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Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models
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Derives conformal Riemannian metrics (LogEBM and InvEBM) from pretrained energy-based models and uses them to compute data-aware geodesics, outperforming RBF and LAND metrics on toy, rotated-character, and AFHQ latent-space data.
Problem
Estimating a Riemannian metric that captures the geometry of high-dimensional curved data manifolds so that shortest paths (geodesics) stay on the data manifold; existing density-based metrics degrade in high dimensions.
Approach
Use the energy function of a pretrained EBM directly to define conformal Riemannian metrics: LogEBM scales the identity by an affine function of the energy, and InvEBM uses the inverse of the unnormalized density, so geodesics are cheap in high-density regions and costly off-manifold.
Key results
- On AFHQ latent space, InvEBM metric achieves FID 16.47 along geodesics vs 42.47 for linear interpolation, 32.67 for slerp, 37.98 for RBF, and 39.17 for LAND.
- LogEBM achieves the highest accumulated probability along geodesics on the UCG toy dataset; InvEBM yields lowest RMSE to its baseline geodesic.
- On rotated characters, EBM-based metrics consistently outperform RBF and LAND on D-RMSE and gamma-RMSE, with LAND showing large deviations.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the inverse EBM metric instead of linear interpolation | Geodesic interpolation in the Stable Diffusion v1 VAE latent space · AFHQ | FID: 42.47 -> 16.47 | Helped |
| Use the log EBM metric instead of linear interpolation | Geodesic interpolation in the Stable Diffusion v1 VAE latent space · AFHQ | FID: 42.47 -> 20.79 | Helped |
| Use the inverse EBM metric instead of RBF | Geodesic interpolation in the Stable Diffusion v1 VAE latent space · AFHQ | FID: 37.98 -> 16.47 | Helped |
| Use the log EBM metric instead of RBF | Geodesic interpolation in the Stable Diffusion v1 VAE latent space · AFHQ | FID: 37.98 -> 20.79 | Helped |
| Use the inverse EBM metric instead of LAND | Geodesic interpolation in the Stable Diffusion v1 VAE latent space · AFHQ | FID: 39.17 -> 16.47 | Helped |
| Use the log EBM metric instead of LAND | Geodesic interpolation in the Stable Diffusion v1 VAE latent space · AFHQ | FID: 39.17 -> 20.79 | Helped |
| Use the inverse EBM metric instead of slerp | Geodesic interpolation in the Stable Diffusion v1 VAE latent space · AFHQ | FID: 32.67 -> 16.47 | Helped |
| Use the log EBM metric instead of slerp | Geodesic interpolation in the Stable Diffusion v1 VAE latent space · AFHQ | FID: 32.67 -> 20.79 | Helped |
| Use EBM-derived metrics instead of RBF or LAND | Geodesic interpolant · URC and BRC | D-RMSE | Helped |
| Use EBM-derived metrics instead of RBF or LAND | Geodesic interpolant · URC and BRC | gamma-RMSE | Helped |
| Use the log EBM metric instead of the inverse EBM metric | Geodesic interpolant · UCG | Accumulated probability | Helped |
| Use the inverse EBM metric instead of the log EBM metric | Geodesic interpolant · UCG | RMSE to matched true-distribution baseline geodesic | Helped |
Surprising findings
- Geodesics move faster in Euclidean terms through high-density regions because constant Riemannian speed is enforced, which is counterintuitive.
- The parametric RBF and LAND metrics produce trajectories only slightly better than linear interpolation on AFHQ, suggesting they fail to scale in high dimensions.
Limitations
- Only conformal (isotropic) metrics are studied; non-conformal metrics such as the Stein metric are left for future work.
- No ground-truth geodesics on AFHQ, so only FID-based on-manifold evaluation is possible.
- Training EBMs is costly and training can be unstable in high dimensions.
Experimental setup
Datasets: Uniform Circular Gaussians (synthetic) Weighted Circular Gaussians (synthetic) Rotated characters (URC, BRC) AFHQ
Benchmarks: Uniform Circular Gaussians Weighted Circular Gaussians Uniform Rotated Characters Biased Rotated Characters AFHQ
Metrics: accumulated probability along geodesic RMSE to baseline geodesic D-RMSE gamma-RMSE FID along geodesics
Baselines: LAND metric RBF metric linear interpolation slerp interpolation true-density metrics on toy data
Builds on: LAND metric RBF metric contrastive divergence geodesic interpolant (Yang and Hayes-style neural geodesics) Stable Diffusion VAE
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