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Learned Reference-based Diffusion Sampler for multi-modal distributions
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Proposes LRDS, a variational diffusion sampler that learns a multi-modal reference distribution (GMM or multi-level EBM) from MCMC samples initialized at known mode locations, and uses it to train a reference-based diffusion sampler for multimodal targets without ground-truth samples.
Problem
Sampling multi-modal unnormalized densities without ground-truth samples; existing variational diffusion samplers are highly sensitive to hyperparameters (e.g., reference scale sigma) that require ground-truth samples to tune, and suffer mode collapse.
Approach
Replace the isotropic Gaussian reference in reference-based diffusion sampling with a learned reference distribution (GMM fit by EM, or a time-dependent energy-based model trained via annealed MCMC) built from MCMC samples initialized at known mode locations, so the reference carries mode-weight information while keeping reference scores tractable (GMM) or estimated (EBM).
Key results
- On 16/32/64-dimensional bi-modal Gaussian mixtures, GMM-LRDS achieves mode-weight errors of 1.7%, 2.7%, and 4.1%, the best in the d=32 and d=64 settings, while several baselines collapse to 66.7% error or uniform weights.
- On the phi^4 model, all competing methods suffer mode collapse while GMM-LRDS stays close to the Laplace-approximation ground truth.
- On the Checkerboard distribution, LRDS is the only sampler that yields samples close to exact and correctly estimates mode weights.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use GMM-LRDS instead of the strongest reported baseline | Bi-modal Gaussian mixture sampling · Bi-modal Gaussian mixture, d = 16, 32, and 64 | absolute mode weight estimation error: d = 16: 0.8% ± 0.6% vs 1.7% ± 0.6%; d = 32: 15.8% ± 8.5% vs 2.7% ± 0.8%; d = 64: 11.7% ± 0.4% vs 4.1% ± 0.6% | Mixed |
| Use a multi-component Gaussian mixture reference instead of a single Gaussian reference (ablation) | LRDS · 16-dimensional bi-modal Gaussian mixture | mode weight recovery | Helped |
| Use EBM-LRDS instead of GMM-LRDS for complex target geometry | LRDS · Complex-geometry target distribution | sample fidelity | Helped |
| Use LRDS instead of competing methods | Multi-modal samplers · 1D phi^4 model | relative mode weight estimation | Helped |
| Use LRDS instead of competing sampling methods | Multi-modal samplers · Checkerboard distribution | sample accuracy | Helped |
| Use LRDS instead of non-diffusion-based approaches | Multi-modal samplers · Rings distribution | mode weight estimation | Helped |
Surprising findings
- Standard variational diffusion samplers (LV-PIS, LV-DDS) with Gaussian references collapse to mode-weight errors above 30% in higher dimensions, showing strong sensitivity to the reference scale hyperparameter.
- A Gaussian reference (G-LRDS) collapses to a single mode even when learned from data, so multi-modal reference structure is necessary.
Limitations
- EBM-LRDS requires costly pre-training of the reference process model via annealed MCMC.
- Requires prior knowledge of target mode locations.
- Competing methods used in the comparison were given the same prior information, so gains depend on that assumption.
Experimental setup
Datasets: synthetic multi-modal targets (high-dimensional Gaussian mixtures, phi^4 field model, Rings, Checkerboard, Bayesian logi
Benchmarks: high-dimensional bi-modal Gaussian mixtures (d=16,32,64) phi^4 model (d=32) Rings Checkerboard Bayesian logistic regression
Metrics: absolute mode weight estimation error sliced KS distance Sinkhorn W2 MMD
Baselines: SMC Replica Exchange LV-PIS LV-DDS LV-DIS LV-CMCD iDEM PDDS
Builds on: Denoising Diffusion Sampler (DDS) Path Integral Sampler (PIS) Log-Variance loss for controlled diffusion samplers Controlled Monte Carlo Diffusion (CMCD) Time-reversed Diffusion Sampler (DIS)
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