We introduce a novel framework for the classification of functional data supported on nonlinear, 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 as 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 prohibitive 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. Through this application, we identify discriminant directions that capture both cortical geometric and thickness predictive features of Alzheimer's disease that are consistent with the existing neuroscience literature.

翻译：我们提出了一种新颖的框架，用于对定义在非线性（可能为随机）流形域上的函数数据进行分类。其动机源于基于受试者大脑皮层表面几何形态及其关联皮层厚度图谱识别阿尔茨海默病患者的应用。所提模型基于将分类问题重构为正则化多元函数线性回归模型，从而可采用直接方法估计最具判别性的方向，并通过适当的微分正则化控制其复杂度。我们的方法无需预先估计函数预测变量的协方差结构——这一过程在应用场景中计算代价过高。我们提供了所提模型样本外预测误差的理论分析，并探索了模拟实验中的有限样本性能。将该方法应用于来自阿尔茨海默病神经影像学倡议与帕金森病进展标志物倡议的合并数据集。通过该应用，我们识别出同时捕捉阿尔茨海默病皮层几何与厚度预测特征的判别方向，这些发现与现有神经科学文献一致。