# Curvature Augmented Manifold Embedding and Learning

arXiv: https://arxiv.org/abs/2403.14813 · HTML: https://mlsanity.com/paper/2403.14813/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 43 references that are in the map.
- First posted on arXiv 21 Mar 2024.

## Builds on (most cited first, 9 of 9)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization](https://mlsanity.com/paper/2012.04456/) (2020)
- [Visualizing Large-scale and High-dimensional Data](https://mlsanity.com/paper/1602.00370/) (2016)
- [Community Detection on Networks with Ricci Flow](https://mlsanity.com/paper/1907.03993/) (2019)
- [TriMap: Large-scale Dimensionality Reduction Using Triplets](https://mlsanity.com/paper/1910.00204/) (2019)
- [Attraction-Repulsion Spectrum in Neighbor Embeddings](https://mlsanity.com/paper/2007.08902/) (2020)
- [From t-SNE to UMAP with contrastive learning](https://mlsanity.com/paper/2206.01816/) (2022)
- [Hierarchical Nearest Neighbor Graph Embedding for Efficient Dimensionality Reduction](https://mlsanity.com/paper/2203.12997/) (2022)
- [Relating tSNE and UMAP to Classical Dimensionality Reduction](https://mlsanity.com/paper/2306.11898/) (2023)

## Built on by (most cited first, 3 of 3)

- [Harnessing Data Asymmetry: Manifold Learning in the Finsler World](https://mlsanity.com/paper/2603.11396/) (2026)
- [IIKL: Isometric Immersion Kernel Learning with Riemannian Manifold for Geometric Preservation](https://mlsanity.com/paper/2505.06288/) (2025)
- [Learning Eigenstructures of Unstructured Data Manifolds](https://mlsanity.com/paper/2512.01103/) (2025)

## Abstract (arXiv)

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.