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BELM: Bidirectional Explicit Linear Multi-step Sampler for Exact Inversion in Diffusion Models
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Proposes BELM, a generic bidirectional explicit linear multi-step formulation that unifies EDICT and BDIA as exact-inversion diffusion samplers, and derives an LTE-optimal O-BELM sampler with exact inversion, zero-stability, and global convergence.
Problem
Diffusion sampling inversion is inexact for DDIM; existing training-free exact-inversion samplers (EDICT, BDIA) are heuristic, have unknown theoretical properties, and often degrade sampling quality.
Approach
A bidirectional explicit linear multi-step (BELM) formulation, in which the update of x_{i-1} uses only x_i, x_{i+1} and the noise prediction at x_i, so the inverse is also explicit; coefficients are chosen by minimizing local truncation error (O-BELM), giving third-order local error.
Key results
- O-BELM reconstruction MSE of 0.004 across 10-100 steps on COCO-14, matching AE lower bound, while DDIM reaches 0.026 at 10 steps.
- O-BELM achieves CIFAR10 FID 10.98 at 10 steps versus 12.27 for BDIA and 17.45 for DDIM.
- On SD-1.5 text-to-image at 50 steps, O-BELM FID 17.96 vs DDIM 18.93, BDIA 19.38, EDICT 21.30.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use O-BELM instead of DDIM | SD-1.5 inversion and reconstruction · COCO-14 validation, 10–100 steps | MSE loss: 0.026 -> 0.004; 0.016 -> 0.004; 0.008 -> 0.004; 0.007 -> 0.004 | Helped |
| Use O-BELM instead of the strongest baseline | Unconditional generation · CIFAR10, 10–100 steps | FID: 12.27 -> 10.98; 7.27 -> 7.17; 5.77 -> 5.24; 5.07 -> 4.18 | Helped |
| Use O-BELM instead of the strongest baseline | Unconditional generation · CelebA-HQ, 10–100 steps | FID: 27.13 -> 19.13; 16.18 -> 11.54; 10.65 -> 10.41; 10.19 -> 10.17 | Helped |
| Use O-BELM instead of DDIM | SD-1.5 text-to-image generation · COCO-14, 10–100 steps | FID: 21.44 -> 18.19; 19.45 -> 17.92; 18.93 -> 17.96; 18.83 -> 18.19 | Helped |
| Use O-BELM instead of DDIM | SD-2.0-base text-to-image generation · COCO-14, 10–100 steps | FID: 20.40 -> 17.01; 18.57 -> 16.53; 17.82 -> 16.52; 17.64 -> 16.75 | Helped |
Surprising findings
- EDICT performs very poorly at few steps (FID 87.11 on CIFAR10 at 10 steps), far worse than DDIM.
Limitations
- Theoretical order of accuracy is second-order for the 2-step sampler, and higher-order variants require more history states.
- Image editing is demonstrated only qualitatively and is not competitive with commercial editing pipelines; the sampler is orthogonal to those methods.
Experimental setup
Datasets: MS-COCO-2014 validation set CIFAR10 CelebA-HQ COCO-14 validation prompts
Benchmarks: COCO-14 reconstruction MSE CIFAR10 FID CelebA-HQ FID COCO-14 text-to-image FID
Metrics: MSE FID
Baselines: DDIM EDICT BDIA AE reconstruction
Builds on: DDIM EDICT BDIA Adams-Bashforth/Adams-Moulton methods
Closest papers by meaning
- Fast samplers for Inverse Problems in Iterative Refinement models (NeurIPS 2024 · similarity 0.89)
- Improving Diffusion-based Inverse Algorithms under Few-Step Constraint via Linear Extrapolation (NeurIPS 2025 · similarity 0.88)
- Adapt and Diffuse: Sample-adaptive Reconstruction via Latent Diffusion Models (ICML 2024 · similarity 0.88)
- Divide-and-Conquer Posterior Sampling for Denoising Diffusion priors (NeurIPS 2024 · similarity 0.88)
- A Unified Sampling Framework for Solver Searching of Diffusion Probabilistic Models (ICLR 2024 · similarity 0.88)
- Differentiable Solver Search for Fast Diffusion Sampling (ICML 2025 · similarity 0.88)
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