# Neural Latent Geometry Search: Product Manifold Inference via Gromov-Hausdorff-Informed Bayesian Optimization

arXiv: https://arxiv.org/abs/2309.04810 · HTML: https://mlsanity.com/paper/2309.04810/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 58 references that are in the map.
- First posted on arXiv 9 Sep 2023.

## Builds on (most cited first, 30 of 32)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Neural Architecture Search with Reinforcement Learning](https://mlsanity.com/paper/1611.01578/) (2016)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Efficient Neural Architecture Search via Parameter Sharing](https://mlsanity.com/paper/1802.03268/) (2018)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Stochastic Beams and Where to Find Them: The Gumbel-Top-k Trick for Sampling Sequences Without Replacement](https://mlsanity.com/paper/1903.06059/) (2019)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [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)
- [Differentiable Graph Module (DGM) for Graph Convolutional Networks](https://mlsanity.com/paper/2002.04999/) (2020)
- [Beltrami Flow and Neural Diffusion on Graphs](https://mlsanity.com/paper/2110.09443/) (2021)
- [Combinatorial Bayesian Optimization using the Graph Cartesian Product](https://mlsanity.com/paper/1902.00448/) (2019)
- [Mat'ern Gaussian Processes on Graphs](https://mlsanity.com/paper/2010.15538/) (2020)
- [Sheaf Neural Networks with Connection Laplacians](https://mlsanity.com/paper/2206.08702/) (2022)
- [Learning Signal-Agnostic Manifolds of Neural Fields](https://mlsanity.com/paper/2111.06387/) (2021)
- [Disentangling by Subspace Diffusion](https://mlsanity.com/paper/2006.12982/) (2020)
- [Gaussian Processes on Graphs via Spectral Kernel Learning](https://mlsanity.com/paper/2006.07361/) (2020)
- [Latent Graph Inference using Product Manifolds](https://mlsanity.com/paper/2211.16199/) (2022)
- [Learning disentangled representations via product manifold projection](https://mlsanity.com/paper/2103.01638/) (2021)
- [Deep Neural Architecture Search with Deep Graph Bayesian Optimization](https://mlsanity.com/paper/1905.06159/) (2019)
- [Product Manifold Learning](https://mlsanity.com/paper/2010.09908/) (2020)
- [Identifying the latent space geometry of network models through analysis of curvature](https://mlsanity.com/paper/2012.10559/) (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)
- [Geometry of Deep Generative Models for Disentangled Representations](https://mlsanity.com/paper/1902.06964/) (2019)

## Built on by (most cited first, 18 of 18)

- [On Vanishing Gradients, Over-Smoothing, and Over-Squashing in GNNs: Bridging Recurrent and Graph Learning](https://mlsanity.com/paper/2502.10818/) (2025)
- [Rough Transformers: Lightweight and Continuous Time Series Modelling through Signature Patching](https://mlsanity.com/paper/2405.20799/) (2024)
- [Uncertainty Modeling in Graph Neural Networks via Stochastic Differential Equations](https://mlsanity.com/paper/2408.16115/) (2024)
- [Neural Snowflakes: Universal Latent Graph Inference via Trainable Latent Geometries](https://mlsanity.com/paper/2310.15003/) (2023)
- [Neural Spacetimes for DAG Representation Learning](https://mlsanity.com/paper/2408.13885/) (2024)
- [A Gaussian Process View on Observation Noise and Initialization in Wide Neural Networks](https://mlsanity.com/paper/2502.01556/) (2025)
- [AMES: A Differentiable Embedding Space Selection Framework for Latent Graph Inference](https://mlsanity.com/paper/2311.11891/) (2023)
- [Mixed-curvature decision trees and random forests](https://mlsanity.com/paper/2410.13879/) (2024)
- [Toward a Unified Geometry Understanding: Riemannian Diffusion Framework for Graph Generation and Prediction](https://mlsanity.com/paper/2510.04522/) (2025)
- [Bayesian Optimization of Functions over Node Subsets in Graphs](https://mlsanity.com/paper/2405.15119/) (2024)
- [Geometric sparsification in recurrent neural networks](https://mlsanity.com/paper/2406.06290/) (2024)
- [Mixed-Curvature Decision Trees and Random Forests](https://mlsanity.com/paper/2406.05227/) (2024)
- [Metric Learning for Clifford Group Equivariant Neural Networks](https://mlsanity.com/paper/2407.09926/) (2024)
- [GaiaFlow: Semantic-Guided Diffusion Tuning for Carbon-Frugal Search](https://mlsanity.com/paper/2602.15423/) (2026)
- [Understanding Latent Diffusability via Fisher Geometry](https://mlsanity.com/paper/2604.02751/) (2026)
- [Classification Fields: Arbitrarily Fine Recursive Hierarchical Clustering From Few Examples](https://mlsanity.com/paper/2605.07119/) (2026)
- [MC-RFM: Geometry-Aware Few-Shot Adaptation via Mixed-Curvature Riemannian Flow Matching](https://mlsanity.com/paper/2605.08557/) (2026)
- [From topology learning to graph generation: A unifying perspective](https://mlsanity.com/paper/2609.02286/) (2026)

## Abstract (arXiv)

Recent research indicates that the performance of machine learning models can be improved by aligning the geometry of the latent space with the underlying data structure. Rather than relying solely on Euclidean space, researchers have proposed using hyperbolic and spherical spaces with constant curvature, or combinations thereof, to better model the latent space and enhance model performance. However, little attention has been given to the problem of automatically identifying the optimal latent geometry for the downstream task. We mathematically define this novel formulation and coin it as neural latent geometry search (NLGS). More specifically, we introduce an initial attempt to search for a latent geometry composed of a product of constant curvature model spaces with a small number of query evaluations, under some simplifying assumptions. To accomplish this, we propose a novel notion of distance between candidate latent geometries based on the Gromov-Hausdorff distance from metric geometry. In order to compute the Gromov-Hausdorff distance, we introduce a mapping function that enables the comparison of different manifolds by embedding them in a common high-dimensional ambient space. We then design a graph search space based on the notion of smoothness between latent geometries and employ the calculated distances as an additional inductive bias. Finally, we use Bayesian optimization to search for the optimal latent geometry in a query-efficient manner. This is a general method which can be applied to search for the optimal latent geometry for a variety of models and downstream tasks. We perform experiments on synthetic and real-world datasets to identify the optimal latent geometry for multiple machine learning problems.