# GLEE: Geometric Laplacian Eigenmap Embedding

## TL;DR

GLEE introduces a novel geometric approach to graph embedding that leverages simplex geometry, outperforming spectral methods like Laplacian Eigenmaps in reconstruction and link prediction tasks.

## Contribution

The paper proposes GLEE, a new graph embedding method based on geometric properties rather than spectral assumptions, improving performance in key graph tasks.

## Key findings

- GLEE outperforms Laplacian Eigenmaps in graph reconstruction.
- GLEE achieves better link prediction accuracy.
- The geometric approach provides more meaningful embeddings.

## Abstract

Graph embedding seeks to build a low-dimensional representation of a graph G. This low-dimensional representation is then used for various downstream tasks. One popular approach is Laplacian Eigenmaps, which constructs a graph embedding based on the spectral properties of the Laplacian matrix of G. The intuition behind it, and many other embedding techniques, is that the embedding of a graph must respect node similarity: similar nodes must have embeddings that are close to one another. Here, we dispose of this distance-minimization assumption. Instead, we use the Laplacian matrix to find an embedding with geometric properties instead of spectral ones, by leveraging the so-called simplex geometry of G. We introduce a new approach, Geometric Laplacian Eigenmap Embedding (or GLEE for short), and demonstrate that it outperforms various other techniques (including Laplacian Eigenmaps) in the tasks of graph reconstruction and link prediction.