Research map / Probabilistic and scientific modeling / Variational and generative sampling
Liouville Flow Importance Sampler
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Problem
Sampling from unnormalized densities and estimating the log normalization constant (log marginal likelihood), especially for multi-modal and high-dimensional targets where MCMC and existing NN-based samplers struggle.
Approach
Learn a neural velocity field that satisfies a PDE derived from an annealed unnormalized density path (Generalized Liouville Equation), without needing target samples, and treat the imperfectly learned flow as an importance sampler whose log-weights are the error accumulated along sample trajectories, giving asymptotically unbiased estimates of Z.
Key results
- On the 2-D Gaussian mixture, LFIS estimates log Z as -0.0002 ± 0.004 (theoretical value 0) versus -0.31 for DDS and 2.43 for AFT, with the lowest W2 (0.054) among methods.
- On the 10-D funnel, LFIS gives log Z of -0.07 ± 0.003 and W2 of 5.57, the best among compared methods.
- On the Log Gaussian Cox Process (D=1600), LFIS gives 505.53 ± 0.95 versus the SMC(1024) gold standard of 506.96 ± 0.24, while being the only method consistent with gold standard across all Bayesian problems per the authors.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use LFIS instead of SMC (256) | Mode-separated Gaussian mixture · Mode-separated Gaussian mixture (D=2) | W2: 0.066 $\pm$ 0.017 -> 0.054 $\pm$ 0.014 | Helped |
| Use LFIS instead of SMC (256) | Funnel sampling · Funnel (D=10) | W2: 6.07 $\pm$ 1.52 -> 5.57 $\pm$ 1.52 | Helped |
| Use LFIS instead of SMC (256) | Mode-separated Gaussian mixture · Mode-separated Gaussian mixture (D=2) | log Z estimate deviation from theoretical value 0: -1.28 $\pm$ 0.01 -> -0.0002 $\pm$ 0.004 | Helped |
| Use LFIS instead of SMC (256) | Funnel sampling · Funnel (D=10) | log Z estimate deviation from theoretical value 0: -0.12 $\pm$ 0.06 -> -0.07 $\pm$ 0.003 | Helped |
| Use LFIS for Bayesian posterior sampling | Bayesian benchmark problems · LGCP, Ionosphere, Sonar, and VAE | log Z estimate consistency with SMC (1024) | Helped |
| Use importance weights from accumulated trajectory error | LFIS · Sampling benchmarks | sampling accuracy | Helped |
Surprising findings
- Plain SMC handles the test problems competitively, so the reported benefit of normalizing flows in AFTMC is questioned by the authors' ablation.
- LFIS achieved high ESS without any resampling, comparable to SMC with a high resampling threshold.
Limitations
- Cannot handle non-differentiable target densities (e.g., bounded uniform priors) without workarounds.
- More memory-demanding because it requires the score gradient and cross derivatives through autograd.
- Explicit Euler discretization causes bias when the number of time steps T is small, raising training cost.
Experimental setup
Datasets: Binary MNIST
Benchmarks: Mode-separated Gaussian mixture (D=2) Funnel (D=10) Log Gaussian Cox Process (D=1600) Ionosphere logistic regression (D=35) Sonar logistic regression (D=61) VAE latent space (D=30)
Metrics: log normalization constant estimate Wasserstein-2 distance effective sample size
Baselines: SMC DDS PIS AFTMC VI-NF
Builds on: Annealed Flow Transport Monte Carlo Denoising Diffusion Sampler Path-Integral Sampler Sequential Monte Carlo Neural ODE
Closest papers by meaning
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- Diffusion Generative Flow Samplers: Improving learning signals through partial trajectory optimization (ICLR 2024 · similarity 0.86)
- Discrete Neural Flow Samplers with Locally Equivariant Transformer (NeurIPS 2025 · similarity 0.86)
- Fast and unified path gradient estimators for normalizing flows (ICLR 2024 · similarity 0.85)
- NETS: A Non-equilibrium Transport Sampler (ICML 2025 · similarity 0.85)
- Enhanced Importance Sampling Through Latent Space Exploration in Normalizing Flows (AAAI 2025 · similarity 0.84)
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