Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models

2025-05-23

AI Summary

High-dimensional data often reside on curved manifolds, where Euclidean distances fail to capture intrinsic geometric structure; moreover, estimating Riemannian metrics in high dimensions remains notoriously challenging. To address this, we propose a novel Riemannian metric construction method grounded in pretrained Energy-Based Models (EBMs), the first to directly derive a data-adaptive Riemannian metric from the EBM energy function—ensuring geodesic lengths vary with local data density and preserving geometric consistency. We further design two low-curvature distortion metric formulations and integrate them with geodesic numerical optimization and VAE latent-space embedding. Evaluated on synthetic manifolds, rotated character datasets, and high-resolution natural image latent spaces, our approach significantly outperforms existing methods: it achieves superior manifold fidelity and higher geodesic path alignment accuracy, particularly in high-dimensional settings.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

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.

Problem

• Estimating Riemannian metrics in high-dimensional curved manifolds
• Deriving Riemannian metrics from Energy-Based Models (EBMs)
• Computing geodesics aligned with data manifold's intrinsic geometry

Innovation

• Derive Riemannian metrics from EBMs
• Compute geodesics using data-aware metrics
• Outperform baselines in high-dimensional settings