Gromov-Hausdorff Distances for Comparing Product Manifolds of Model Spaces

TL;DR

This paper introduces a method to compare and select optimal product manifold geometries for latent spaces in machine learning models using Gromov-Hausdorff distances, enabling principled geometry choice.

Contribution

It presents a new approach to quantify differences between candidate latent geometries and an algorithm to compute these distances, aiding in optimal manifold selection.

Findings

Algorithm for computing Gromov-Hausdorff distance between model spaces

Graph search method for selecting best latent geometry

Implementation details for practical computation

Abstract

Recent studies propose enhancing machine learning models by aligning the geometric characteristics of the latent space with the underlying data structure. Instead of relying solely on Euclidean space, researchers have suggested using hyperbolic and spherical spaces with constant curvature, or their combinations (known as product manifolds), to improve model performance. However, there exists no principled technique to determine the best latent product manifold signature, which refers to the choice and dimensionality of manifold components. To address this, we introduce a novel notion of distance between candidate latent geometries using the Gromov-Hausdorff distance from metric geometry. We propose using a graph search space that uses the estimated Gromov-Hausdorff distances to search for the optimal latent geometry. In this work we focus on providing a description of an algorithm to compute the Gromov-Hausdorff distance between model spaces and its computational implementation.