Non-linear Embeddings in Hilbert Simplex Geometry

TL;DR

This paper explores the use of Hilbert geometry for embedding graphs in the standard simplex, showing it is a competitive, fast, and robust alternative to hyperbolic and Euclidean embeddings.

Contribution

It introduces Hilbert simplex geometry as a new embedding space for graphs, demonstrating its effectiveness and computational advantages.

Findings

Hilbert simplex geometry is competitive with hyperbolic and Euclidean embeddings.

The approach is fast and numerically robust.

It effectively embeds graph distance matrices.

Abstract

A key technique of machine learning and computer vision is to embed discrete weighted graphs into continuous spaces for further downstream processing. Embedding discrete hierarchical structures in hyperbolic geometry has proven very successful since it was shown that any weighted tree can be embedded in that geometry with arbitrary low distortion. Various optimization methods for hyperbolic embeddings based on common models of hyperbolic geometry have been studied. In this paper, we consider Hilbert geometry for the standard simplex which is isometric to a vector space equipped with the variation polytope norm. We study the representation power of this Hilbert simplex geometry by embedding distance matrices of graphs. Our findings demonstrate that Hilbert simplex geometry is competitive to alternative geometries such as the Poincar\'e hyperbolic ball or the Euclidean geometry for embedding tasks while being fast and numerically robust.