Diffusion-Based Inverse Solvers

• Diffusion model-based inverse solvers are algorithms that use learned diffusion priors to recover unknown signals from noisy, incomplete measurements via Bayesian inference.
• They integrate measurement constraints into the reverse generative process using methods like Noise Combination Sampling (NCS) and Conditional Posterior Mean Estimation (DCS) for robust reconstruction.
• Recent advances, including variational mode-seeking and measurement optimization, enhance both reconstruction fidelity and computational efficiency in high-dimensional imaging tasks.

Diffusion model-based inverse problem solvers are a class of algorithms that leverage learned diffusion priors to solve linear and nonlinear inverse problems, most notably in high-dimensional domains such as image restoration, compressed sensing, and scientific inference. By combining Bayesian inference principles with generative diffusion processes, these solvers reconstruct unknown signals from indirect, noisy, or incomplete measurements, often outperforming classical and supervised methods both in fidelity and in the expressiveness of recoverable distributions.

1. Mathematical Foundation and Bayesian Formulation

Diffusion model-based inverse solvers formalize the inverse problem as the recovery of a latent signal $x_0\in\mathbb{R}^d$ from a measurement $y \in \mathbb{R}^n$ through a (typically non-invertible) observation operator $A$ (or general non-linear map $\mathcal{A}$) and additive noise: $$y = A x_0 + n, \quad n \sim \mathcal{N}(0, \sigma^2 I).$$ The Bayesian posterior combines the measurement likelihood and the prior: $p(x_0 | y) \propto p(y | x_0) p(x_0)$ where $p(x_0)$ is implicitly represented by a pretrained diffusion model, defining a Markov noising chain and a learned reverse (denoising) process (Chung et al., 4 Aug 2025).

The core challenge is to sample from (or approximate the mode of) the often intractable posterior $p(x_0|y)$. Diffusion priors, which learn data distributions via iterative denoising from Gaussian noise, offer strong regularization for severely ill-posed inverse problems, generalizing to both MAP and MMSE estimators, as well as providing a means to draw full-posterior samples in highly multimodal regimes.

2. Algorithmic Taxonomy and Core Methodologies

A systematic taxonomy of diffusion-based inverse solvers highlights several dominant approaches, each differing in how measurement information is injected into the sampling or optimization pipeline (Chung et al., 4 Aug 2025, Patsenker et al., 5 Aug 2025).

Method	Guidance Approach	Posterior Coverage / Typical Use
DPS, MPGD, DDNM	Posterior-score (approx.)	Empirical Bayes, sampling, MAP
SMC, DAPS	Ensemble/decoupled GD	Full posterior / multimodal
RED-diff, VML-MAP, ProjDiff	Variational/energy descent	MAP, regularized optimization
Noise Combination Sampling (NCS)	Noise subspace embedding	Hyperparameter-free, robust MAP
Deep Data Consistency (DDC), CoSIGN	Learned constraint (deep)	Fast, few-step high-fidelity

Posterior-score Replacement: Many solvers, starting from the DDPM or continuous SDE/ODE framework, replace the unconditional score with an approximation of the conditional one: $$\nabla_{x_t}\log p(x_t|y) \approx \nabla_{x_t}\log p_t(x_t) + \nabla_{x_t}\log p(y | x_t)$$ with $\nabla_{x_t}\log p(y | x_t)$ approximated via Tweedie’s denoising formula or by plugging in the MMSE estimate $\hat{x}_{0|t}$ (Chung et al., 4 Aug 2025, Su et al., 24 Oct 2025, Patsenker et al., 5 Aug 2025).

Variational (MAP/ELBO) Algorithms: Alternately, algorithms such as RED-diff and VML-MAP minimize explicit variational objectives (e.g., reverse-KL or ELBO-based losses), directly regularizing both fidelity to $y$ and proximity to the diffusion prior at each step (Mardani et al., 2023, Gutha et al., 11 Dec 2025).

Ensemble/Monte Carlo Domain: SMC, AFDPS, and related methods propagate weighted ensembles of particles through the reverse process, iteratively reweighting by measurement likelihood and resampling, yielding asymptotically correct posterior samples (Chung et al., 4 Aug 2025, Chen et al., 4 Jun 2025).

3. Recent Methodological Advances

Recent advances in the field address limitations of prior samplers—namely instability, tuning difficulty, and suboptimal integration of measurement information—by introducing principled algorithmic innovations.

Noise Combination Sampling (NCS)

NCS (Su et al., 24 Oct 2025) reframes posterior guidance by constructing the stochastic noise term in the reverse DDPM update as a linear combination of $K$ pre-sampled standard normals. The optimal combination $\gamma^*$,

$$\gamma^* = \frac{E_t^\top c}{\|E_t^\top c\|_2}, \qquad \epsilon_t^* = E_t \gamma^*$$

is chosen to optimally align with the measurement score $c = \nabla_{x_t}\log p(y | x_t)$. This closed-form embedding eliminates step-size hyperparameters and preserves the $\mathcal{N}(0,I)$ law for $\epsilon^*_t$. NCS yields robust, hyperparameter-free solvers outperforming DPS and MPGD backbones, particularly at low reverse-step count ($T\sim20-100$).

Conditional Posterior Mean Estimation (DCS)

DCS (Patsenker et al., 5 Aug 2025) explicitly estimates $\mathbb{E}[x_0|x_t, y]$ by fitting a single correction parameter $\epsilon_y$ per diffusion step through a maximum likelihood update. This removes the need for computationally expensive gradient-based conditioning and enables a one-pass, memory/minimal-overhead solver, maintaining or improving quality under high noise—addressing a notorious failure mode of traditional projection-based methods.

Variational Mode-Seeking Loss (VML-MAP)

VML-MAP (Gutha et al., 11 Dec 2025) derives a tractable, closed-form loss to align the diffusion process’s conditional distribution $p(x_0|x_t)$ with the desired $p(x_0|y)$, yielding: $$\mathrm{VML}_s(x_t) = -\log p(x_t) + \frac{1}{2\sigma_t^2}\|x_t - D(x_t, t)\|^2 + \frac{1}{2\sigma_y^2}\|y - H D(x_t, t)\|^2 + C$$ Minimizing this via stochastic gradient descent at each reverse step enables both MAP estimation and a controlled trade-off between fidelity and computational cost, surpassing earlier solvers in speed and/or sample quality.

Measurements Optimization (MO)

MO (Chen et al., 5 Dec 2024) enhances sampling efficiency by interleaving multi-step SGLD moves on the measurement loss $\frac{1}{2}\|y - A(x)\|^2$ with projections back onto the diffusion manifold via denoising. This dramatically reduces neural-function evaluations (by $\sim10$–40× versus prior baselines) while achieving SOTA metrics on both linear and nonlinear imaging tasks.

4. Stability, Robustness, and Practical Consequences

A major challenge in diffusion-based inverse solvers is balancing measurement consistency with adherence to the data manifold encoded by the prior. Prior methods typically require careful step-size/guidance tuning; over-integration of data constraints can lead to off-manifold samples, loss of image realism, or instability.

NCS (Su et al., 24 Oct 2025) and related methods (TMPD (Boys et al., 2023), DCS (Patsenker et al., 5 Aug 2025)) obviate this tuning by embedding measurement scores directly into the noise process or conditional posterior mean, backed by theoretical Gaussianity and closed-form optimality proofs. These approaches guarantee stability even when the number of reverse steps is aggressively subsampled ($T=20$), a regime where standard DPS and MCG guidance heuristics degrade or fail.

Robustness to choices of codebook size ($K$), noise levels, and runtime parameters is empirically demonstrated, with runtime overhead dominated by neural network evaluation rather than guidance computations. Across FFHQ/ImageNet tasks and metrics (PSNR, FID, LPIPS), recent methods consistently outperform, especially where computational efficiency is top priority.

5. Systematic Benchmarks and Comparisons

Exhaustive experimental benchmarks (Su et al., 24 Oct 2025, Patsenker et al., 5 Aug 2025, Gutha et al., 11 Dec 2025, Chen et al., 5 Dec 2024) highlight the current performance landscape.

Technique	Reverse Steps (T)	PSNR↑	FID↓	LPIPS↓	Overhead	Notes
DPS	1000	12.5–25.9	33–104	0.16–0.19	2–3×	Step-size tuning critical
NCS-DPS	20	19.2	19.4	0.137	negligible	NCS backbone, hyperparameter-free
DCS	50	30.1–34.8	19–26	0.024–0.137	1×	Conditional posterior mean
VML-MAP	20 (×GD)	—	38–62	0.136–0.146	2–8× faster	KL-driven, preconditioned variant
MO (DPS-MO)	50–100	29–31	—	0.110–0.184	10–40× faster	SGLD + denoiser projection

These methods deliver state-of-the-art results in sample quality and tractability, with the modern trend being towards drastic reduction in neural evaluations (from 1000s to tens or hundreds) and minimal or no post-hoc guidance parameter tuning.

6. Limitations and Open Problems

Despite substantial progress, several open questions remain:

• Nonlinearity and Model Misspecification: Most closed-form guidance methods and theoretical results assume linear $A$ and Gaussian noise. Extension to truly nonlinear (e.g., phase retrieval, general PDEs) or black-box forward models remains an active challenge (Su et al., 24 Oct 2025, Chen et al., 5 Dec 2024).
• Codebook Design in NCS: The optimal scaling of codebook size $K$ relative to ambient or manifold dimension is under-explored, as are benefits of jointly training codebooks (Su et al., 24 Oct 2025).
• Blind Inverse Problems and Unknown Noise: Extensions to the “blind” case (estimating measurement operator and latent jointly) require parallel diffusion chains or joint score modeling, increasing compute and instabilities (Chung et al., 4 Aug 2025).
• Uncertainty Quantification: While SMC/ensemble methods approach full posterior sampling, most efficient solvers focus on MAP or high-density point estimates, raising questions about propagation and representation of uncertainty.
• Global Convergence and Error Bounds: Although NCS and TMPD provide theoretical guarantees in Gaussian or near-Gaussian regimes, practical solvers on real data operate far from these assumptions, lacking nonasymptotic error bounds.

7. Theoretical and Practical Implications

The recent developments in diffusion-based inverse problem solvers redefine the landscape by unifying and simplifying the integration of measurement constraints into generative priors:

• Conceptual Unification: NCS and similar methods subsume traditional “add gradient–then denoise” schemes into a single noise synthesis step, grounded by the geometry of the noise subspace and the locally optimal alignment with measurement scores (Su et al., 24 Oct 2025).
• Hyperparameter-Free Robustness: The elimination of tunable guidance strengths and step sizes democratizes practical adoption and increases reproducibility across imaging domains and datasets.
• Accelerated Inverse Solvers: Sampling budgets as low as $T=20$–100 are now feasible without quality loss, with negligible computational overhead, confirming that (approximately) all required posterior information can be efficiently injected via the noise (Su et al., 24 Oct 2025, Chen et al., 5 Dec 2024).

A plausible implication is that further architectural and algorithmic co-design—linking learned codebooks, adaptive measurement integration, and direct posterior loss formulations—will yield even more robust and general-purpose solvers able to handle nonlinear, black-box, or blind settings with minimal intervention.

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References

• "Noise is All You Need: Solving Linear Inverse Problems by Noise Combination Sampling with Diffusion Models" (Su et al., 24 Oct 2025)
• "Injecting Measurement Information Yields a Fast and Noise-Robust Diffusion-Based Inverse Problem Solver" (Patsenker et al., 5 Aug 2025)
• "Mode-Seeking for Inverse Problems with Diffusion Models" (Gutha et al., 11 Dec 2025)
• "Diffusion models for inverse problems" (Chung et al., 4 Aug 2025)
• "Enhancing and Accelerating Diffusion-Based Inverse Problem Solving through Measurements Optimization" (Chen et al., 5 Dec 2024)
• "Tweedie Moment Projected Diffusions For Inverse Problems" (Boys et al., 2023)