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Exact Posterior Score Estimation for Solving Linear Inverse Problems

NeurIPS 2026 · Image generation › Fast diffusion sampling · approach: Diffusion posterior sampling · 0 citations (OpenAlex)

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Derives the exact posterior score for linear Gaussian inverse problems and turns it into a denoising objective (EPS) that regresses onto clean images at a measurement-dependent anisotropic pivot, usable from scratch or by fine-tuning an EDM backbone.

Problem

Sampling from the posterior p(x0|y) for linear inverse problems with a pretrained diffusion/flow prior requires the posterior score, but pretrained denoisers only provide the unconditional score; existing approximate guidance introduces bias and training-based conditional methods discard the denoisi

Approach

For linear Gaussian problems the posterior score equals a denoiser queried at a measurement-dependent pivot mu_star under an anisotropic covariance Sigma_star; EPS trains a standard-style denoiser on (mu_star, y, t) targeting x0, so inference reuses the backbone sampler with no likelihood gradients.

Key results

• On ImageNet-64 random inpainting, EPS at 20 NFE reaches 24.87 PSNR and FID 77.06 versus 23.95 PSNR and 99.60 FID for ΠGDM at 100 NFE.
• EPS at 1 NFE (posterior mean) achieves 26.60 PSNR on ImageNet random inpainting, the highest pointwise score reported, at the cost of higher distributional error.
• EPS outperforms Palette, the conditional baseline sharing the same backbone and budget, on every reported task at 100 NFE (e.g., FID 79.60 vs 81.88 on random inpainting).

Surprising findings

• A single denoiser evaluation at the highest noise level gives a posterior-mean estimate that achieves the best PSNR and SSIM, but worse distributional metrics.
• Ablation shows that replacing the raw input x_t with the pivot mu_star explains most of the gain over Palette.

Limitations

• Derivation assumes a linear forward operator and Gaussian observation noise; nonlinear operators are not covered by the closed form.
• Latent diffusion backbones make pixel-space linear operators nonlinear in latent space, requiring extra care.
• Training is required per task/operator family, though amortization across tasks is studied in the appendix.

Experimental setup

Datasets: FFHQ ImageNet

Benchmarks: FFHQ-64 ImageNet-64

Metrics: PSNR SSIM LPIPS FID MMD CRPS

Baselines: DPS DAPS DDNM ΠGDM MPGD Palette

Builds on: EDM Palette Tweedie's formula DPS

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