Curvature Augmented Manifold Embedding and Learning

TL;DR

CAMEL introduces a physics-inspired manifold embedding method that incorporates curvature and multi-body forces, improving data visualization and dimensionality reduction across various learning tasks.

Contribution

The paper presents a novel DR approach formulating the problem as a physics-based force field model with non-pairwise interactions, inspired by lattice physics and topology.

Findings

CAMEL outperforms existing methods like tSNE, UMAP, TRIMAP, and PacMap in visual and metric evaluations.

The method effectively handles unsupervised, supervised, semi-supervised, and inverse learning tasks.

Code and demonstrations are publicly available for reproducibility.

Abstract

A new dimensional reduction (DR) and data visualization method, Curvature-Augmented Manifold Embedding and Learning (CAMEL), is proposed. The key novel contribution is to formulate the DR problem as a mechanistic/physics model, where the force field among nodes (data points) is used to find an n-dimensional manifold representation of the data sets. Compared with many existing attractive-repulsive force-based methods, one unique contribution of the proposed method is to include a non-pairwise force. A new force field model is introduced and discussed, inspired by the multi-body potential in lattice-particle physics and Riemann curvature in topology. A curvature-augmented force is included in CAMEL. Following this, CAMEL formulation for unsupervised learning, supervised learning, semi-supervised learning/metric learning, and inverse learning are provided. Next, CAMEL is applied to many benchmark datasets by comparing existing models, such as tSNE, UMAP, TRIMAP, and PacMap. Both visual comparison and metrics-based evaluation are performed. 14 open literature and self-proposed metrics are employed for a comprehensive comparison. Conclusions and future work are suggested based on the current investigation. Related code and demonstration are available on https://github.com/ymlasu/CAMEL for interested readers to reproduce the results and other applications.