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Exact Posterior Score Estimation for Solving Linear Inverse Problems

This paper introduces Exact Posterior Score (EPS), a novel training objective that derives a closed-form exact posterior score for linear inverse problems, enabling high-fidelity sampling with standard denoising architectures while significantly reducing computational cost compared to gradient-based methods.

Original authors: Abbas Mammadov, Ozgur Kara, Kaan Oktay, Iskander Azangulov, Adil Kaan Akan, Hyungjin Chung, James Matthew Rehg, Yee Whye Teh

Published 2026-06-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Abbas Mammadov, Ozgur Kara, Kaan Oktay, Iskander Azangulov, Adil Kaan Akan, Hyungjin Chung, James Matthew Rehg, Yee Whye Teh

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a jigsaw puzzle, but someone has taken a few pieces, smeared the picture with fog, and handed you a blurry, incomplete version. Your goal is to reconstruct the original, sharp image. This is what scientists call a "linear inverse problem."

For a long time, AI models have been great at guessing what a complete picture looks like based on what they've seen before (like knowing what a cat usually looks like). But when you give them a blurry, broken photo, they often get confused. They might try to force the blurry parts to look like a cat, even if the blurry parts actually belong to a dog, or they might just smooth everything out until it looks like a fuzzy blob.

This paper introduces a new method called EPS (Exact Posterior Score) that acts like a super-smart puzzle guide. Here is how it works, using simple analogies:

The Old Way: Guessing and Correcting

Imagine you are trying to fix that blurry puzzle.

The "Training-Free" Method: You have a very talented artist (the AI) who knows how to draw cats perfectly. You show them the blurry photo and say, "Make this look like a cat." The artist draws a cat, but it doesn't quite fit the blurry lines. So, you grab a ruler and force the artist's drawing to align with the blurry lines. You do this over and over, adjusting the drawing step-by-step. It works okay, but it's slow, and you often end up with a drawing that looks a bit stiff or "hallucinated" (fake details that don't match the clues).
The "Training-Based" Method: Instead of using the existing artist, you hire a new artist and show them thousands of examples of "blurry photos + the correct answer." They learn to draw the answer directly. This is fast, but you have to hire a new artist for every type of puzzle (one for blurry photos, one for missing pieces, one for upside-down photos). It's expensive and doesn't reuse the talent of the original artist.

The New Way: EPS (The "Smart Pivot")

The authors of this paper realized that the math behind the puzzle has a secret shortcut. They found that you don't need to force the artist to correct their drawing, nor do you need to hire a new artist.

Instead, you just need to change the question you ask the original artist.

The "Pivot" (The Shifted Query):
Normally, you show the artist the blurry photo and ask, "What does the clean version of this specific blurry mess look like?"
EPS changes the question. It takes the blurry photo and the clues (the parts that aren't missing or smeared) and mathematically combines them into a new, smarter starting point (called the "pivot").
- Analogy: Imagine the artist is standing in a foggy room. Instead of asking them to guess where the furniture is based on the fog, you walk them to the exact spot where the furniture must be based on the floor plan, and then ask, "Now, what does the furniture look like from this specific angle?"
The "Anisotropic" Noise (The Directional Fog):
In the old way, the AI assumes the "fog" (noise) is the same in every direction. But in reality, some parts of the photo are very clear (low fog), and some are very blurry (high fog).
EPS teaches the AI to understand that the "fog" is directional. It knows exactly which parts of the image are trustworthy and which parts are a guess.
- Analogy: Instead of telling the artist, "It's foggy everywhere," EPS says, "The left side is crystal clear, but the right side is thick fog. Focus your guessing only on the right side."

Why This is a Big Deal

It's Exact: The authors proved mathematically that this "shifted question" is the perfect way to solve the puzzle. No more guessing or approximating.
It's Fast: Because the question is so well-structured, the AI doesn't need to take 100 tiny steps to fix the image. It can solve it in about 20 steps, which is much faster than other methods that need 100 or 250 steps.
It's Reusable: You can take an AI that was already trained to remove noise from photos (a "pretrained denoiser") and just give it this new "pivot" question. You don't need to retrain the whole brain; you just tweak how you talk to it.
Better Results: In tests on faces (FFHQ) and general objects (ImageNet), EPS produced sharper, more realistic images that matched the clues better than any other method, whether that method was free (no training) or required training a new model.

The "One-Step" Trick

The paper also discovered a cool side effect. If you ask this "pivot" question when the image is extremely blurry (at the very start of the process), the AI instantly gives you the best possible average guess of the original image.

Analogy: If you ask the artist, "What is the most likely shape of the object in this total fog?" they can give you a very accurate, sharp outline immediately, without needing to do any step-by-step guessing. This is great if you just want a clear, sharp picture and don't care about generating different variations.

Summary

The paper says: "We found a mathematical trick to turn a difficult 'fix the broken photo' problem into a simple 'remove the noise' problem. By shifting the starting point and accounting for the direction of the blur, we can use existing AI models to solve these problems perfectly, quickly, and without needing to retrain them from scratch."

Technical Summary: Exact Posterior Score Estimation for Solving Linear Inverse Problems

Problem Statement

Linear inverse problems involve recovering an unknown signal $x_0$ from noisy linear measurements $y = Ax_0 + \eta$ , where $A$ is a known forward operator and $\eta$ is observation noise. These problems are pervasive in imaging (e.g., inpainting, super-resolution, deblurring) and often ill-posed, meaning multiple signals are consistent with the same observation. The goal is to sample from the posterior distribution $p(x_0|y)$ rather than finding a single point estimate, to capture uncertainty and balance data fidelity with prior plausibility.

While diffusion and flow-based models provide powerful data priors by learning to reverse Gaussian corruption, they inherently provide the unconditional score $\nabla_{x_t} \log p(x_t)$ . Solving inverse problems requires the posterior score $\nabla_{x_t} \log p(x_t|y)$ . Existing approaches fall into two categories with significant limitations:

Training-free methods (e.g., DPS, DDNM, $\Pi$ GDM) keep a pretrained denoiser fixed and apply approximate measurement-matching corrections (e.g., gradient steps, projections) at each reverse step. These introduce bias, often leading to oversmoothing, hallucinated structures, or poorly calibrated uncertainty.
Training-based methods (e.g., Palette, InDI) train a new conditional model from scratch or via distillation. While expressive, they often abandon the denoising structure of the prior, requiring the network to learn the operator dependence end-to-end, which prevents efficient warm-starting from pretrained checkpoints and increases training costs.

Methodology: Exact Posterior Score (EPS)

The paper derives a closed-form expression for the exact posterior score under linear Gaussian inverse problems and general Gaussian interpolants. The core insight is that posterior sampling remains a denoising problem but with a measurement-aware input and an operator-dependent anisotropic noise covariance.

Theoretical Derivation

The authors show that the posterior score at time $t$ can be expressed as:
$\nabla_{x_t} \log p(x_t|y) = \frac{1}{\beta_t^2} \left( \alpha_t D_{\Sigma^\star(t)}(\mu^\star(x_t, y, t)) - x_t \right)$
where:

$D_{\Sigma^\star(t)}$ is an anisotropic denoiser that estimates $E[x_0 | x_0 + \xi = \mu^\star]$ with $\xi \sim \mathcal{N}(0, \Sigma^\star(t))$ .
$\mu^\star(x_t, y, t)$ is the posterior pivot, a precision-weighted fusion of the current noisy state $x_t$ and the measurement $y$ .
$\Sigma^\star(t)$ is the posterior covariance, derived from the inverse problem parameters ( $A, \sigma_y$ ) and the diffusion schedule ( $\alpha_t, \beta_t$ ).

Crucially, $\mu^\star$ acts as a sufficient statistic for $(x_t, y)$ , and $\Sigma^\star$ captures the anisotropic uncertainty: directions observed by $A$ become more certain, while unobserved directions (in the nullspace of $A$ ) remain uncertain.

The EPS Training Objective

The authors define Exact Posterior Score (EPS) as a training objective that leverages this identity:

Input Transformation: Instead of training on the standard noisy input $x_t$ , the network is trained on the pivot $\mu^\star(x_t, y, t)$ .
Anisotropic Noise: The training targets the denoising of $\mu^\star$ under the specific anisotropic covariance $\Sigma^\star(t)$ .
Efficient Simulation: A key theoretical contribution (Proposition 3) shows that the required anisotropic corruption is induced naturally by the closed-form construction of $\mu^\star$ . Thus, one can train using standard isotropic noise sampling ( $x_t = \alpha_t x_0 + \beta_t \epsilon$ ) and compute $\mu^\star$ analytically, avoiding the need to sample anisotropic noise directly.
Architecture: The network architecture is identical to standard pretrained denoisers, with the input channels expanded to concatenate the pivot $\mu^\star$ and the raw observation $y$ . This allows EPS to be trained from scratch or, more efficiently, fine-tuned from a pretrained unconditional denoiser.

Inference

At inference, EPS uses the exact same sampler (e.g., Euler ODE) as the underlying backbone. The only modification is replacing the standard denoiser call $D_\theta(x_t, t)$ with $D_\theta(\mu^\star(x_t, y, t), y, t)$ . This requires no likelihood gradients, no projections, and no inner optimization loops during sampling.

Key Contributions

Exact Posterior Score Identity: The derivation of a closed-form posterior score for linear Gaussian inverse problems, showing that posterior sampling reduces to denoising at a shifted pivot under an anisotropic covariance.
EPS Algorithm: A training objective and sampling algorithm that preserves the input/output structure of standard pretraining. It enables efficient fine-tuning from pretrained checkpoints and uses the backbone's sampler unchanged at inference.
Efficiency and Performance: EPS achieves state-of-the-art results with significantly fewer denoiser evaluations (NFE) than gradient-based samplers. It converges in $\sim20$ steps, whereas baselines like DPS or $\Pi$ GDM often require 100+ steps to approach similar performance.
High-Noise Limit: The method provides a one-step estimator for the posterior mean $E[x_0|y]$ in the high-noise limit, offering a fast, MMSE-optimal point estimate without iterative sampling.

Experimental Results

The authors evaluate EPS on five linear inverse problems (random inpainting, box inpainting, 4 $\times$ super-resolution, Gaussian deblurring, motion deblurring) across FFHQ and ImageNet datasets.

Metrics: EPS outperforms both training-free (DPS, DAPS, DDNM, $\Pi$ GDM, MPGD) and training-based (Palette) baselines across pointwise fidelity (PSNR, SSIM), perceptual quality (LPIPS, FID), and distributional calibration (CRPS, MMD).
Sampling Efficiency: EPS reaches its performance plateau within $\sim20$ NFE. In contrast, sampling-based baselines continue to improve slowly up to 100 NFE and often fail to reach the EPS asymptote.
Training Efficiency: When fine-tuned from a pretrained checkpoint, EPS converges in a fraction of the iterations required by other training-based methods (e.g., Palette), attributed to the preservation of the pretrained denoising geometry.
Runtime: EPS is the fastest inverse problem solver among the compared methods, running at essentially the same wall-clock speed as the unconditional pretrained denoiser (approx. 1.006 $\times$ ), as it avoids the backward passes required by gradient-based methods.
Ablations: Experiments confirm that the shifted pivot $\mu^\star$ is the critical factor for performance, and that the anisotropic geometry learned by EPS is essential, as zero-shot pivoting (using a pretrained denoiser without fine-tuning) performs poorly.

Significance and Claims

The paper claims that EPS provides a principled and calibrated approach to posterior sampling for linear inverse problems. By deriving the exact posterior geometry, it bridges the gap between the efficiency of training-free methods and the expressiveness of training-based methods.

Structural Alignment: Unlike conditional training methods that force the network to learn a new conditional path from scratch, EPS adapts the pretrained prior to the specific anisotropic geometry induced by the measurement. This allows for rapid fine-tuning and the reuse of powerful unconditional priors.
Transparency: The closed-form pivot and covariance offer a transparent handle for analyzing the sampler, separating measured and unobserved directions explicitly.
Practicality: The method requires no complex sampling loops or gradient computations during inference, making it suitable for low-latency applications.

The authors acknowledge limitations: the exact derivation assumes a linear forward operator and Gaussian observation noise. Nonlinear operators or non-Gaussian noise would require local linearization or training against the true likelihood, for which the closed form does not directly apply. Additionally, latent diffusion backbones require care as the decoder may render linear pixel-space operators nonlinear in latent space.