[Paper Review] Benchmarking Uncertainty Quantification of Plug-and-Play Diffusion Priors for Inverse Problems Solving

Xiaoyu Qiu, Taewon Yang+3 more|arXiv (Cornell University)|Feb 4, 2026

Probabilistic and Robust Engineering Design0 citations

TL;DR

The paper benchmarks uncertainty quantification (UQ) of plug-and-play diffusion prior (PnPDP) solvers for inverse problems, showing that similar reconstruction quality can hide vastly different posterior uncertainty, and proposes a UQ-driven taxonomy and diagnostic framework.

ABSTRACT

Plug-and-play diffusion priors (PnPDP) have become a powerful paradigm for solving inverse problems in scientific and engineering domains. Yet, current evaluations of reconstruction quality emphasize point-estimate accuracy metrics on a single sample, which do not reflect the stochastic nature of PnPDP solvers and the intrinsic uncertainty of inverse problems, critical for scientific tasks. This creates a fundamental mismatch: in inverse problems, the desired output is typically a posterior distribution and most PnPDP solvers induce a distribution over reconstructions, but existing benchmarks only evaluate a single reconstruction, ignoring distributional characterization such as uncertainty. To address this gap, we conduct a systematic study to benchmark the uncertainty quantification (UQ) of existing diffusion inverse solvers. Specifically, we design a rigorous toy model simulation to evaluate the uncertainty behavior of various PnPDP solvers, and propose a UQ-driven categorization. Through extensive experiments on toy simulations and diverse real-world scientific inverse problems, we observe uncertainty behaviors consistent with our taxonomy and theoretical justification, providing new insights for evaluating and understanding the uncertainty for PnPDPs.

Motivation & Objective

• Motivate the need for uncertainty-aware evaluation of PnPDP solvers in ill-posed inverse problems.
• Propose a UQ-driven categorization of PnPDP methods based on their ability to approximate the Bayesian posterior.
• Develop toy-model diagnostics to quantify calibration of uncertainty and compare methods.
• Demonstrate consistent uncertainty behaviors across real-world scientific inverse problems.
• Provide recommendations for evaluating and understanding uncertainty in diffusion-based inverse solvers.

Proposed method

• Define the posterior target p(x|y) and distinguish posterior-targeting, heuristic, and MAP-like PnPDP solvers.
• Introduce empirical posterior variance from repeated samples as a proxy for solver-induced uncertainty.
• Design toy experiments with known ground-truth posterior to calibrate AU and EU and validate UQ metrics.
• Evaluate real-data inverse problems (linear scattering, sparse-sampling MRI, sparse-view CT) across multiple PnPDP methods.
• Provide a UQ-driven taxonomy linking methods to their posterior-targeting capabilities and theoretical guarantees.

Figure 1 : Illustration of the Accuracy Trap phenomenon and three types of uncertainty behaviors. Blue contours show the ground-truth posterior $p(x\mid y)$ , which can be multi-modes. The red star denotes the ground-truth $x^{*}$ ; $\hat{x}_{1},\hat{x}_{3}$ are posterior-plausible reconstructions,

Experimental results

Research questions

• RQ1 Can stochastic PnPDP solvers recover the posterior p(x|y) and its uncertainty under ill-posed forward models?
• RQ2 How do different PnPDP solvers compare in terms of calibrated uncertainty, not just reconstruction accuracy?
• RQ3 Does a UQ-driven categorization reflect observed uncertainty behaviors across toy and real-data tasks?
• RQ4 How does measurement sparsity or out-of-distribution data affect solver uncertainty?
• RQ5 What are the limitations and practical gaps between asymptotic guarantees and real-world implementations of posterior-targeting solvers?

Key findings

• Uncertainty behavior of PnPDP solvers can vary widely even when reconstruction accuracy is similar.
• Posterior-targeting solvers (e.g., MCG-Diff, FPS-SMC, PnPDM) show calibrated uncertainty in toy and some real-data tests, but can still exhibit bias or degeneracy in practice.
• MAP-like solvers (e.g., REDDiff) yield near-zero variance, consistent with point estimation.
• Heuristic solvers display diverse uncertainty patterns and can fail to calibrate uncertainty under certain forward models.
• Uncertainty generally increases with measurement sparsity, and accuracy and uncertainty are complementary assessments.
• A UQ-driven taxonomy aligns with observed behaviors and complements algorithm-structure classifications.

Figure 2 : Similar reconstruction with distinct uncertainty. Comparison of PnPDP solvers on linear inverse scattering reconstruction with $K=100$ times reconstruction on each solver. Top two Row: These methods produce similar reconstruction quality (in PSNR). Bottom Row: The pixel-wise variance maps