Exact Posterior Score Estimation for Solving Linear Inverse Problems

2026-06-15 · Source: Computer Vision and Pattern Recognition · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Computer Vision & Pattern Recognition · Depth: Expert, quick

Summary

Exact Posterior Score (EPS) is a novel method for solving linear inverse problems using diffusion and flow-based models. It addresses the challenge of adapting unconditional prior scores for posterior sampling by deriving the exact posterior score in closed form for linear Gaussian inverse problems under general Gaussian interpolants. EPS reframes posterior sampling as a denoising problem at an operator-dependent shifted pivot with anisotropic noise. This identity underpins EPS's training objective, which preserves the input/output structure of standard denoiser pretraining, allowing training from scratch or fine-tuning. During inference, EPS uses the backbone's sampler without likelihood gradients or projections. Evaluated on five linear inverse problems across FFHQ and ImageNet, EPS outperforms training-free and training-based baselines on fidelity, perceptual, and distributional metrics, using roughly an order of magnitude fewer denoiser evaluations than gradient-based posterior samplers.

Key takeaway

For Machine Learning Engineers working on linear inverse problems with diffusion models, you should consider adopting Exact Posterior Score (EPS). This method allows you to leverage pretrained denoisers more effectively by providing an exact posterior score, significantly improving fidelity and perceptual metrics. Your inference processes will also become more efficient, requiring roughly an order of magnitude fewer denoiser evaluations compared to gradient-based samplers. Evaluate EPS for tasks like image restoration to enhance both quality and computational performance.

Key insights

Exact Posterior Score (EPS) transforms linear inverse problems into efficient denoising by deriving the precise posterior score.

Principles

• Posterior sampling reduces to a denoising problem.
• Exact posterior scores improve inverse problem solutions.

Method

EPS derives the exact posterior score in closed form, reframing posterior sampling as denoising at an operator-dependent shifted pivot with anisotropic noise. It trains a denoiser from scratch or fine-tunes it, then uses the backbone sampler for inference.

In practice

• Solve linear inverse problems with EPS.
• Fine-tune pretrained denoisers using EPS.
• Achieve faster inference with fewer evaluations.

Topics

• Exact Posterior Score
• Linear Inverse Problems
• Diffusion Models
• Posterior Sampling
• Denoising
• Image Restoration

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

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• A Stabilized Path-Space Approach to Diffusion-Based Posterior Sampling — A stabilized path-space framework improves diffusion posterior sampling by connecting it to stochastic optimal control and removing bias.
• SP$^3$: Spherical Priors for Plug-and-Play Restoration — Spherical Encoders can replace denoisers in Plug-and-Play image restoration for faster, high-quality results.

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Editorial summary, takeaway, and curation by AIssential. Original article published by Computer Vision and Pattern Recognition.