# Gromov-Hausdorff Distances for Comparing Product Manifolds of Model Spaces

arXiv: https://arxiv.org/abs/2309.05678 · HTML: https://mlsanity.com/paper/2309.05678/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 20 references that are in the map.
- First posted on arXiv 9 Sep 2023.

## Builds on (most cited first, 12 of 12)

- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Continuous Hierarchical Representations with Poincar'e Variational Auto-Encoders](https://mlsanity.com/paper/1901.06033/) (2019)
- [Mixed-curvature Variational Autoencoders](https://mlsanity.com/paper/1911.08411/) (2019)
- [Beltrami Flow and Neural Diffusion on Graphs](https://mlsanity.com/paper/2110.09443/) (2021)
- [Sheaf Neural Networks with Connection Laplacians](https://mlsanity.com/paper/2206.08702/) (2022)
- [Latent Graph Inference using Product Manifolds](https://mlsanity.com/paper/2211.16199/) (2022)
- [Product Manifold Learning](https://mlsanity.com/paper/2010.09908/) (2020)
- [Projections of Model Spaces for Latent Graph Inference](https://mlsanity.com/paper/2303.11754/) (2023)
- [Latent Variables on Spheres for Autoencoders in High Dimensions](https://mlsanity.com/paper/1912.10233/) (2019)
- [The Complexity of the Hausdorff Distance](https://mlsanity.com/paper/2112.04343/) (2021)

## Abstract (arXiv)

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.