Weak Diffusion Priors Can Still Achieve Strong Inverse-Problem Performance

Abstract: Can a diffusion model trained on bedrooms recover human faces? Diffusion models are widely used as priors for inverse problems, but standard approaches usually assume a high-fidelity model trained on data that closely match the unknown signal. In practice, one often must use a mismatched or low-fidelity diffusion prior. Surprisingly, these weak priors often perform nearly as well as full-strength, in-domain baselines. We study when and why inverse solvers are robust to weak diffusion priors. Through extensive experiments, we find that weak priors succeed when measurements are highly informative (e.g., many observed pixels), and we identify regimes where they fail. Our theory, based on Bayesian consistency, gives conditions under which high-dimensional measurements make the posterior concentrate near the true signal. These results provide a principled justification on when weak diffusion priors can be used reliably.

• The paper demonstrates that in high-information regimes, inverse problem reconstruction is robust to weak or mismatched diffusion priors due to strong posterior concentration.
• It introduces novel techniques like AdamSphere and HoldoutTopK, yielding competitive PSNR and improved stability across various imaging tasks.
• Empirical results reveal that while weak priors perform well with informative measurements, they fail under extreme conditions such as large mask inpainting or high upscaling factors.

Robustness of Inverse Problems to Weak Diffusion Priors

Introduction

This paper, "Weak Diffusion Priors Can Still Achieve Strong Inverse-Problem Performance" (2601.22443), conducts a rigorous investigation into the conditions under which diffusion-based generative priors, even when weak (i.e., highly truncated samplers or trained on mismatched distributions), deliver competitive or state-of-the-art solutions to inverse problems. Challenging prevailing assumptions, the study demonstrates that, in data-informative regimes, the quality of reconstruction depends primarily on the informativeness of observations, not the strength or match of the diffusion prior. The paper makes significant contributions encompassing theoretical characterization, robust empirical validation, and algorithmic innovations, and also outlines clear boundaries of this prior-robustness phenomenon.

Theoretical Foundations: When Prior Matters Less

The paper formalizes the notion of a "weak prior" as either a diffusion generator executed with a low number of steps (e.g., 3-step DDIM, which generates low-fidelity, blurry images), or trained on a distribution heterogenous with respect to the target data. The central question addressed is: When do such weak priors suffice, and when do they fail?

A key result is the theoretical analysis rooted in Bayesian consistency. The authors employ a surrogate model where the generative prior is represented as a Gaussian mixture. For a single high-dimensional observation (typical in image inverse problems), they prove that, given sufficiently rich measurements (i.e., high $m$ in $y=Ax+\epsilon$), the posterior sharply concentrates on the best matching candidate, essentially "washing out" the impact of the prior choice. This is captured by the following main finding:

Figure 1: Posterior concentrates around $x^*$ as observation dimensionality $N$ grows, demonstrating insensitivity to prior in the high-information regime.

Theorem 1 establishes that, under mild identifiability and regularity conditions, the probability that the posterior does NOT concentrate on the true solution decays exponentially in $m$. Thus, for high-dimensional, information-rich problems, substantial prior mismatch has negligible effect on inference quality. This explains numerous empirical observations, such as successful out-of-domain MRI reconstructions with a brain prior [jalal2021robust].

Failure modes are also predicted by the theory: when the data is non-informative (e.g., large contiguous image regions are missing in inpainting, or extremely low-resolution observations are available), the exponential separation vanishes, and the reconstruction strongly depends on the prior quality and match.

Algorithmic Contributions

To leverage weak priors effectively in initial-noise optimization, the paper introduces the AdamSphere optimizer and the HoldoutTopK early stopping strategy. AdamSphere enforces optimization on the typical shell of the Gaussian distribution, counteracting the tendency of unconstrained optimizers to leave the high-probability region of the prior. HoldoutTopK, inspired by validation-set selection, retains the most generalizable iterates via a held-out subset of observations.

Empirical ablations demonstrate that these innovations stabilize and improve reconstruction in the weak prior regime, outperforming both default Adam and variance-based early stopping and yielding consistent improvements over prior initial-noise optimization methods (e.g., DMPlug [wang2024dmplug]).

Empirical Evaluation

Extensive experiments substantiate the theoretical conclusions, covering the following aspects:

• Cross-domain priors: Reconstruction is performed using both in-domain and severely out-of-domain priors (e.g., a bedroom-trained model for faces or churches) on inpainting, Gaussian and nonlinear deblurring, and super-resolution tasks. Visual and quantitative results show that, with informative measurements, weak priors match or consistently outperform strong-prior baselines (e.g., DPS [chung2023diffusion]), even in highly mismatched source-target pairs.

  Figure 2: Visual comparison on box inpainting and super-resolution. Despite severe prior mismatch (e.g., using a bedroom prior for a face image), both reach highly accurate reconstructions in the high-information regime.

• Quantitative improvements: The method achieves notable PSNR/LPIPS/SSIM gains relative to DPS and DMPlug in the informative regime. For instance, in inpainting with 70% pixels masked, a 3-step DDIM prior trained out-of-domain achieves PSNR $32.76$ (bedroom prior on CelebA) versus $31.98$ for in-domain DPS.
• Computation-matched benchmarks: Compute efficiency, when measured by denoiser evaluation count, confirms that initial-noise optimization using weak priors remains superior or competitive in final metric quality.

Figure 3: Compute-matched comparison: initial-noise optimization with weak priors efficiently matches or exceeds DPS at fixed compute budget.

• Latent diffusion backbone: Application to highly general priors, such as Stable Diffusion v2.1 and DiT on ImageNet, show strong results consistent with those observed for pixel-space diffusion, reinforcing that prior-robust regimes extend to modern latent diffusion foundations.

  Figure 4: Reconstruction on ImageNet with latent diffusion priors (Stable Diffusion 2.1, DiT): accuracy persists despite severely mismatched priors.

Failure Modes and Empirical Delineation

Examination of the boundaries is central to this study. Two archetypal failure modes are identified both theoretically and experimentally:

• Large contiguous mask inpainting: As masking increases (e.g., $\ge 0.5\times 0.5$), the LPIPS and visual fidelity of weak-prior methods degrade rapidly, with reconstructions reflecting semantic content from the prior rather than the information in the measurement.

  Figure 5: LPIPS metric divergence: the gap between our method and DPS grows with increasing masked fraction, identifying the measurement-informative boundary.

• Extreme super-resolution: For large upscaling factors (e.g., $\times 16$), observation dimension $m$ becomes too small for posterior concentration to a unique plausible image; reconstructions become prior-dominated, and out-of-domain priors fail catastrophically.

  Figure 6: Super-resolution failure cases: strong prior matches content, while weak or mismatched priors produce semantically irrelevant or heavily biased imagery.

Implications and Future Directions

This work has significant practical implications: in imaging, scientific, and medical domains where matched high-fidelity diffusion models are unavailable, the use of weak, possibly mismatched, priors is robustly justified provided the measurements are sufficiently informative. This lowers the barrier to deploying powerful generative priors in real-world inverse problems, including scenarios with limited compute (few-step priors), scarce data, or broad domain variation.

Theoretically, the results motivate further investigation of the minimal measurement-information threshold necessary for prior-robustness, as well as the development of more efficient hybrid or measurement-injection/initial-noise optimization algorithms for weak-prior settings.

Looking forward, sharper characterization of the transition from data- to prior-dominated regimes would unify these insights with the broader theory of Bayesian inference under model mismatch in high dimensions.

Conclusion

The paper provides a formal, empirical, and algorithmic foundation for understanding and exploiting the surprising robustness of inverse-problem solvers to weak diffusion priors. This work rebalances prior assumptions in the field: in data-informative regimes, the precise form and quality of the generative prior is less central to reconstruction performance than previously assumed, enabling the practical use of weak or mismatched priors in diverse settings. The study also delineates the domains where this robustness fails, setting clear guidelines for prior design and use in the next generation of diffusion-based inverse problem solvers.