Waisman Core grant P30 HD00335245. Collins was supported by a CIBM fellowship (NLM 5T15LM007359). The contents do not represent views of the Dept. of Veterans Affairs or the United States Government. We are grateful to Anqi Qiu (National University of Singapore) for explaining various mathematical details of ODF construction with code examples, which were useful in setting up the experimental analysis.
