Exact Posterior Score Estimation for Solving Linear Inverse Problems

2026-06-16 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Computer Vision · Depth: Expert, extended

Summary

Exact Posterior Score (EPS) is a novel method for solving linear inverse problems using diffusion and flow-based generative models. It addresses the challenge of needing a posterior score for sampling, rather than the unconditional prior score provided by standard diffusion training. EPS derives the exact posterior score in closed form for linear Gaussian inverse problems, demonstrating that posterior sampling can be reframed as a denoising task at an operator-dependent shifted pivot, μ★, under an anisotropic noise covariance, Σ★. This identity forms the basis of the EPS training objective, which maintains the input/output structure of standard denoising pretraining, allowing for efficient training from scratch or fine-tuning from pretrained denoisers. At inference, EPS reuses the underlying backbone's sampler without requiring likelihood gradients or projections. Evaluated on five linear inverse problems across FFHQ and ImageNet datasets, EPS consistently outperforms existing training-free and training-based baselines in fidelity, perceptual quality, and distributional metrics, achieving comparable or superior results with roughly an order of magnitude fewer denoiser evaluations, often plateauing within ∼20 NFE.

Key takeaway

For Machine Learning Engineers developing solutions for linear inverse problems, EPS offers a significant advancement over existing diffusion-based methods. You should consider fine-tuning your pretrained denoisers with the EPS objective to achieve superior reconstruction fidelity, perceptual quality, and distributional calibration. This approach provides faster inference, often plateauing within ∼20 NFE, and enables a 1-NFE option for MMSE-optimal pointwise estimates, streamlining your development and deployment of robust inverse problem solvers.

Key insights

For linear Gaussian inverse problems, exact posterior sampling reduces to anisotropic denoising at a measurement-aware shifted pivot.

Principles

• Exact posterior scores for linear Gaussian inverse problems have a closed form.
• Posterior sampling can be reframed as anisotropic denoising at a measurement-aware pivot.
• The posterior pivot μ★ acts as a sufficient statistic for the measurement and current state.

Method

EPS trains a denoiser Dθ by regressing onto clean targets x₀ using a measurement-dependent pivot μ★(xₛ, y, t) as input. Inference reuses the backbone's sampler with Dθ(μ★,y,t).

In practice

• Fine-tune pretrained denoisers with the EPS objective for linear inverse problems.
• Use 1-NFE EPS for MMSE-optimal pointwise estimates (high PSNR/SSIM).
• Train amortized EPS models to handle multiple inverse problem types.

Topics

• Exact Posterior Score
• Linear Inverse Problems
• Diffusion Models
• Denoising
• Anisotropic Noise
• Image Restoration
• Generative Models

Best for: Research Scientist, AI Scientist, Machine Learning Engineer, Computer Vision Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.