CAMEL: Curvature-Augmented Manifold Embedding and Learning

TL;DR

CAMEL is a novel Riemannian manifold embedding method that enhances high-dimensional data analysis by capturing topological and local structures, providing interpretability and outperforming existing techniques.

Contribution

Introduces CAMEL, a Riemannian metric-based embedding technique that combines topology, curvature, and local orthogonal projections for improved data visualization and classification.

Findings

Outperforms state-of-the-art methods on benchmark datasets

Provides interpretable embeddings with physical insights

Demonstrates high scalability and robustness

Abstract

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.