Source-linked AI summary
Beyond Best-of-$k$: Prior-Consistent DAPS for Ambiguous Nonlinear Inverse Problems
Merve Kocabas, Benjamin A. Burns, Andreas Geiger, Kent Gauen, Sara Fridovich-Keil
TL;DR
Ambiguous inverse problems can have multiple measurement-consistent solutions with different prior support, and existing samplers can commit to an unsupported mode. PC-DAPS maintains and prunes multiple candidates by prior consistency within one sampling run. It reduces prior-inconsistent samples and improves nonlinear reconstruction quality while preserving performance on linear tasks.
Problem
For ambiguous inverse problems, multiple measurement-consistent solutions can differ in prior support, making it challenging to sample solutions supported by both the measurement and prior.
Source: §1 · p. 2Method
PC-DAPS maintains multiple particles and prunes or replaces weaker candidates according to prior consistency as annealing noise decreases, without ground-truth access.
Source: §1 · pp. 2–3Results
PC-DAPS reduces prior-inconsistent samples and improves nonlinear reconstruction quality while preserving performance on linear tasks.
Source: §1 · p. 3Takeaways & Limitations
PC-DAPS performs prior-based sample selection within a single run, delaying mode selection until the prior becomes more informative.
Source: §1 · p. 2PC-DAPS shows a smaller improvement on structurally diverse ImageNet than on FFHQ because the prior landscape is more complex.
Source: §6 · p. 10
Abstract
from arXiv · showhide
Diffusion models have become strong priors for solving inverse problems. However, nonlinear inverse problems such as phase retrieval remain challenging when multiple measurement-consistent solutions are distinguishable only by the prior; we call such measurements ambiguous. Existing diffusion-based samplers rely on local approximations during reconstruction. For an ambiguous measurement, these approximations can allow measurement guidance to favor a measurement-consistent mode that is close to a prior-supported mode but is itself not supported by the prior. Evaluation often masks this type of failure by reporting the best-of-$k$ runs, where the best is selected by comparison to the ground truth. We introduce Prior-Consistent DAPS (PC-DAPS), a variant of Decoupled Annealing Posterior Sampling (DAPS) designed to address this failure case by maintaining multiple candidate solutions within a single sampling run. As annealing noise decreases, PC-DAPS prunes and replaces weaker candidates according to their prior consistency, without any ground-truth access. PC-DAPS reduces the incidence of prior-inconsistent samples and improves sample quality on nonlinear inverse problems while preserving sample quality and diversity on linear inverse problems.
1 INTRODUCTION
Ambiguous inverse problems can have measurement-consistent solutions that differ in prior support, while trajectory-based samplers may commit prematurely to an unsupported mode. PC-DAPS keeps multiple candidates and selects among them using prior consistency, improving nonlinear reconstruction results without ground-truth-based selection.
1 INTRODUCTION
- In phase retrieval, rotated or color-inverted images can match the Fourier magnitude measurement yet be implausible under a prior favoring upright faces.Measurement-equivalent solutions can have the same likelihood but different posterior probabilities because of their prior consistency.Source: §1 · p. 2
- Trajectory-based samplers can prematurely commit to a prior mode whose nearest measurement-consistent mode is unsupported by the prior.Best-of-k evaluation selects using ground truth unavailable at test time and does not ensure prior-consistent samples.Source: §1 · p. 2
- PC-DAPS maintains a bag of particles and uses a Gaussian-mixture denoiser posterior to prune candidates by prior consistency without ground-truth access.As annealing noise decreases, weaker candidates are pruned and replaced by more prior-consistent survivors.Source: §1 · pp. 2–3
- PC-DAPS reduces prior-inconsistent samples and improves nonlinear reconstruction quality while preserving performance on linear tasks.The evaluation spans FFHQ and ImageNet, with three nonlinear and seven linear inverse problems.Source: §1 · pp. 2–3
2 RELATED WORK
Related work addresses inverse problems through likelihood-guided diffusion, split-and-annealed sampling, particle-based methods, and post-sampling diagnostics. PC-DAPS modifies DAPS by using prior-based selection during sampling to reduce prior-inconsistent hallucinations.
Likelihood-guided diffusion sampling
- Likelihood-guided diffusion methods approximate the intractable likelihood term, while several use a single trajectory.DPS evaluates likelihood at the one-step posterior mean [Chung et al., 2023]; Jin et al. (2026) stabilize the prior update but remain committed to a single trajectory.Source: §2 · p. 3
Split and annealed posterior sampling
- Split-and-annealed methods alternate prior and measurement steps; PC-DAPS retains DAPS’s variable-splitting structure but adds prior-based selection.DAPS uses repeated denoising-renoising, while RePS adds restarts and DAPS++ reports that DAPS reconstruction is largely driven by measurement consistency.Source: §2 · p. 3
Particle-based posterior sampling
- Particle-based methods typically target ensemble fidelity or particle diversity, whereas PC-DAPS uses particles to explore the posterior landscape and reduce prior-inconsistent returns.The cited methods propagate, reweight, and resample particles to improve measurement consistency.Source: §2 · p. 3
Diagnosing hallucination and prior misfit
- Existing hallucination diagnostics are applied after sampling, while PC-DAPS uses prior support during sampling to decide which reconstructions survive.The targeted failure is a measurement-consistent mode unsupported by the prior, also described by Burns & Fridovich-Keil (2026).Source: §2 · p. 3
3 PRELIMINARIES
The section frames inverse problems as posterior sampling that must balance measurement consistency with prior plausibility. It explains how DAPS can become trapped in a prior-inconsistent mode and motivates maintaining multiple candidate modes.
3.1 BAYESIAN INVERSE PROBLEMS
- Bayesian inverse problems seek signals consistent with an observed measurement and plausible under a prior, represented jointly by the posterior p(x | y).The measurement operator may be linear or nonlinear, and the observation includes additive Gaussian noise.Source: §3.1 · p. 4
- When measurement-consistent signals are well separated, they can have equal likelihoods but sharply different posterior probabilities because their prior plausibility differs.Existing reconstruction methods may sample such signals at roughly equal frequencies, misrepresenting their posterior probabilities.Source: §3.1 · p. 4
- In Fourier phase retrieval, visually distinct images can produce identical Fourier magnitude measurements, including rotated or color-inverted faces that conflict with an upright-face prior.The section identifies reducing prior-inconsistent samples while retaining quality and diversity of measurement- and prior-consistent samples as the goal.Source: §3.1 · p. 4
3.2 DECOUPLED ANNEALING POSTERIOR SAMPLING (DAPS)
- DAPS alternates denoising, likelihood-consistency refinement with MCMC, and forward-process renoising to move toward a measurement-consistent clean sample.Its local Langevin updates may not explore distant prior modes, so renoising is intended to permit larger moves between iterations.Source: §3.2 · pp. 4–5
- In the toy problem, DAPS puts majority mass on a prior-inconsistent mode absent from the true posterior, while PC-DAPS prunes it; neither adequately samples the likelihood-inconsistent atom.The exact posterior assigns positive weight to both prior atoms, with majority weight on the likelihood-consistent atom.Source: Figure 2 · p. 5
- DAPS can enter a nearby likelihood-consistent region unsupported by the prior, then become trapped there through repeated renoising and local refinement.Its Gaussian approximation discourages large deviations from the denoiser estimate, while subsequent updates can keep returning the particle to the same region.Source: §3.2 · p. 5
- PC-DAPS maintains multiple candidate measurement-consistent modes to identify and remove trapped prior-inconsistent particles.In the toy example it places mass on the prior-supported, measurement-consistent mode and avoids the prior-inconsistent region.Source: §3.2 · p. 6
4 METHOD
PC-DAPS corrects prematurely committed particles within a shared K-particle run, avoiding post-hoc rejection that wastes computation and requires a way to identify bad samples. Its two core changes are prior-consistency pruning and a bag-informed multimodal approximation of the denoiser posterior.
4 METHOD
- PC-DAPS corrects prematurely committed particles during sampling, avoiding post-hoc rejection that wastes computation and requires ground truth or posterior access to choose samples.It maintains K particles with shared annealing and a pretrained denoiser, but different initializations and update noise.Source: §4 · p. 6
4.1 PRUNING SCHEME
- At the annealing stage when the prior becomes discriminative, PC-DAPS checks for multiple likelihood modes, scores particle clusters by prior consistency, and replaces below-threshold clusters with copies from stronger ones.The score averages Tweedie-based consistency over noise scales; clusters near the prior data manifold are expected to have small Stein scores.Source: §4.1 · pp. 6–7
4.2 GAUSSIAN MIXTURE APPROXIMATION OF THE DENOISER POSTERIOR
- PC-DAPS replaces DAPS’s unimodal denoiser-posterior approximation with a Gaussian mixture informed by particle diversity, using clean estimates as centers before pruning and cluster statistics afterward.A singleton cluster uses the full bag’s diagonal covariance.Source: §4.2 · p. 7
- Within each mode, PC-DAPS shares a bag-mean center, yielding a variance-reduced prior estimate that improves linear-task reconstruction quality and uses data-dependent rather than heuristic geometry-independent covariance.The mixture score is available in closed form for the DAPS Langevin update, which discourages motion away from a prior mode identified by a particle cluster.Source: §4.2 · p. 7
5 RESULTS
PC-DAPS improves phase-retrieval outcomes, especially on FFHQ, while remaining competitive across the evaluated tasks. It also preserves variation among its within-run particles, and its phase-retrieval comparison with baselines accounts for differing evaluation protocols.
Main results
- Across experiments, PC-DAPS is competitive and often achieves the highest sample quality, with its largest improvements in phase retrieval.Table 1 compares FFHQ and ImageNet across seven linear and three nonlinear tasks using PSNR, SSIM, and LPIPS.Source: §5 · p. 7, Table 1 · p. 8
- For phase retrieval, PC-DAPS’s four within-run samples are averaged, while baselines use the ground-truth-selected best sample from four independent runs.The table caption identifies the PC-DAPS mean-over-four versus baseline best-of-four protocol.Source: §5 · p. 8, Table 1 · p. 8
Prior-inconsistent hallucinations
- On FFHQ, upside-down/color-inverted reconstructions fall from 14.5% to 1.5% and faithful reconstructions rise from 79.5% to 97.5%; ImageNet improvements are smaller.On ImageNet, faithful reconstructions increase from 38.5% to 45.0%, unrecoverable runs/bags drop from 25% to 15%, and upside-down/color-inverted reconstructions decrease from 42.0% to 30.5%.Source: Table 2 · p. 9, §5 · p. 9
Main results
- Despite pruning and replacement, PC-DAPS’s four samples retain subtle within-run diversity comparable to four independent DAPS runs.Figure 3 also shows that PC-DAPS resolves upside-down orientation inconsistencies seen in DAPS best-of-k runs.Source: Figure 3 · p. 9, §5 · p. 9
6 DISCUSSION
Best-of-k evaluation selects runs using ground truth, while PC-DAPS uses learned prior information to reduce variation without a ground-truth reference. The discussion also identifies more structurally diverse priors as an open challenge and argues for assessing prior consistency alongside standard metrics.
6 DISCUSSION
- Best-of-k results select runs using ground truth, an evaluation approach associated with run-to-run inconsistency and variable single-sample quality.Source: §6 · p. 9
- PC-DAPS uses learned prior information to reduce unwanted sampling variance without a ground-truth reference.The discussion links existing run-to-run instability to large quality variation in single-sample results.Source: §6 · p. 10
- The improvement is smaller on structurally diverse ImageNet, where the prior landscape is more complex.The authors identify eliminating prior-inconsistent reconstructions under less-structured priors as future work.Source: §6 · p. 10
- Prior consistency should be considered alongside PSNR, SSIM, and LPIPS because those metrics compare samples with a single reference.The authors note that these metrics can penalize multiple plausible solutions that differ from the fixed reference.Source: §6 · p. 10
A.1 FULL ALGORITHM
The algorithm takes a measurement, denoiser, measurement operator, noise schedule, and particle-bag settings as inputs. It includes Langevin refinement and a one-shot cluster-based replacement step before returning K samples.
A.1 FULL ALGORITHM
- The required inputs include measurement y, denoiser Dθ, operator A, measurement noise βy, a decreasing noise schedule, bag size K, Langevin steps N, and threshold γ.Source: §A.1 · p. 13
- The displayed steps include constructing q(x0 | xt) using Equations (11) and (12), with clustering in the alternate branch.Source: §A.1 · p. 13
- The algorithm includes a Langevin refinement stage when i > 1, then loops over j = 0, . . . , N − 1 and computes successive ˆx(k,j+1) values.Source: §A.1 · p. 13
- When clustering yields multiple clusters and the pruning condition holds, the algorithm scores clusters and replaces particles from weaker clusters with copies from selected clusters.The procedure marks the bag as pruned after this one-shot replacement.Source: §A.1 · p. 13
- The algorithm returns the K clean-signal estimates indexed by k.Source: §A.1 · p. 13
A.2 TOY EXPERIMENT DETAILS
The toy experiment specifies a two-atom prior and a quadratic measurement operator, with a closed-form posterior. It also identifies a measurement-equivalent point outside the prior support as a prior-inconsistent hallucination.
A.2 TOY EXPERIMENT DETAILS
- The toy prior has two atoms at ±au, with reference vector u set to the first standard basis vector e1.Source: §A.2 · p. 13
- For the specified prior, DAPS and PC-DAPS use the closed-form denoiser instead of a trained network for the probability-flow ODE and Tweedie estimates.Source: §A.2 · p. 13
- The measurement operator is quadratic in the inner product, and observations add Gaussian noise with variance βy^2.Figure 2 uses βy = 0.05, b = 0.1, and κ = 2, giving c = 0.1.Source: §A.2 · pp. 13–14
- The ground-truth posterior is given in closed form as a mixture of point masses at −au and au, with mixture weight π defined from their likelihoods.Source: §A.2 · p. 14
- Although A((a + b)u) = A(−au), the point (a + b)u has no prior mass and is identified as a prior-inconsistent hallucination.Source: §A.2 · p. 14
A.3 MORE DETAILS ON GAUSSIAN MIXTURE APPROXIMATION
The toy comparison examines Langevin refinement under four configurations, including DAPS and PC-DAPS approximations. The Gaussian mixture can draw refinement toward prior-consistent regions identified by other particles, unlike the local Gaussian behavior described for DAPS.
A.3 MORE DETAILS ON GAUSSIAN MIXTURE APPROXIMATION
- Figure 4 compares four panes: DAPS, PC-DAPS with the DAPS Gaussian, PC-DAPS with the PC-DAPS Gaussian mixture, and a ground-truth denoiser-and-Langevin iteration.The example uses a multimodal prior on R2 and a measurement operator selecting x2; panes depict high-probability level sets.Source: §A.3 · p. 14
- The PC-DAPS Gaussian mixture is attracted toward prior-consistent regions of the measurement preimage identified by other particles, even when distant from the Langevin initialization.Its component covariances also adapt better to the geometry of prior support than the isotropic DAPS Gaussian.Source: §A.3 · p. 14
- With the DAPS Gaussian approximation, Langevin refinement searches for measurement-consistent points close to initialization and can end at a posterior-inconsistent point.Source: §A.3 · p. 14
A.4 ADDITIONAL IMPLEMENTATION CHOICES
PC-DAPS implementation choices balance sample quality and computational cost while preserving prior-supported modes during refinement. Ablations show how particle count, ODE integration, annealing schedules, and pruning affect reconstruction.
Skipping the last Langevin refinement
- At the final annealing time, PC-DAPS can skip Langevin refinement and use the ODE endpoint because the near-zero-noise step is tiny and added noise can reduce quality.Source: §A.4 · p. 14
Adaptive ODE steps
- Adaptive ODE stepping uses one Euler step at high noise and five at lower noise, where finer integration reduces discretization error near the data manifold.This choice limits unnecessary ODE evaluations in the particle-based sampler.Source: §A.4 · p. 14
Additional implementation choices
- PC-DAPS’s Gaussian mixture represents multiple prior-supported regions, unlike Gaussian approximations that can yield measurement-consistent but prior-inconsistent Langevin targets.The mixture allows refinement toward a prior-supported mode associated with the measurement.Source: §A.4 · p. 15
Clustering technique
- Particles are downsampled, centered, and projected onto principal components explaining 90% of variance before a BIC-selected Gaussian mixture assigns them to components.Source: §A.4 · p. 15
Pruning threshold
- Pruning is triggered only when the stated test fires and clustering yields multiple clusters; weaker-scoring clusters are removed and their particles copied from survivors.The score threshold includes γ, calibrated to the 95th percentile of within-cluster relative score differences across 10 calibration images per dataset.Source: §A.4 · p. 15
Ablation of ODE steps and annealing noise scheduling steps
- Denser annealing noise schedules are particularly beneficial for nonlinear inverse problems, while ODE-step trends are consistent with those reported for DAPS.Source: §A.4 · p. 15
- More ODE steps generally improve perceptual quality (LPIPS) but can reduce PSNR, with the trade-off depending on the inverse problem.Source: Figure 5 · p. 16
Number of particles in a bag
- On phase retrieval, increasing bag size improves PSNR and LPIPS rapidly through K = 8–16, with smaller gains beyond that range.PC-DAPS is reported to achieve higher reconstruction quality than DAPS best-of-k while using less sampling time for larger bag sizes.Source: §A.4 · p. 16
A.5 TIME COST
PC-DAPS is evaluated across inverse-problem tasks, with comparisons covering reconstruction quality, sampling time, and qualitative sample behavior. The setup also specifies benchmark conditions and baseline protocols.
A.5 TIME COST
- PC-DAPS propagates K particles in one trajectory, whereas DAPS requires K separate runs; clustering, mixture fitting, and pruning add only small overhead relative to ODE and Langevin updates.Source: §A.5 · p. 16
- For phase retrieval, the DAPS timing includes four runs, while PC-DAPS forms and selects its particle set in one run; super-resolution takes longer with PC-DAPS.The comparison reports sampling time and LPIPS for 100 FFHQ images; multiple particles do not always entail much higher wall-clock cost for the same quality metric.Source: §A.5 · pp. 16–17, Figure 8 · p. 18
A.6 INVERSE PROBLEM SETUP
- The benchmark uses 100 test images per dataset at 256×256 resolution, with eight DAPS tasks plus 16× super-resolution and high-pass filtering.The stated default measurement noise is Gaussian with standard deviation βy = 0.05.Source: §A.6 · p. 17
- Under mean-of-k phase-retrieval evaluation, PC-DAPS achieves higher reconstruction quality than other baselines.All methods are evaluated on the mean across k reconstructions rather than selecting the reconstruction closest to ground truth.Source: Table 4 · p. 18
A.6.1 BASELINE DETAILS
- For phase retrieval, baselines other than DAPS use best-of-four ground-truth selection, while PC-DAPS metrics average four reconstructions; this gives baselines test-time ground-truth information unavailable to PC-DAPS.Source: §A.6.1 · pp. 18–19
A.7 MORE QUALITATIVE SAMPLES
- PC-DAPS samples retain diversity on 16× super-resolution and 190×190 box inpainting, and provide competitive reconstructions across linear and nonlinear tasks on FFHQ and ImageNet.ImageNet phase-retrieval samples still include failed noisy or diverged particles and underperform compared with FFHQ.Source: Figure 10 · p. 21, Figure 11 · p. 22, Figure 12 · p. 23