Research map / Probabilistic and scientific modeling / Variational and generative sampling
Neural Sampling from Boltzmann Densities: Fisher-Rao Curves in the Wasserstein Geometry
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Shows that Boltzmann Fisher-Rao curves are absolutely continuous Wasserstein curves, exhibits an exploding velocity norm for linear interpolation, and proposes a gradient-flow interpolation f_t=(T-t)/T f_D + t psi_t with fixed Langevin velocity field for neural sampling from unnormalized densities.
Problem
Sampling from an unnormalized Boltzmann density by learning a curve of Boltzmann densities from a simple prior; existing interpolations (linear, learned) have ill-behaved velocity fields with unclear Wasserstein regularity.
Approach
Prove that Fisher-Rao curves of Boltzmann densities are Wasserstein absolutely continuous under a partial Poincaré inequality, show the linear interpolation's velocity norm can explode, and propose a gradient-flow interpolation that fixes v_t = grad(f_t - f_Z) (Langevin/Fokker-Planck) and learns only psi_t, sampling via an ODE.
Key results
- On the 40-mode Gaussian mixture, gradient flow interpolation achieves ESS 0.992 and energy distance 0.0021 versus 0.183 ESS and 0.247 energy distance for linear interpolation.
- On the 8-dimensional many well distribution, learned interpolation reaches ESS 0.999 and energy distance 9e-5, gradient flow ESS 0.991 with energy distance 8e-4, linear ESS 0.985 with energy distance 0.278.
- An analytic 1D example shows the L2 norm of the linear-interpolation velocity field explodes near t=1, growing roughly exponentially with the distance m of the second mode.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use gradient flow interpolation instead of learned interpolation (σ=30) | Boltzmann sampling · Gaussian Mixture Model (40 modes) | ESS: 0.922 -> 0.992 | Helped |
| Use gradient flow interpolation instead of learned interpolation (σ=20) | Boltzmann sampling · Gaussian Mixture Model (40 modes) | NLL: 6.894 -> 6.864 | Helped |
| Use gradient flow interpolation instead of learned interpolation (σ=20) | Boltzmann sampling · Gaussian Mixture Model (40 modes) | Energy Distance: 0.058 -> 0.0021 | Helped |
| Use gradient flow interpolation instead of learned interpolation | Boltzmann sampling · Many Well (8D) | ESS: 0.999 -> 0.991 | Hurt |
| Use gradient flow interpolation instead of learned interpolation | Boltzmann sampling · Many Well (8D) | NLL: 6.876 -> 6.989 | Hurt |
| Use gradient flow interpolation instead of learned interpolation | Boltzmann sampling · Many Well (8D) | Energy Distance: 9e-5 -> 8e-4 | Hurt |
| Increase latent standard deviation from σ=20 to σ=30 for learned interpolation (ablation) | Learned interpolation · Gaussian Mixture Model (40 modes) | ESS: 0.605 -> 0.922 | Helped |
| Increase latent standard deviation from σ=20 to σ=30 for learned interpolation (ablation) | Learned interpolation · Gaussian Mixture Model (40 modes) | Energy Distance: 0.058 -> 0.121 | Hurt |
Surprising findings
- The linear interpolation's velocity norm can explode, with the growth rate exponential in mode separation.
- On the many well target, the learned interpolation outperforms the proposed gradient flow interpolation in NLL and energy distance.
Limitations
- Methods require knowledge of the target domain.
- Solving PDEs / loss sampling is challenging in high dimensions.
- Theoretical regularity of the gradient flow path and the norm of its velocity field remain open.
Experimental setup
Datasets: Gaussian mixture model (40 modes, 2D) 8-dimensional many well distribution
Benchmarks: Gaussian mixture model 40 modes many well distribution
Metrics: effective sample size negative log likelihood energy distance
Baselines: linear interpolation learned interpolation (Máté and Fleuret)
Builds on: Máté and Fleuret learned interpolation Benamou-Brenier formula score-based diffusion (Song et al.)
Closest papers by meaning
- Sampling in Unit Time with Kernel Fisher-Rao Flow (ICML 2024 · similarity 0.87)
- Wasserstein Gradient Flows and Forward-Only Diffusion Are Not Enough for Multimodal Sampling (NeurIPS 2026 · similarity 0.86)
- Wasserstein residuals: Learning Gradient Flows from Population Dynamics (NeurIPS 2026 · similarity 0.86)
- Fast training and sampling of Restricted Boltzmann Machines (ICLR 2025 · similarity 0.86)
- Wasserstein Convergence of Critically Damped Langevin Diffusions (NeurIPS 2025 · similarity 0.85)
- Minimizing $f$-Divergences by Interpolating Velocity Fields (ICML 2024 · similarity 0.85)
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