What is the shortest path between two data points lying in a high-dimensional space? While the answer is trivial in Euclidean geometry, it becomes significantly more complex when the data lies on a curved manifold--requiring a Riemannian metric to describe the space's local curvature. Estimating such a metric, however, remains a major challenge in high dimensions. In this work, we propose a method for deriving Riemannian metrics directly from pretrained Energy-Based Models (EBMs)--a class of generative models that assign low energy to high-density regions. These metrics define spatially varying distances, enabling the computation of geodesics--shortest paths that follow the data manifold's intrinsic geometry. We introduce two novel metrics derived from EBMs and show that they produce geodesics that remain closer to the data manifold and exhibit lower curvature distortion, as measured by alignment with ground-truth trajectories. We evaluate our approach on increasingly complex datasets: synthetic datasets with known data density, rotated character images with interpretable geometry, and high-resolution natural images embedded in a pretrained VAE latent space. Our results show that EBM-derived metrics consistently outperform established baselines, especially in high-dimensional settings. Our work is the first to derive Riemannian metrics from EBMs, enabling data-aware geodesics and unlocking scalable, geometry-driven learning for generative modeling and simulation.

Many manifold learning algorithms aim to create embeddings with low or no distortion (isometric). If the data has intrinsic dimension d, it is often impossible to obtain an isometric embedding in ddimensions, but possible in s > ddimensions. Yet, most geometry preserving algorithms cannot do the latter. This paper proposes an embedding algorithm to overcome this. The algorithm accepts as input, besides the dimension d, an embedding dimension s d.

The key quantity of interest here is the Riemannian metric, which characterizes the Riemannian geometry and defines straight lines and derivatives on the manifold.

However, practical posteriors are often, even locally, highly non-Gaussian, and empirical performance deteriorates.