Weak Diffusion Priors

• Weak diffusion priors are generative models using diffusion processes that underfit fine details due to limited capacity, data mismatches, or truncated inference.
• They are applied in sparse CT reconstruction, dataset distillation, and class-conditioned generation, often balancing global structure recovery with loss of fine features.
• Theoretical analyses reveal that while weak priors can guarantee convergence in data-rich regimes, they risk hallucinations and mode collapse in ill-posed inverse problems.

Weak diffusion priors are generative diffusion models whose capacity, data support, or inference constraints render them unable to fully capture or exploit the fine structure of the target data distribution within a given inverse problem or generative task. Such priors can result either from insufficient network capacity, suboptimal training data, mismatched model domains, severely truncated samplers, or deliberately restricted forms (e.g., class-only conditioning or coarse latent models). Despite their limitations, weak diffusion priors display distinct operational regimes. They may be sufficient for certain high-level inferences, fail catastrophically in other settings, and admit precise theoretical and experimental characterization. This article surveys the definition, mathematical structure, application regimes, theoretical guarantees, and critical limitations of weak diffusion priors, with focus on inverse imaging, dataset distillation, 3D reconstruction, and high-dimensional Bayesian settings.

1. Mathematical Formalism and Definition

Diffusion priors are parameterized generative models $p_\theta(x)$ learned via score-based diffusion processes. For continuous-state problems (e.g., imaging or 3D point clouds), the forward process $dx_t = f(x_t, t)\,dt + g(t)\,dw_t$, $x_{t=0}\sim q(x_0)$, destroys structure by increasing noise; the reverse process inverts this, using a neural score network $s_\theta(x, t) \approx \nabla_x \log p_t(x)$ to drive denoising (Feng et al., 2023, Möbius et al., 2024).

A weak diffusion prior arises when $p_\theta(x)$ is suboptimal in one or more aspects:

• Domain mismatch: $p_\theta(x)$ is learned on data far from the true $x^\star$ (e.g., faces vs. bedrooms) (Jia et al., 30 Jan 2026).
• Low representational capacity: the network architecture underfits global or local data structure.
• Restricted conditioning: conditional priors use only minimal guidance (e.g., class label embedding) (Yue et al., 2024).
• Truncated inference: only a few reverse steps (e.g., 3-step DDIM or highly compressed chains) are used to decode a sample, resulting in low-fidelity outputs (Jia et al., 30 Jan 2026).

In formal inverse problems, $y = A x + \epsilon$, a weak diffusion prior forms the Bayesian posterior $p(x|y) \propto p(y|x) p_\theta(x)$, often augmented with measurement or data-fidelity gradients in analysis-by-synthesis or plug-and-play loops (Cheung et al., 4 Feb 2025, Leong et al., 24 Sep 2025, Aguila et al., 16 Oct 2025, Möbius et al., 2024).

2. Operational Regimes and Applications

Weak diffusion priors have been empirically and theoretically delineated across a variety of tasks:

• Sparse inverse problems: In sparse-view computed tomography (CT), diffusion priors markedly outperform classical analytical alternatives (e.g., Tikhonov or total variation) for ultralow projection counts ($n_\text{proj} \lesssim 10$–$15$), capturing global anatomy but missing fine detail. Their performance plateaus beyond this regime: classical methods excel once enough data is present, and diffusion-based reconstructions stagnate—failing to recover small structures or added information (Cheung et al., 4 Feb 2025).
• Dataset distillation: In dataset distillation, vanilla diffusion priors confer strong diversity (avoiding mode collapse) and generalization (via stochastic regularization), but are weak in "representativeness": sampled $\mathcal{S}_\text{syn}$ sets may fail to closely cover the original data manifold unless external feature-space or kernel-based guidance is injected (Su et al., 20 Oct 2025).
• Class-conditioned generation: Class-only conditioning (category embeddings) in diffusion transformers constitutes a "weak prior," providing minimal guidance on geometry or texture. This manifests as slow convergence and persistent deficits in sample fidelity; adding visual priors from prior diffusion outputs significantly strengthens conditional generation (Yue et al., 2024).
• Truncated diffusion sampling: Truncating samplers to few (1–4) reverse steps, whether for computational efficiency or resource constraints, yields weak priors expressed in the output space as mode-averaged (blurry) and low-frequency reconstructions. Notably, empirical work shows these weak priors can nonetheless succeed in data-informative regimes (Jia et al., 30 Jan 2026).
• Coarse/factorized priors: Two-stage frameworks such as Residual Prior Diffusion explicitly construct a coarse (weak) prior for global structure and delegate fine-grained detail to a second-stage residual diffusion process (Kutsuna, 25 Dec 2025).

3. Theoretical Analysis and Recovery Guarantees

Weak diffusion priors introduce distinct analytical regimes in inverse problems:

• Consistency in data-informative limits: When the forward operator $A$ reveals enough independent information (e.g., $m$ observed pixels in imaging, $m$ large), even severely mismatched or weak priors yield posteriors that collapse to the true $x^\star$ at an exponential rate in $m$. In high-dimensional settings, one can model $p_\theta(x)$ as a Gaussian mixture; the posterior converges to the true mode provided the per-pixel selection gap is nonvanishing (Jia et al., 30 Jan 2026).
• Projected gradient descent interpretation: The action of a diffusion prior in inverse problems can be mathematically viewed as a time-varying projection operator onto a learned data manifold $\Sigma$; as the noise schedule descends, this projection becomes sharper. When the score network is accurate, these projections drive convergence provided $A$ satisfies a restricted isometry property over $\Sigma$ (classical CS regime) (Leong et al., 24 Sep 2025).
• Failure on ill-conditioned or undersampled tasks: When measurements are noninformative (e.g., large contiguous missing regions, extreme super-resolution), weak priors are unable to recover + the data does not identify $x^\star$ over the prior entropy—leading to hallucinations or mode averaging (Cheung et al., 4 Feb 2025, Jia et al., 30 Jan 2026).

4. Failure Modes, Hallucinations, and Representativeness

Weak diffusion priors risk critical pathologies depending on the task structure:

• Hallucinations: In sparse reconstructions, diffusion models inject high-frequency plausible textures into missing regions that do not correspond to reality; such hallucinations can mimic anatomical detail, misleading downstream analysis or clinical decision-making (Cheung et al., 4 Feb 2025).
• Mode aliasing: In 3D novel-view synthesis, Score Distillation Sampling with sparse views can fall into incorrect high-density modes of the diffusion prior. The rendered distributions lack entropy, promoting convergence to wrong manifold regions unless visual inline priors or rectified distributions are introduced (e.g., via inpainting diffusion guided by geometric priors) (Wang et al., 2024).
• Representativeness deficit: Vanilla diffusion-based dataset distillation samples may be diverse and generalize across modes, but remain unrepresentative of the true training set. Mercer-kernel (e.g., feature-space) priors are required to correct for this deficit (Su et al., 20 Oct 2025).
• Support and discretization mismatch: In variational diffusion inference, simple Gaussian priors force large drift corrections in the SDE, increasing discretization errors and leading to poor exploration of target modes. Overly weak priors exacerbate mode collapse in multimodal targets unless generalized to mixtures that can support manifold-spanning exploration (Blessing et al., 1 Mar 2025).

5. Control, Mitigation, and Hybrid Schemes

Several interventions strengthen or compensate for the limitations of weak diffusion priors:

• Kernel-based representativeness guidance: Injecting a kernel (e.g., linear kernel over feature space) gradient into reverse steps during distillation can enforce representativeness while preserving the diversity and generalization inherent to diffusion (Su et al., 20 Oct 2025).
• Hybrid two-stage models: Residual Prior Diffusion and related frameworks decompose modeling into stagewise priors—a coarse latent stage for global geometry and a residual diffusion for fine texture—yielding fast convergence and fidelity with relatively weak first-stage priors (Kutsuna, 25 Dec 2025).
• Partial measurement/early stopping: Optimizing the generative latent strictly over the sphere and stopping with a holdout set of measurements minimizes overfitting when solving for $x=G(z)$ under $A\,x \approx y$ with a weak prior (Jia et al., 30 Jan 2026).
• Iterative refinement: In variational inference, iteratively adding Gaussian components to the prior (e.g., via MALA or heuristic selection) improves exploration and coverage, overcoming the local support of weak single-Gaussian priors (Blessing et al., 1 Mar 2025).
• Visual-prior injection: Passing the intermediate output of an earlier diffusion stage as an additional high-dimensional prior into subsequent denoising stages, as in Diffusion on Diffusion (DoD), markedly improves conditional sample fidelity and convergence speed compared to class-only weak priors (Yue et al., 2024).

6. Empirical Benchmarks and Quantitative Regimes

Weak diffusion priors can yield competitive or even superior results under specific regimes, detailed below:

Application Domain	Advantageous Regime (Weak Priors)	Limiting Regime (Classical/Strong Needed)
Sparse CT reconstruction	n_proj ≤ 10–15 (outperform TV, L2, Dice ↑)	n_proj ≥ 20–25 (diffusion plateaus, TV/L2 dominate) (Cheung et al., 4 Feb 2025)
Dataset distillation	Diversity/generalization (IPC ≪ N)	Low representativeness; fixed by kernel guidance (Su et al., 20 Oct 2025)
Image inpainting/superres	m large, random locations (PSNR/SSIM ↑)	Structured missing (boxes), extreme superres (failure) (Jia et al., 30 Jan 2026)
3D point cloud Bayesian	Moderate α (prior/likelihood balance)	Too strong/weak: overfit or unrealistic structures (Möbius et al., 2024)

In summary, the operational effectiveness of weak diffusion priors depends critically on measurement informativeness and signal-prior alignment. When measurements saturate the latent variable entropy, the posterior contracts and prior mismatch is largely forgiven. When data is ambiguous or ill-posed, weak priors induce hallucination, mode collapse, or non-representative outputs.

7. Prospects and Open Directions

The study of weak diffusion priors has sharpened several directions:

• Characterizing phase transitions between data-dominated and prior-dominated regimes under varying measurement strength and prior mismatches (Jia et al., 30 Jan 2026, Cheung et al., 4 Feb 2025).
• Developing automated tuning protocols (early stopping, guidance strength, hybrid inference) to exploit weak priors when feasible, and gracefully fail or defer to analytical/strong priors otherwise (Kutsuna, 25 Dec 2025).
• Understanding fundamental limitations of current diffusion architectures in settings requiring explicit control of the support, diversity, and representativeness simultaneously (distillation, scientific imaging) (Su et al., 20 Oct 2025, Blessing et al., 1 Mar 2025).
• Combining weak priors with geometric or domain-knowledge-informed guidance (e.g., inline priors, kernel supervision) to mitigate mode aliasing and hallucination (Wang et al., 2024, Yue et al., 2024).

This synthesis is based on precise empirical and theoretical results from recent literature, including (Cheung et al., 4 Feb 2025, Jia et al., 30 Jan 2026, Su et al., 20 Oct 2025, Leong et al., 24 Sep 2025, Yue et al., 2024, Kutsuna, 25 Dec 2025, Feng et al., 2023, Blessing et al., 1 Mar 2025, Aguila et al., 16 Oct 2025, Möbius et al., 2024), and (Wang et al., 2024).