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Broadening Target Distributions for Accelerated Diffusion Models via a Novel Analysis Approach
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Proposes a Hessian-based accelerated stochastic DDPM sampler and a tilting-factor/Tweedie analysis giving O(1/eps) convergence for Gaussian mixtures, finite-variance targets (early-stopped), and targets with Lipschitz Hessian of log-density.
Problem
Existing accelerated diffusion sampler guarantees hold only for restrictive target classes (smoothness along the whole sampling path or bounded support) and carry high dimension dependence.
Approach
A new accelerated stochastic DDPM sampler whose reverse covariance includes the Hessian of log-density, estimated via a Hessian matching regression; analysis uses a tilting-factor representation of the error and Tweedie's formula to compute posterior moments, requiring smoothness only of q_0.
Key results
- Hessian-accelerated sampler achieves O(d^1.5 min{d,N}^1.5 / eps) steps for Gaussian mixtures, the first accelerated guarantee for that class.
- For finite-variance targets with early stopping, O(d^1.5 log^1.5(1/delta)/eps) steps, improving bounded-support prior results by O(d^1.5).
- In a Gaussian mixture experiment (N=d=4), the Hessian-based sampler achieves the best KL convergence among the compared accelerated samplers in non-asymptotic regimes.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the Hessian-accelerated DDPM sampler instead of prior accelerated DDPM samplers | Finite-variance targets with early stopping · Finite-variance targets (early-stopped) | steps to ε-accuracy in total variation: 𝒪(d^3/ε) -> 𝒪(d^{1.5}/ε) | Helped |
| Use the Hessian-accelerated DDPM sampler | Targets with M-Lipschitz Hessian of the log-density · Smooth log-density with M-Lipschitz Hessian | steps to ε-accuracy in total variation: 𝒪(d^{1.5} log^{1.5} M / ε) | Helped |
| Use the Hessian-accelerated DDPM sampler | Gaussian-mixture targets · Gaussian mixture (N ≤ d) | steps to ε-accuracy in total variation: 𝒪(d^{1.5} N^{1.5} / ε) | Helped |
Limitations
- Dimension dependence for general (non-smooth) targets still d^1.5 and not linear
- Requires score and Hessian estimation at higher accuracy than regular samplers
- Square-root matrix computation for the accelerated covariance is costly, motivating an approximate sampler
Experimental setup
Datasets: Gaussian mixture synthetic targets (N=4, d=4)
Benchmarks: Gaussian mixture KL divergence simulation
Metrics: KL divergence total variation bound number of steps computation time
Baselines: regular DDPM sampler accelerated stochastic sampler (Li et al. accelerated DDPM) PF-ODE deterministic sampler
Builds on: DDPM Ozaki discretization score matching Girsanov-based SDE analysis
Closest papers by meaning
- Accelerating Convergence of Score-Based Diffusion Models, Provably (ICML 2024 · similarity 0.92)
- O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions (ICLR 2025 · similarity 0.90)
- Diffusion Path Samplers via Sequential Monte Carlo (NeurIPS 2026 · similarity 0.87)
- Faster Diffusion Sampling with Randomized Midpoints: Sequential and Parallel (ICLR 2025 · similarity 0.87)
- Unified Convergence Analysis for Score-Based Diffusion Models with Deterministic Samplers (ICLR 2025 · similarity 0.87)
- Lipschitz Singularities in Diffusion Models (ICLR 2024 · similarity 0.87)
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