Stable Geodesic Update on Hyperbolic Space and its Application to Poincare Embeddings

TL;DR

This paper introduces a new geodesic update rule for hyperbolic space that guarantees convergence and outperforms Euclidean gradient descent, improving Poincare embeddings for complex network modeling.

Contribution

It presents an explicit geodesic update algorithm in hyperbolic space with proven convergence and better rate, addressing bias issues in existing Riemannian gradient methods.

Findings

Algorithm converges with theoretical guarantees.

Outperforms Euclidean gradient descent in convergence rate.

Demonstrates superior performance in Poincare embeddings.

Abstract

A hyperbolic space has been shown to be more capable of modeling complex networks than a Euclidean space. This paper proposes an explicit update rule along geodesics in a hyperbolic space. The convergence of our algorithm is theoretically guaranteed, and the convergence rate is better than the conventional Euclidean gradient descent algorithm. Moreover, our algorithm avoids the "bias" problem of existing methods using the Riemannian gradient. Experimental results demonstrate the good performance of our algorithm in the \Poincare embeddings of knowledge base data.