Exact Posterior Score Estimation for Solving Linear Inverse Problems

Researchers have derived a closed-form expression for the exact posterior score in linear Gaussian inverse problems, enabling a new training objective called Exact Posterior Score (EPS) that preserves the structure of standard denoiser pretraining [1]. The work, presented at the SPIGM workshop at ICML, addresses a core limitation in using diffusion and flow-based models for inverse problems such as image restoration [4]. These models learn data priors by training a denoiser to reverse Gaussian corruption, but the score they provide is unconditional — not the posterior score needed to incorporate measurement information [2]. Existing approaches either steer a fixed pretrained denoiser with approximate measurement-matching corrections or train a conditional restoration model that abandons the denoising structure of the prior [3]. The authors show that for linear Gaussian inverse problems under general Gaussian interpolants, posterior sampling reduces to a denoising problem at an operator-dependent shifted pivot under an anisotropic noise covariance [5]. This structural insight yields the EPS objective, whose target and loss match those of standard pretraining, with the input replaced by a measurement-dependent pivot [3]. EPS can be trained from scratch or fine-tuned from a pretrained denoiser [1]. At inference, EPS uses the same deterministic or stochastic sampler as the underlying diffusion backbone, replacing every denoiser call with a measurement-conditioned variant [5]. No likelihood gradient, projection, or inner optimization is required during sampling [3]. The per-step overhead is negligible relative to a denoiser forward pass because the pivot is obtained through a structured linear solve [5]. This contrasts with gradient-based posterior samplers, which typically rely on repeated likelihood evaluations — a computational pattern related to maximum likelihood estimation, where parameters are chosen to maximize the probability of observed data under an assumed model [7]. The method was evaluated on five linear inverse problems across the FFHQ and ImageNet datasets [1]. EPS outperformed both training-free and training-based baselines on fidelity, perceptual, and distributional metrics [2]. It also used roughly an order of magnitude fewer denoiser evaluations than gradient-based posterior samplers [4]. The work was authored by Abbas Mammadov, Ozgur Kara, Kaan Oktay, Iskander Azangulov, Adil Kaan Akan, Hyungjin Chung, James Matthew Rehg, and Yee Whye Teh, and was recognized with an oral presentation at the SPIGM workshop [4].