Gromov-Hausdorff Distances for Comparing Product Manifolds of Model Spaces

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 suggest using hyperbolic and spherical spaces with constant curvature, or their combinations (product manifolds), to improve model performance. However, determining the best latent product manifold signature, which refers to the choice and dimensionality of manifold components, lacks a principled technique. To address this, we introduce a novel notion of distance between candidate latent geometries using Gromov-Hausdorff from metric geometry. We propose using a graph search space that utilizes computed 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.