Gromov-Hausdorff Distances for Comparing Product Manifolds of Model
  Spaces
Haitz Saez de Ocariz Borde Alvaro Arroyo Ismael Morales Ingmar Posner Xiaowen Dong
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.