Classification is a core topic in functional data analysis. A large number of functional classifiers have been proposed in the literature, most of which are based on functional principal component analysis or functional regression. In contrast, we investigate this topic from the perspective of manifold learning. It is assumed that functional data lie on an unknown low-dimensional manifold, and we expect that better classifiers can be built upon the manifold structure. To this end, we propose a novel proximity measure that takes the label information into account to learn the low-dimensional representations, also known as the supervised manifold learning outcomes. When the outcomes are coupled with multivariate classifiers, the procedure induces a family of new functional classifiers. In theory, we show that our functional classifier induced by the $k$-NN classifier is asymptotically optimal. In practice, we show that our method, coupled with several classical multivariate classifiers, achieves outstanding classification performance compared to existing functional classifiers in both synthetic and real data examples.

翻译：分类是函数型数据分析中的一个核心课题。文献中已提出大量函数型分类器，其中大多数基于函数型主成分分析或函数型回归。相比之下，我们从流形学习的角度研究这一课题。假设函数型数据位于未知的低维流形上，我们期望基于流形结构能够构建更好的分类器。为此，我们提出了一种新颖的邻近度量方法，该方法在低维表示（亦称为监督流形学习结果）的学习过程中纳入标签信息。当这些结果与多元分类器结合时，该过程衍生出一系列新的函数型分类器。在理论上，我们证明了由$k$-NN分类器导出的函数型分类器具有渐近最优性。在实际应用中，我们通过合成数据与真实数据案例表明，相较于现有函数型分类器，本方法与多种经典多元分类器结合后能取得卓越的分类性能。