Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Curvature Augmented Manifold Embedding and Learning

Yongming Liu

TL;DR

Curvature-Augmented Manifold Embedding and Learning (CAMEL) tackles dimensionality reduction by treating data-point interactions as a physics-inspired force system that includes a curvature-driven term. The method unifies existing force-based DR approaches and extends them with a many-body interaction model and a fast curvature surrogate, enabling unsupervised, supervised, semi-supervised, metric, and inverse learning. CAMEL introduces a curvature similarity metric and dedicated evaluation measures, and demonstrates competitive performance on benchmark datasets while offering improved interpretability and efficiency. The work provides a publicly available implementation to facilitate reproducibility and further applications.

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.

Curvature Augmented Manifold Embedding and Learning

TL;DR

Curvature-Augmented Manifold Embedding and Learning (CAMEL) tackles dimensionality reduction by treating data-point interactions as a physics-inspired force system that includes a curvature-driven term. The method unifies existing force-based DR approaches and extends them with a many-body interaction model and a fast curvature surrogate, enabling unsupervised, supervised, semi-supervised, metric, and inverse learning. CAMEL introduces a curvature similarity metric and dedicated evaluation measures, and demonstrates competitive performance on benchmark datasets while offering improved interpretability and efficiency. The work provides a publicly available implementation to facilitate reproducibility and further applications.

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.
Paper Structure (30 sections, 28 equations, 20 figures, 1 table, 5 algorithms)

This paper contains 30 sections, 28 equations, 20 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Method: t-SNE
  4. Method: LargeVis
  5. Method: UMAP
  6. Method: PaCMAP
  7. Method: TRIMAP
  8. Short summary for DR methods
  9. CAMEL method and algorithm
  10. Many body potential and force
  11. Curvature in topology
  12. Proposed force field model
  13. Algorithm and Implementation
  14. Unsupervised learning
  15. Supervised learning
  16. ...and 15 more sections

Figures (20)

  • Figure 1: Schematic illustration of neighbor forces of a hexagon lattice unit cell
  • Figure 2: Schematic illustration of different curvature for a pair of points $i$ and $j$
  • Figure 3: Schematic illustration of force field model for pair of points $i$ and $j$
  • Figure 4: Schematic illustration of same edge distance with different curvature in a graph
  • Figure 5: Visual comparison of 5 models with 8 datasets
  • ...and 15 more figures