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DAPS++: Rethinking Diffusion Inverse Problems with Decoupled Posterior Annealing

Conference: ECCV2026
Paper: ECCV paper page
Area: Image Generation
Keywords: diffusion model; inverse problem; posterior annealing; Langevin sampling; measurement consistency

TL;DR

DAPS++ uses two empirical measurements (the gradient ratio and the gradient inner product) to show that the diffusion prior in inverse-problem solvers actually acts only as a warm initializer, and on that basis fully separates diffusion-based initialization from likelihood-driven Langevin refinement; across super-resolution, inpainting, deblurring and HDR on FFHQ / ImageNet it matches or beats DAPS-1K with roughly 1/10 to 1/20 of the NFEs.

Background & Motivation

From a Bayesian viewpoint, solving an inverse problem means performing inference under the posterior \(p(x_0\mid y)\propto p(y\mid x_0)p(x_0)\): the likelihood describes the measurement process and the prior encodes structural assumptions about the signal. Classical solvers constrain the solution space with Tikhonov, total-variation or wavelet priors, while the modern mainstream replaces them with the far more expressive score-based diffusion prior — DPS and its relatives inject an approximate likelihood gradient \(\nabla_{x_t}\log p(y\mid x_t)\) at every diffusion timestep, so that the reverse SDE, or its deterministic ODE counterpart, drifts toward the observation while it generates. Decoupled variants such as DCDP and DAPS refine this at the algorithmic level: diffusion updates and measurement updates alternate, each data-consistency step pushes the estimate toward the observation, and the subsequent re-noising step restores the appropriate noise level, so that the time-marginal distributions \(\pi(x_t\mid y)\) are supposed to approach the true posterior level by level.

In practice, however, that trajectory does not hold. The authors measure two quantities. First, the inner product between the likelihood gradient and the prior score is empirically close to zero under high noise, i.e. the two gradients are approximately orthogonal. Second, the ratio of their magnitudes stays far above 1 throughout the diffusion process and grows with the noise-to-measurement ratio \(\sigma_t/\gamma\) (the appendix formalizes this with a Lipschitz bound of the form \(\kappa_t \gtrsim \sigma_t|r_t|/(\gamma^2 C + \sigma_{\min}(A))\); ⚠️ reconstructed from the paper's typesetting — refer to the original appendix for exact symbol meanings). In other words, the likelihood gradient dominates the update dynamics: each intermediate distribution \(\pi(x_t\mid y)\) is an isolated approximation of the posterior rather than part of a coherent marginal sequence, the path from \(x_T\) to \(x_0\) is a set of loosely connected estimates, and a strictly Bayesian reading is insufficient to explain how these solvers behave. The sharpest piece of evidence is that deleting the prior term from the time-marginal update and keeping only the data-consistency term leaves reconstruction quality essentially unchanged — continuous prior guidance can therefore be replaced by a one-time diffusion initialization.

This reassigns the roles of prior and likelihood: the diffusion stage places the estimate near the data manifold and constrains the feasible solution space, while the likelihood gradient satisfies the measurement inside that space — a "initialize first, then correct" relationship that never required the two to be entangled. Core idea: rewrite diffusion-based inverse-problem solving explicitly as a two-stage structure in which the prior performs a one-time initialization and the likelihood owns the refinement — Stage 1 pushes pure noise to the manifold with a single Tweedie step (switching to one RK4 step at low noise), and Stage 2 runs likelihood-only ULA on that image-space estimate with re-noising at the end of every cycle, removing the time-marginal constraint DAPS relies on and cutting the NFEs by roughly 90% at essentially no quality cost.

Method

Overall Architecture

The input is an observation \(y\), a known forward operator \(\mathcal{A}\) and the noise variance \(\gamma^2\); the output is a clean estimate \(\hat{x}_0\). Nothing is trained — the method simply wraps a pre-trained pixel-space diffusion model. Inference runs \(K\) cycles, each with two non-overlapping stages: Stage 1 starts from the current noisy state \(x_{\mathrm{in}}\) and uses the score network only to produce a clean estimate \(\hat{x}_0\) (one Tweedie step when the noise level exceeds a threshold \(\bar\sigma\), one RK4 step below it); Stage 2 hands \(z^{(0)}=\hat{x}_0\) to \(J\) steps of likelihood-only ULA refinement, yielding \(\tilde{x}_0\); at the end of the cycle \(\tilde{x}_0\) is re-noised back to \(x_{\mathrm{in}}\leftarrow \tilde{x}_0+\sigma_k\epsilon_k\) at the current noise level and the next cycle begins, with \(\sigma_k\) decaying as a power law in \(k\). Structurally the key fact is that the score network is touched only once per cycle in Stage 1, whereas Stage 2 never accesses the prior; the two stages exchange nothing but "a clean image close to the data manifold", so there is no need to maintain the time-marginal distribution at every annealing level as DAPS does. The number in the method name is the annealing step count \(K\): DAPS++-50 on FFHQ and DAPS++-100 on ImageNet.

Key Designs

1. Demoting the prior to a one-time warm start: rewriting "posterior sampling" as "initialization"

This is the argumentative backbone of the paper and the point where it parts ways with every method that treats the diffusion prior as a guide throughout inference. Rather than debating what the algorithm nominally samples, the authors measure two quantities. The first is the inner product between the likelihood gradient and the prior score, \(\langle \nabla_{x_t}\log p_t(y\mid x_t;\sigma_t),\,\nabla_{x_t}\log p_t(x_t;\sigma_t)\rangle\): under high noise it is empirically close to zero, so the two gradients are approximately orthogonal — which also explains why DPS can add the two terms without destructive interference. The second is the ratio of their magnitudes, \(\kappa_t\), the one custom quantity the argument needs:

\[\kappa_t=\frac{\bigl\|\nabla_{x_t}\log p_t(y\mid x_t;\sigma_t)\bigr\|}{\bigl\|\nabla_{x_t}\log p_t(x_t;\sigma_t)\bigr\|}\]

On FFHQ Gaussian blur, \(\kappa_t\) stays well above 1 across the whole noise range and grows monotonically with \(\sigma_t/\gamma\); the Lipschitz analysis in the appendix turns this into a lower bound that grows with the noise-to-measurement ratio. Read together, the two measurements say something unambiguous: once initialization is done, the prior gradient is both too small (a magnitude gap of more than an order of magnitude) and approximately orthogonal to the likelihood gradient, so it can neither accelerate nor redirect the measurement-driven correction — the prior's actual function is to place the initial estimate near the data manifold. The authors verify this directly: removing the prior term from the time-marginal update and keeping only data consistency leaves reconstruction quality almost unchanged. Continuous prior guidance is thus replaced by a one-time diffusion initialization, which is the premise of every structural simplification that follows.

2. Thresholded Tweedie / single-step RK4 initialization: global structure cheaply, without losing detail

To fix the form of the initialization the authors start from EDM's probability-flow ODE: a single Euler step from \(x_t\) recovers exactly Tweedie's formula, i.e. a one-step score-based mean estimate of the clean image

\[\hat{x}_0(x_t)=\mathbb{E}[x_0\mid x_t]\approx x_t+\sigma_t^2\,s_\theta(x_t,\sigma_t)\]

This is a first-order mean estimator — smooth, cheap, and already recovering the dominant structure of the signal (large-scale geometry, color distribution, semantic content). More importantly, Stage 2 only needs the starting point to be near the data manifold, not the sampling path to be precise, so higher-order ODE solvers do not improve Stage 2 convergence (Section S1 of the supplementary material validates this by comparing initialization strategies). The price of pure Tweedie, however, is an overly smooth image with insufficient detail, so a threshold \(\bar\sigma\) splits the initialization in two: Tweedie denoising for fast initialization when \(\sigma_t>\bar\sigma\), and a single RK4 step when \(\sigma_t\le\bar\sigma\) to preserve fine details at minimal extra cost, with \(\bar\sigma=10\gamma\) chosen as the point where the first-order truncation error starts to dominate the measurement residual. The ablation shows what this buys: inside DAPS's prior-plus-likelihood refinement framework, swapping the 2-step Euler initialization for Tweedie raises SSIM from 0.716 to 0.741, so initialization quality alone translates into metrics.

3. Likelihood-only ULA refinement: turning measurement consistency from step-wise guidance into Langevin sampling in image space

Stage 2 is where the method diverges most from DPS and DAPS. Given the initialization \(\hat{x}_0\), the authors run the unadjusted Langevin algorithm (ULA) in image space, with state \(z\):

\[z^{(j)}=z^{(j-1)}+\eta_j\,\nabla_z\log p(y\mid z^{(j-1)})+\sqrt{2\eta_j}\,\xi_j,\qquad \nabla_z\log p(y\mid z)=\frac{1}{\gamma^2}\,J_{\mathcal{A}}(z)^{\top}\bigl(y-\mathcal{A}(z)\bigr)\]

which is exactly the pseudo-code step "compute the residual \(r=y-\mathcal{A}(z)\), then multiply by a vector-Jacobian product (VJP) of the forward operator". In standard MCMC the prior gradient \(\nabla_z\log p(z)\) acts as a regularizer, but the diffusion initialization already places \(\hat{x}_0\) in a region where \(\|\nabla\log p(x_0)\|=O(\varepsilon)\), so this term is numerically negligible and is dropped outright, leaving the likelihood alone. Dropping the prior saves more than one network forward pass: under the Gaussian measurement model of variance \(\gamma^2\) the likelihood gradient is globally Lipschitz and the log-density is strongly concave, so ULA comes with established convergence guarantees, and convergence is governed purely by the geometry of the likelihood — far fewer steps suffice than are needed to maintain a time-marginal distribution. The same step also explains DPS's efficiency bottleneck: DPS differentiates the residual \(\|y-\mathcal{A}(\hat{x}_0(x_t))\|_2^2\) through the Tweedie estimator and back through the score network at every diffusion timestep, one network-traversing Jacobian-vector product per step, which is expensive and makes the method structurally incompatible with higher-order ODE solvers; moving likelihood optimization out of the diffusion process leaves one VJP of the measurement operator per step and removes that constraint entirely.

4. Re-noising plus polynomial annealing: trading generative stochasticity for regularization

Pure likelihood refinement drives the estimate toward a measurement-consistent solution but also squeezes out the variability of noise-corrupted or entirely unobserved components (the missing region in inpainting, for instance). At the end of every refinement cycle the authors re-inject noise, \(x_{\mathrm{in}}\leftarrow\tilde{x}_0+\sigma_k\epsilon_k\), so the next cycle explores from a fresh state: this restores the variability that the likelihood flattened, prevents overfitting to measurement noise, and distinguishes the procedure from a MAP estimate that collapses onto a single mode — the whole pipeline samples stochastically in the latent space defined by the diffusion model. The injected noise schedule must stay below the measurement noise, otherwise it amplifies noise: \(\sigma\) decays as a power law from \(\sigma_{\max}=100\) to \(\sigma_{\min}=0.1\) with \(\gamma=0.05\). The other annealing knob is the step size: \(\eta_j\) decays polynomially with the iteration index, with the exponent falling in the range \(-4\) to \(-7\) (EDM-inspired; ⚠️ the paper only states the exponent range and never gives the exact correspondence to EDM's power-law parameter \(\rho\)), concentrating iterations in the low-noise regime where fine-scale refinement is most effective; the noise schedule itself follows EDM's POLY(-7) discretization. The annealing step count is \(K=50\) on FFHQ and \(K=100\) on ImageNet, with \(J=4\) to \(8\) MCMC refinement steps per cycle, except nonlinear deblurring which needs about 50 steps because the curvature near the initialization is inhomogeneous.

A Worked Example

Take one Gaussian-blur case on FFHQ (a \(61\times61\) kernel with \(\sigma=3.0\), \(\gamma=0.05\)). Starting from pure noise \(x_T\), the noise level is \(\sigma_{\mathrm{in}}=100\gg\bar\sigma=10\gamma=0.5\), so Stage 1 performs a single Tweedie step and already yields an estimate with the rough facial contours and color layout in place. Stage 2 starts from that estimate and runs \(J=4\) to \(8\) ULA steps, each computing one residual \(y-\mathcal{A}(z)\) and multiplying by one VJP of the forward operator, pulling the estimate toward the observation; then the state is re-noised at the current \(\sigma_k\) and the next cycle begins. Over the \(K=50\) cycles, \(\sigma_k\) decays as a power law from 100 to 0.1, so the re-noising perturbation shrinks and the ULA corrections become progressively finer; only once \(\sigma_k\) falls below the threshold 0.5 do the last few cycles switch Stage 1 from Tweedie to a single RK4 step to recover detail (exactly how many cycles that is depends on the POLY(-7) discretization; ⚠️ the paper does not list it cycle by cycle). Across the whole pipeline the score network is called on the order of \(K\) times, while the measurement-operator VJPs number \(K\times J\) — this is where the roughly 90% NFE reduction relative to DAPS-1K comes from.

Loss & Training

The method is entirely training-free: no fine-tuning, no distillation, just a pre-trained pixel-space diffusion model reused at inference time. Stage 2 optimizes the implicit measurement-residual objective \(\|y-\mathcal{A}(z)\|_2^2/(2\gamma^2)\), implemented through the VJP above. Only a handful of quantities are set by hand: the noise-schedule range \(\sigma_{\max}=100,\sigma_{\min}=0.1\), the threshold \(\bar\sigma=10\gamma\), the annealing step count \(K\) (50 on FFHQ / 100 on ImageNet), the refinement steps per cycle \(J\) (typically 4 to 8, about 50 for nonlinear deblurring), and the learning rate \(\eta\), which the paper tunes individually per task; hyper-parameter sensitivity analyses live in the appendix.

Key Experimental Results

Main Results

Evaluation uses the \(256\times256\) versions of FFHQ and ImageNet, with 100 validation images each and all images normalized to \([0,1]\) for comparability. Tasks cover linear problems (\(\times4\) super-resolution, box and random-mask inpainting, Gaussian deblurring, motion deblurring) and nonlinear ones (HDR, nonlinear deblurring); both blur kernels are \(61\times61\) with standard deviations 3.0 and 0.5, inpainting uses the standard \(128\times128\) box mask, and every measurement carries additive white Gaussian noise with \(\gamma=0.05\). Metrics are SSIM↑ / LPIPS↓ / FID↓. Baselines are DPS, DDRM, DDNM, DiffPIR (written "DiPIR" in the paper), DCDP, DAPS-100 and DAPS-1K; SVD-projection methods such as DDRM and DDNM apply only to linear operators, so their nonlinear columns are empty. Phase retrieval is discussed only in Section S2.2 of the supplementary material and is not evaluated in the main tables, and \(\times8\) super-resolution appears as a qualitative result (Fig. 3f) while the main tables use \(\times4\).

Table 1: reconstruction quality over 100 FFHQ validation images (SSIM↑ / LPIPS↓ / FID↓, \(\gamma=0.05\)). DAPS++ uses only 50 annealing steps while DAPS-1K uses 1000.

Inverse problem DPS DAPS-1K DAPS++-50
Super-resolution \(\times4\) 0.591 / 0.357 / 81.1 0.782 / 0.192 / 55.5 0.781 / 0.176 / 46.0
Inpainting 0.727 / 0.259 / 75.6 0.747 / 0.176 / 50.1 0.812 / 0.141 / 42.1
Gaussian blur 0.647 / 0.285 / 73.3 0.786 / 0.179 / 52.7 0.784 / 0.171 / 51.1
Motion blur 0.588 / 0.327 / 76.8 0.836 / 0.137 / 38.4 0.829 / 0.136 / 37.9
Nonlinear blur 0.648 / 0.281 / 74.3 0.762 / 0.191 / 57.8 0.745 / 0.194 / 54.9
HDR 0.693 / 0.284 / 77.6 0.839 / 0.163 / 41.1 0.834 / 0.169 / 42.3

For reference, the remaining baselines on FFHQ peak at 0.820 (DDNM super-resolution SSIM) and 0.810 (DDNM inpainting SSIM) on linear tasks, but with markedly higher FID (76.1 and 152.9 respectively); DiffPIR peaks at 0.553 / 0.257 / 52.6 (motion blur), and DCDP stays around 0.64 to 0.76 SSIM on linear tasks with 0.803 / 0.160 / 44.8 on nonlinear blur.

Table 2: reconstruction quality over 100 ImageNet validation images (SSIM↑ / LPIPS↓ / FID↓). DAPS++-100 uses 100 annealing steps.

Inverse problem DPS DAPS-1K DAPS++-100
Super-resolution \(\times4\) 0.454 / 0.474 / 208.4 0.638 / 0.295 / 109.1 0.661 / 0.276 / 114.2
Inpainting 0.623 / 0.343 / 157.5 0.715 / 0.229 / 114.0 0.771 / 0.195 / 105.3
Gaussian blur 0.526 / 0.355 / 155.6 0.658 / 0.268 / 107.6 0.663 / 0.273 / 122.9
Motion blur 0.542 / 0.365 / 161.4 0.769 / 0.175 / 47.1 0.763 / 0.179 / 56.1
Nonlinear blur 0.520 / 0.365 / 151.5 0.720 / 0.212 / 75.9 0.718 / 0.211 / 75.7
HDR 0.349 / 0.552 / 213.8 0.824 / 0.171 / 44.7 0.807 / 0.191 / 48.5

Ablation Study

Table 3: decoupling ablation on FFHQ inpainting, all configurations under the same 100 diffusion-NFE budget, isolating initialization form and whether refinement uses the prior.

Config Initialization Refinement SSIM↑ / LPIPS↓ Time (s)
DAPS-100 2-step Euler Prior + likelihood 0.716 / 0.194 7.4
DAPS-100 w/o prior gradient 2-step Euler Likelihood only 0.716 / 0.194 6.8
DAPS + Tweedie init. Tweedie Prior + likelihood 0.741 / 0.186 10.1
DAPS + Tweedie/RK4 init. Tweedie/RK4 Prior + likelihood 0.712 / 0.192 5.9
DAPS++ Tweedie/RK4 Likelihood only 0.812 / 0.141 2.4

Table 4: noise tolerance (FFHQ Gaussian blur, \(\sigma_{\min}\) matched to \(\gamma\)).

Config \(\gamma=0.05\) \(\gamma=0.1\) \(\gamma=0.2\) \(\gamma=0.4\)
DAPS-1K 0.786 / 0.179 0.615 / 0.330 0.410 / 0.495 0.192 / 0.633
DAPS++-50 0.784 / 0.171 0.728 / 0.191 0.674 / 0.229 0.579 / 0.303

Key Findings

  • Removing the prior gradient is the cleanest and most convincing cut: rows 1 and 2 of Table 3 score identically (0.716 / 0.194) and differ only by 0.6 s of runtime, so keeping the diffusion prior during refinement yields essentially no quality gain while paying for one extra score-network gradient. This is what backs the "prior = warm initializer" reading.
  • Initialization form matters more than whether refinement uses the prior: under the same DAPS prior-plus-likelihood refinement, replacing the 2-step Euler initialization with Tweedie lifts SSIM from 0.716 to 0.741. Note that the Tweedie/RK4 initialization scores only 0.712 / 0.192 inside that framework, below plain Tweedie's 0.741 — that row does not cleanly isolate initialization form, and the authors' stated reason for the threshold is that plain Tweedie is overly smooth with insufficient detail; only after switching refinement to likelihood-only does the full DAPS++ reach 0.812 / 0.141 at 2.4 s, roughly 3 times faster than DAPS-100.
  • The efficiency-quality trade-off favors DAPS++ across the board: in a budget sweep from 20 to 200 diffusion NFEs (against DAPS at 50 to 1K), DAPS++ attains lower LPIPS at equal wall-clock cost, and even the extreme DAPS++-20 setting converges to competitive results on linear degradations; on FFHQ the 50-step DAPS++-50 already beats the 100-step DAPS-100.
  • The advantage grows with noise: as \(\gamma\) rises from 0.05 to 0.4, DAPS-1K's SSIM collapses from 0.786 to 0.192 (essentially unusable) while DAPS++-50 only drops from 0.784 to 0.579, and it is most stable when \(\sigma_{\min}\) is matched to \(\gamma\) — the re-noising structure and the reduced refinement step count jointly suppress noise overfitting.
  • Refinement step count is governed by the operator: well-conditioned, near-quadratic operators converge extremely fast (super-resolution needs 2 steps because downsampling preserves most signal energy; per-pixel tone mapping in HDR needs about 5), Gaussian deblurring needs more because of its broader eigenvalue spread, nonlinear deblurring needs about 50 because the signal-dependent kernel yields inhomogeneous curvature, and in phase retrieval the likelihood has multiple equivalent minima, so a wrong basin at initialization cannot be recovered by local refinement.
  • Boundaries worth noting: on ImageNet, DAPS++-100 is broadly on par or better in SSIM / LPIPS but its FID remains higher than DAPS-1K on most tasks (super-resolution 114.2 vs 109.1, Gaussian blur 122.9 vs 107.6, motion blur 56.1 vs 47.1), so distribution-level metrics still lag at heavily reduced step counts; on FFHQ there are individual SSIM losses too (motion blur 0.829 vs 0.836, HDR 0.834 vs 0.839) though LPIPS / FID are better.

Highlights & Insights

  • It turns "what is the diffusion prior actually doing" into a measurable question. The gradient ratio \(\kappa_t\) and the gradient inner product are cheap to compute yet separate the nominal "posterior sampler" from the actual "initialize-then-refine" behaviour; this falsify-first-then-redesign line of argument is more convincing than simply proposing a new module, and it incidentally explains why DPS can mix the two gradients without destructive interference.
  • The payoff of decoupling is compound, not just compute. Stage 2 no longer differentiates through the score network, so it escapes DPS's per-step back-propagation bottleneck and becomes naturally compatible with higher-order ODE solvers; and because the time-marginal constraint is abandoned, larger step sizes and fewer iterations become viable, which in turn resists overfitting to measurement noise — speed and robustness come from the same structural change, the paper's clearest "aha" moment.
  • Re-noising is a reusable regularization device: generative stochasticity replaces explicit prior regularization or early stopping, translating "don't overfit the noise" into "inject less noise than the measurement noise", with a simple, interpretable knob (the relation between \(\sigma_{\min}\) and \(\gamma\)).
  • A transferable diagnostic: any plug-and-play framework that alternates a generative prior with data consistency can first compute \(\kappa_t\); if the prior gradient really is small and near-orthogonal during refinement, the prior can be demoted to a one-time initialization and the remaining iterations handed to explicit likelihood optimization. Whether this holds for latent diffusion or higher-resolution inverse problems is an open and worthwhile question.

Limitations & Future Work

  • Narrow experimental scope: validation is limited to the 256-resolution FFHQ and ImageNet benchmarks with pixel-space pre-trained diffusion models; latent diffusion (e.g. the Stable Diffusion family), higher resolutions, real-world degradations, and the scientific imaging settings the introduction emphasizes (astronomy, biomedical imaging) are all absent.
  • No guarantee for non-convex operators: the authors acknowledge that phase retrieval, with its multiple equivalent minima, depends on the initialization landing in the right basin, which the thresholded Tweedie/RK4 initialization does not guarantee.
  • Distribution-level metrics still lag: FID on most ImageNet tasks is worse than DAPS-1K, so "few steps plus strong decoupling" has not fully closed the gap at the sample-distribution level, and the paper does not analyze this.
  • Hyper-parameters and tuning cost: \(\bar\sigma=10\gamma\), the polynomial exponent, and a per-task learning rate all need to be set by hand; when \(\gamma\) is unknown or the measurement noise is not white Gaussian, the paper does not discuss how to satisfy the constraint that the injected re-noising level must stay below the measurement noise.
  • Little quantitative evidence that it really samples the posterior: the paper criticizes DPS/DAPS for behaving more like MAP iterations, yet Stage 2 runs very few ULA steps (4 to 8 for most tasks), so it may itself behave more like regularized optimization with random perturbations; substantiating "exploring the posterior rather than collapsing to one mode" would need evidence at the level of likelihood-surface coverage or sample diversity.
  • Theory lives in the appendix: both the initialization residual magnitude (\(O(|\mathcal{A}|\sigma_t)+O(\gamma\sqrt{d_y})\)) and the convergence bound are stated only as conclusions, so the derivations cannot be checked from the main text.
  • vs DPS: DPS performs "Tweedie initialization + one likelihood correction" at every diffusion timestep, with the two terms coupled so that the likelihood gradient must be back-propagated through the Tweedie estimator into the score network — one Jacobian-vector product per timestep. That is its main computational bottleneck and it makes the method structurally incompatible with higher-order ODE solvers. This paper moves likelihood correction out of the diffusion process entirely and drops the prior term; it gives up the "step-wise guidance" narrative in exchange for needing only one VJP of the measurement operator per step.
  • vs DAPS: DAPS already decouples at the algorithmic level (alternating Tweedie estimation with Langevin dynamics applied directly in image space, without back-propagating through the score function), but it re-invokes the score network at each annealing level to preserve the time-marginal \(p(x_t\mid y)\), which forces many Langevin steps per level to approximately sample from \(p(x_0\mid x_t,y)\); if those steps do not converge, the guarantee breaks down. This paper removes the constraint altogether — the score network enters only Stage 1 — which is the direct source of the 90% NFE reduction.
  • vs DCDP: DCDP also takes a decoupled route of diffusion purification plus data consistency, but its data-consistency update remains a step embedded in the diffusion loop. The difference here is that the entire likelihood-optimization phase is isolated with its own iteration budget (\(J\) steps) and its own annealing step sizes, so the effort can be scaled to the conditioning of the operator.
  • vs DDRM / DDNM: these use an SVD of the forward operator to enforce a strict measurement-consistency projection — theoretically clean, restricted to linear operators, and "harder" in how they handle noise. This paper uses a gradient-based likelihood that naturally covers nonlinear operators (HDR, nonlinear deblurring) at the cost of no projection-style consistency guarantee.
  • vs DiffPIR: DiffPIR embeds PnP denoising and proximal data consistency inside diffusion sampling, so every step still passes through the diffusion model; this paper uses the diffusion model only in Stage 1 and leaves the rest to the likelihood, pushing the NFE count an order of magnitude lower at comparable quality.

Rating

  • Novelty: ⭐⭐⭐⭐ Novel perspective — two cheap empirical measurements recast diffusion inverse-problem solvers as "initialization plus refinement"; the individual components (Tweedie initialization, ULA refinement, re-noising, polynomial annealing) are all known, so the contribution is structural and explanatory.
  • Experimental Thoroughness: ⭐⭐⭐⭐ Two datasets, six inverse-problem types, 100 validation images and seven baselines under a unified protocol, with an ablation that directly falsifies the role of the prior during refinement; the gaps are the absence of latent diffusion, real-world data, and phase retrieval from the main tables.
  • Writing Quality: ⭐⭐⭐⭐ The argument chain is clear (measure, reinterpret, redesign, then compare against DPS/DAPS point by point) and the pseudo-code is complete; some equations are damaged by typesetting and the ablation table does not cleanly isolate the initialization form.
  • Value: ⭐⭐⭐⭐ Training-free and plug-and-play, matching or beating DAPS-1K at roughly 1/10 of the NFEs, which is attractive for resource-limited or real-time imaging; its answer to "is the prior gradient still worth computing" also has methodological value.