Riemannian Nonlinear Mixed Effects Models: Analyzing Longitudinal Deformations in Neuroimaging

Statistical machine learning models that operate on manifold-valued data are being extensively studied in vision, motivated by applications in activity recognition, feature tracking and medical imaging. While non-parametric methods have been relatively well studied in the literature, efficientformulations for parametric models (which may offer benefits in small sample size regimes) have only emerged recently. Sofar, manifold-valued regression models (such as geodesic regression) are restricted to the analysis of crosssectional data, i.e., the so-called “fixed effects” in statistics. But in most “longitudinal analysis” (e.g., when a participant provides multiple measurements, over time) the application offixed effects models is problematic. In an effort to answer this need, this paper generalizes non-linear mixed effects model to the regime where the response variable is manifold-valued, i.e., f : R
^d
→ M. We derive the underlying model and estimation schemes and demonstrate the immediate benefits such a model can provide - both for group level and individual level analysis - on longitudinal brain imaging data. The direct consequence of our results is that longitudinal analysis of manifold-valued measurements (especially, the symmetric positive definite manifold) can be conducted in a computationally tractable manner.