Riemannian Nonlinear Mixed Effects Models: Analyzing Longitudinal
Deformations in Neuroimaging
Abstract
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,
efficient formulations for parametric models (which may offer benefits in small sample size regimes) have only emerged
recently. So far, 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 of fixed 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 : Rd ? 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