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 superior classifiers can be developed based on 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 new family of functional classifiers. In theory, we prove 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 highly competitive classification performance compared to existing functional classifiers across both synthetic and real data examples. Supplementary materials are available online.

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开发出更优的分类器。为此，我们提出了一种新颖的邻近度量方法，该方法通过考虑标签信息来学习低维表示（亦称为监督流形学习结果）。当这些结果与多元分类器结合时，该流程催生出一类新的函数型分类器。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，我们通过合成与真实数据案例表明，本方法与多种经典多元分类器结合后，相较于现有函数型分类器能取得极具竞争力的分类性能。补充材料可在线获取。