  We introduce a novel framework for the classification of functional data
supported on non-linear, and possibly random, manifold domains. The motivating
application is the identification of subjects with Alzheimer's disease from
their cortical surface geometry and associated cortical thickness map. The
proposed model is based upon a reformulation of the classification problem into
a regularized multivariate functional linear regression model. This allows us
to adopt a direct approach to the estimation of the most discriminant direction
while controlling for its complexity with appropriate differential
regularization. Our approach does not require prior estimation of the
covariance structure of the functional predictors, which is computationally not
feasible in our application setting. We provide a theoretical analysis of the
out-of-sample prediction error of the proposed model and explore the finite
sample performance in a simulation setting. We apply the proposed method to a
pooled dataset from the Alzheimer's Disease Neuroimaging Initiative and the
Parkinson's Progression Markers Initiative, and are able to estimate
discriminant directions that capture both cortical geometric and thickness
predictive features of Alzheimer's Disease, which are consistent with the
existing neuroscience literature.

Methodology;Applications