A novel method, named Curvature-Augmented Manifold Embedding and Learning (CAMEL), is proposed for high dimensional data classification, dimension reduction, and visualization. CAMEL utilizes a topology metric defined on the Riemannian manifold, and a unique Riemannian metric for both distance and curvature to enhance its expressibility. The method also employs a smooth partition of unity operator on the Riemannian manifold to convert localized orthogonal projection to global embedding, which captures both the overall topological structure and local similarity simultaneously. The local orthogonal vectors provide a physical interpretation of the significant characteristics of clusters. Therefore, CAMEL not only provides a low-dimensional embedding but also interprets the physics behind this embedding. CAMEL has been evaluated on various benchmark datasets and has shown to outperform state-of-the-art methods, especially for high-dimensional datasets. The method's distinct benefits are its high expressibility, interpretability, and scalability. The paper provides a detailed discussion on Riemannian distance and curvature metrics, physical interpretability, hyperparameter effect, manifold stability, and computational efficiency for a holistic understanding of CAMEL. Finally, the paper presents the limitations and future work of CAMEL along with key conclusions.

翻译：针对高维数据分类、降维与可视化任务，本文提出了一种名为曲率增强的流形嵌入与学习（CAMEL）的新方法。CAMEL利用定义在黎曼流形上的拓扑度量，并采用独特的黎曼度量同时处理距离与曲率，从而增强其表达能力。该方法还在黎曼流形上运用了平滑单位分割算子，将局部正交投影转化为全局嵌入，从而同时捕捉整体拓扑结构与局部相似性。局部正交向量为聚类的显著特征提供了物理解释。因此，CAMEL不仅提供低维嵌入，还阐释了该嵌入背后的物理含义。CAMEL已在多个基准数据集上进行了评估，尤其在处理高维数据集时展现了优于现有最优方法的性能。该方法的核心优势在于其高表达性、可解释性与可扩展性。本文对黎曼距离与曲率度量、物理可解释性、超参数影响、流形稳定性及计算效率进行了详细讨论，旨在全面理解CAMEL。最后，本文总结了CAMEL的局限性、未来研究方向及关键结论。