RODS: Robust Optimization Inspired Diffusion Sampling for Detecting and Reducing Hallucination in Generative Models
Authors: Yiqi Tian
Pengfei Jin
Mingze Yuan
Na Li
Bo Zeng
Quanzheng Li
Venue: NeurIPS 2025 poster ()
Keywords: diffusion model
robust optimization
hallucination
Primary area: deep_learning

Abstract: Diffusion models have achieved state-of-the-art performance in generative modeling, yet their sampling procedures remain vulnerable to hallucinations—often stemming from inaccuracies in score approximation. In this work, we reinterpret diffusion sampling through the lens of optimization and introduce RODS (Robust Optimization–inspired Diffusion Sampler), a novel method that detects and corrects high-risk sampling steps using geometric cues from the loss landscape.  RODS enforces smoother sampling trajectories and \textit{adaptively} adjusts perturbations, reducing hallucinations without retraining and at minimal additional inference cost. Experiments on AFHQv2, FFHQ, and 11k-hands demonstrate that RODS maintains comparable image quality and preserves generation diversity. More importantly, it improves both sampling fidelity and robustness, detecting over 70\% of hallucinated samples and correcting more than 25\%, all while avoiding the introduction of new artifacts. We release our code at https://github.com/Yiqi-Verna-Tian/RODS.

RODS: Robust Optimization Inspired Diffusion Sampling for Detecting and Reducing Hallucination in Generative Models
Yiqi Tian *
Pengfei Jin
Mingze Yuan mingzeyuangg harvard.
ytianO@mgh. harvard. edu yit300pitt.edi
pjini@mgh. harvard.
Na Li mali@seas hwr.arrd.dee... edu
Bo Zeng
Quanzheng Lit quanzheng@ngh. harvard.
bzeng@pitt.ed. edu
Abstract
Diffusion models have achieved oitte-of-hhe-rr art performance in generative mod- eling, yet their sampling procedures remain vulnerable to hallucinations often stemming from inaccuracies in approximation. In this work, reinterpret diffusion sampling through the lens of optimization and introduce RODS (Robust Optimization-inspen inspiired Diffusion Sampler), novel method that detects and corrects high-risk sampling steps using geometric cues from the loss landscape. RODS enforces smoother sampling trajectories and adaptively adjusts perturbations, ducing hallucinations without retraining and at minimal additional inference cost. Experiments AFHQv2, FFHQ, and Ilk-aands demonstrate that RODS maintains comparable image quality and preserves generation diversity. More importantly, it improves both sampling fidelity and robustness, detecting over 70% hallucinated samples and correcting more than 25%, all while avoiding the introduction of new artifacts. We release our code at https://github.co www.b..co//iiii.i..... /iiii/iiii-i Merna-Tia/RODS.
Introduction
Diffusion models [19 25 also known as score-based generative models, have achieved remarkable success across various domains such as image generation [16] [3), audio synthesis 29 10]. and
video production 20 48 These models generate data through an iterative denoising process that progressively transforms noise input structured outputs, offering a flexible trade-off between computational cost and sample quality (19 25). Beyond their generative capabilities, diffusion models have proven effective solving complex inverse problems such as image inpainting [34, (6], colorization [45] 31) and medical imaging applications [3 40). This wide range of applications underscores the significant potential of diffusion models to transform different fields by generating high-quality, diverse synthetic data that can be tailored for specific needs.
Despite their impressive performance, diffusion models are prone to ballucinations--uutee outputs that are unfounded, unfaithful, not grounded in the underlying data [4 30 While extensively studied in text 4 [4] hallucination equally problematic in visual domains. For instance, diffusion models
"Equal contribution.
'Center for Advanced Medical Computing and Analysis, Massachusetts General Hospital and Harvard
Medical School, Boston, MA 02114
I Department Industrial Engineering, University Pittsburgh, Pittshurgh, PA 15261 School of Engincering and Applied Sciences, Harvard University, Boston, MA 02138
Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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may synthesize anatomically implausible humans 2 38] or generate unrealistic scene compositions.
In high-stakes applications such as medical imaging or radar sensing, these errors can lead to A key difficulty is that hallucination lacks
false diagnoses or misinformed decisions 27
formal definition, making both detection and mitigation inherently challenging [24]. Nonetheless, its importance has sparked growing research to understand and address these failures.
Recent mitigation strategies in diffusion models include trajectory variance filtering, which detects off-manifold samples via prediction instability in the denoising path [2]; local diffusion techniques that isolate out-of-distribute regions and process them separately [27]; and early stopping methods that truncate sampling before overfitting to noise occurs [51]. Additionally, fine-tuning methods have also shown promise, particularly in hallucination-pri prone areas like hands and faces, using structure- aware losses [32 or subject-specific adaptation [4.. While these methods address specific aspects of problem, hallucination mitigation remains an open challenge. Existing approaches vary scope,
assumptions, and evaluation criteria, and the field still lacks standardized benchmarks for systematic comparison.
To understand how diffusion samplers can fail, it is helpful to view sampling as an optimization process. Consider the probability flow ordinary differential equation (PF-ODE) as guiding climber down a mountain ridge in dense fog, At each time I, the learned score function provides direction of descent; time progresses 4 0), the fog lifts and the terrain becomes clearer. This step-by-step refinement closely resembles the continuation method in numerical optimization, where one solves a sequence of gradually harder problems locally approach final objective. However, when the score function is imperfect-as as is often the practice case-it can point in misleading directions. In benign regions, such errors are tolerable, but in sensitive areas, they may flip the intended descent direction entirely, sending the trajectory off- off-manifold and leading to hallucinated outputs [46 4 2 39
Diffusion Sampling Process
Optimization Problem Solving Process
Curvature Change Detection
(b1)
(a)
Step
Sapmoon
H((r.)
Inacourate approddmation Pu(X)?
RODS Optimizution Diffusion Sampler
(b)
Robust Update Correction
(b2)
Euler
H(x:)
Euter update
Fi
Zi+1
RODS
H(n)) 2 Robust update
(c)
Time step
Figure The roadmap of our paper: (a) Section 3 formulates the diffusion sampling process as an optimization problem solved based on the continuation method. (b) Section 4 introduces the RODS framework address the inaccurate approximation of the score function: (bl) Section 4.1 details how we detect high-risk regions based on local curvature changes (highlighted in green). (b2) Section 4.2 describes how robust updates are performed mitigate potential hallucinations.
(c) Section 5 presents experimental results and analysis. illustrating improvements in hallucination detection and correction.
This observation motivates simple yet powerful idea: sample as cautious optimizer. Instead of blindly following the indicated direction, probe the surrounding neighborhood for signs instability, such regions with rapidly changing curvature. This perspective naturally leads to the principle of robust optimization (RO)-a strategy that accounts for worst-case perturbations to ensure progress even under uncertainty. We operationalize this idea through plug-and and--an--lay method called the Robust

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Optimization- inspiied Diffusion Sampler (RODS). At each step. RODS: (i) explores small local neighborhood of radius p detect directions where the score landscape exhibits high curvature or instability, and (fi) adjusts the update direction to maintain objective descent under worst-case local perturbations. Importantly, RODS requires no retraining of the diffusion model and introduces minimal computational overhead.
The roadmap of our paper is summarized in Figure follows:
The key contributions of this work are as
We reinterpret diffusion sampling through the lens of continuation methods in numerical optimization, which frames each sampling step progressively refined sub-problem and opens the door for leveraging various optimization tools.
We propose RODS, plug-and-pen sampling framework that integrates RO principles into the diffusion sampling process. RODS includes novel curvature- based change detection step that probes local geometry to identify sudden variations in score field behavior. While not detecting hallucinations directly, these changes often signal high-risk regions that correlate with failure modes such hallucination. According the defection of the high- risk regions, RODS then dynamically adjusts sampling directions based on worst-case local perturbations, enhancing robustness and reliability in uncertain or error-prone regions.
We conduct extensive numerical experiments on high-dimensional, real-world image datasets- includiig AFHQv2, FFHQ, and Hand dataset-the that go beyond synthetie 1D or 2D Gaussian examples. Our results show that while maintaining comparable image quality and generation diversity, RODS successfully detects over 70% of hallucinated samples and corrects more than 25%, all without introducing new artifacts requiring any modification to the pre-trained diffusion model.
Preliminary
2.1 Diffusion Model
Diffusion models are probabilistic generative models that transform data into noise via forward stochastic process and then reconstruct it by reversing that process. Formally, the forward process is defined by the stochastic differential equation (SDE):
dæ; = p t) dt + (t) dwi,
where We is standard Brownian motion and & E [0, T Under mild conditions, this SDE admits deterministic counterpart- the probability flow ODE 49):
dx; = (((ret)) 2 (t) (t) log (ex)) dt,
which preserves the same marginal distributions (z).
In EDM method 25 the forward process defined by zero drift, u(x,,() and variance (t) 22t, resulting Gaussian smoothing Pt Po * (0, f2). where denotes the convolution operator. To generate samples, one trains a neural network to approximate the score function so s0 t) Vog (x and substitutes it into the reverse ODE. This yields different parameterizations that are mathematically equivalent but tailored to different training objectives and architectural designs:
dx. -==000(010)) -co( di
2t Do(2c;t)
where 8p, <o. and Da denote the score, noise prediction, and denoising function, respectively, as used in different methods. Starting from Gaussian distribution large one iteratively removes noise recover samples from Po- This framework has shown strong empirical results across image, audio, and other generative tasks.
2.2 Continuation Method
The continuation method is a powerful optimization technique designed handle challeng- ing, nonconvex problems by progressively transforming the optimization landscape. It begins by
3

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introducing a regularization term XR(x)) to the objective function:
min f(x) ) + AR(x),
where f(x:) is the original objective function, and R(x) imposes prior or penalty the solution. The parameter À controls the influence of the regularization term.
The key idea is to simplify the problem early in the optimization process by setting À to a large value, allowing the regularization term to dominate and smooth out the optimization landscape. As the optimization proceeds, À is gradually reduced, making the problem increasingly faithful to the original objective f(x)). At each step, the solution to the simpler problem is used as the starting point for the next iteration. This iterative reduction in continues until it approaches zero, at which point the algorithm converges to a solution of the original problem. See Appendix [..I for algorithm details. This method is particularly well-suited for nonconvex problems, where poor local minima can hinder direct optimization approaches. This method balances tractability and accuracy, smoothing out local irregularities in the optimization landscape while progressively honing in the true objective.
2.3 Robust Optimization (RO)
a fundamental framework optimization that seeks solutions which remain effective under arbitrary perturbations with defined set. In its standard formulation, RO expressed as:
Robust optimization (RO)
3
min max f u), well
where uf denotes an uncertainty that defined to encapsulate potential variations disturbances. Rather than optimizing a fixed objective, RO explicitly accounts for randomness within 44 thereby ensuring that the solution is resilient adverse shifts in the problem data. This approach is particularly valuable in situations where model parameters or environmental conditions are uncertain, as it helps prevent performance degradation when faced with unexpected changes.
3 An Optimization View of Diffusion Sampling Process
Recent research has explored connections between dif- fusion models and optimization methods, broadly clas- sified into two categories. The first category interprets Score-Matching Marching Langevin Dynamics (SMLD) diffusion models as variants of stochastic gradient descent (SGD) _0].The The second category leverages diffusion mod- els as learned priors for sol solving optimization tasks, such
Prodmed Sappedtorss
lango
49 52 50
planning or inverse problems 23
12 35). Our work aligns
more closely with the first category but distinctly frames diffusion sampling as a continuation or homotopy method from numerical optimization. Unlike conventional inter- pretations based on fixed-noise-level SMLD, we explicitly regard the entire diffusion trajectory as solving a sequence of progressively refined optimization subproblems. Fur- ther theoretical comparisons and detailed discussions are
small
Image held
Score fleid Ineratiive updato sumpling
Comiimation Mothed
provided in Appendix(D D
Figure 2: The equivalence between the diffusion sampling process and the opti- mization continuation method.
Our discussion so far treats diffusion sampling via an ODE perspective, writing
dæ
di = - =x(
(5)
and integrating from time T down to Here, the neural network co is trained such that V fe(x) x - -2 !), where fe(x) = log (x).
Hence, each step of 5 resembles gradient descent update on fx()). In this section, we reinterpret the same steps as continuation method optimization, show that each Fuler step can viewed as solving local subproblem via proximal operator, and finally present unified pscudocode.

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Analogy in Diffusion. The continuation method aims to solve challenging objective
min f(x) + XR(x)
by first introducing a strong regularization R(xx) with a large smoothing parameter X, and then decreasing gradually. One thereby transitions from a simpler (highly smoothed) problem the original. more complex objective.
In diffusion models. the noise level t plays the role of "smoothing strength." At high noise t, the convolved distribution (z( = (r( t2) is more diffuse than (o(x), so its negative log-density fa(x)
log (x is relatively smooth and has fewer local minima than fo(z) = log po (r)) As t decreases, fa) transitions from a highly smoothed version to the original, sharper landscape. This parallels the continuation principle: one "continues" from easy subproblem at high noise (t 2 T) to the more difficult, exact - log (x)) at ( = 0. Such procedure equivalence formalized in Theorem with the proof detailed in the Appendix A.
Theorem 1 (Procedure Equivalence). Assume the image distribution can be approximated by mixture of Gaussians, iee Po(x)) = >im w; wxN(x (8), Σ wi = =. Define
[((x) logro(z) log
wx(x) exp aa- 1=-
201
where Wi (x) = ER my (alpssaatt
Ein (x)x)) 1. Then min f1( min
log m leg))
Moreover, fe(z) shifts continuously from a heavily smoothed version = T toward Po()) at t= 0, aligning with continuation.
Analogy in Euler Step. Taking closer look at the procedure, when discretizing the diffusion ODE, we move from L; to 1:-1 l Al simple Euler step:
It. At ff. (zr.).
(6)
This update is precisely one step of gradient descent on ft, (Ir), under a local trust-region interpretation where the step size At controls how far we move from Tt.- Below, formally show that this one-step gradient can also be viewed as solving a local subproblem with proximal (quadratic) penalty.
Theorem 2 (Euler One-Step Gradient Proximal Update). Define the local subproblem at time t;:
1 min (sa(s)) + 24t lly 22,
A single gradient descent step on ft, of size At solves approximately, with solution y satisfying
y" prox fú; (FL.)
y" -r, At Vl,
and if we approximate V St. (y*) R V (Tt:), this yields the Fuler step (6)
This shows that each Euler move is single gradient descent update on ft, (z), equivalently inter- pretable as a proximal (trust-region) solution at time t;.
Overall, because diffusion lowers the noise level t in discrete steps, we can view the entire sampling process as a continuation method: each subproblem corresponds to minimizing fe, starting from ty = 'T' (high noise, smooth landscape) and ending at to 0 (true but sharp - log Po (x)). Algorithm [ details this procedure.
numerically identical to solving the ODE (5 with an Fuler integrator but emphasizes the iterative optimization interpretation. Furthermore, the proximal operator on each subproblem (7) matches the Fuler updates, as illustrated in Figure More sophisticated ODE solvers (Heun, RK4) correspond to more elaborate ways of approximating y² in each subproblem.
5

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Algorithm 1 Diffusion Sampling as Continuation-Methoed Optimization
Algorithm 2 Robust Optimization Diffu- sion Sampler (RODS)
Require:
Time steps {t, {t,]i=N with
Require: Score network (a(=; σ), time steps t; E [O....,N] NN. perturbation set M, curvature threshold é
IN = and to = 0,
Step size At (so 4i-1 = 11-1400 At),
1: Sample z0 N(, EgT
Neural network <0(:,-) satisfying
2: for ...... - d
(f((r) = eo (2, (x,t)/ /t.
3: d; Dolacit)
1: Initialize: Pty ~ N0, I)
4: u(x:)) <- de/t,
2: for N, N.....! do
3: Compute Gradient:
5: Compute curvature index:
(zr, = co (Z t.)
H(x:) - max V 1,/1/ + 6)1l Va, (o(x))) BE.M
4
6:
if' H(;;) then
4: Proximal Subproblem:
7:
0; - arg maxseM It. (Ti,
8:
I; + 6; d; B.-Do(2
Ttp-1 proxxa /s, (Ie,)
9:
4
5: (Euler Equivalent):
10:
1i+1 t- Iz (li+1 - 1.)) di
11: else
Tte It, Al V. (114.)
12:
T1+1 +- Hy (li+1 - 14)) d4
6: end for
13: end if'
7: Return: Tto
14:
end for
15:
return TN
4 Robust Optimization inspired Diffusion Sampler (RODS)
Recent studies have shown that diffusion models sometimes produce unrealistic or "hallucinated" outputs 38 [8. especially when sampling paths pass through low-density regions where the model's estimates become unreliable [2]. Standard sampling methods closely follow the model's estimated direction but lack built-in safeguards against sudden instabilities sharp changes in behavior. Therefore, our proposed hallucination detection builds on the insight that hallucinated behavior closely correlated with model uncertainty, which can be quantitatively inferred from the local curvature change (rapid fluctuations) of the score field v(x) V. log (1r). For example, sharp increases in curvature proxy often signal transitions into unstable poorly approximated zones, where hallucinations are more likely to occur.
From the optimization perspective, we could conduct robust optimization idea to improve the reliability of sampling. Instead of directly minimizing the estimated potential function fa()) at each timestep (i.e. each subproblem), consider such function via more cautious approach:
min max ft(x), f.F.F
where F, represents set of plausible variations around the estimated function. This setup asks: what the worst-case outcome in the surrounding region, and how can we choose x to avoid bad surprises? Intuitively speaking, by using RO method, we peek at the place where the model is most uncertain. borrow a direction that could most effectively mitigate potential errors, and use it to update the current one. As result, anticipate that, the sampling process becomes more stable, particularly in regions where the model's approximation is noisy or inconsistent.
4.1 Curvature Change Detection
To decide when RODS should switch from a standard to robust update, we need a fast signal that a sampling step is entering unstable territory. As mentioned [49], generalization errors arise during the diffusion reverse process in low-density regions. This results in the learned score function deviating from the sharp. high-gradient structure of the ground truth score, and leads to potential hallucination effects in the diffusion model, where generated samples may deviate from realistic

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outputs. [2] proposed detecting such errors by measuring prediction instability along the temporal direction of the denoising trajectory. In contrast, our approach focuses on identifying instability in the spatial domain. High local curvature in the score field v(x)) serves as an indicator of model uncertainty and, consequently, hallucination risk. To detect landscapes of the score field, we propose using an index H(xx), inspired by the maximum eigenvalue of the second derivative of the score norm:
H(xx)
V 1o + 6)ll V=[lo(=)]|||s 8 = arg max || + 6)1|, 1|8|=p
where p is the detection radius. Intuitively, H(xx) measures the degree of fluctuation in the magnitude of the score along direction 8. For theoretical insights, please refer to Appendix [A2 We consider the current step x to have entered high-risk region when H(x) > e, where e is a predefined detection threshold. In practice, the choice of E reflects trade-off: a lower threshold increases sensitivity and enables the model to detect more hallucinated samples, but may also lead to false positives by flagging benign regions unstable. When the surrounding region is reliable (e.g., visually simple), smaller p is preferred, reflecting higher trust in local information. Conversely, a larger p makes the approach more conservative, expanding the search when the neighborhood may be misleading.
More importantly, the threshold can be adapted different applications. For example, in high-stakes domains such as medical imaging, or more challenge generation tasks, lower may be preferred to prioritize sensitivity (Appendix (..2.. Furthermore, all of our experiments show that applying updates to false positive samples does not degrade output quality at the cost of mild computation (as shown in the results section), indicating that RODS remains robust even in the presence of false positives.
4.2 Robust Sampling Process
Once we detect that the current sampling step lies in a high-risk region, we apply a robust update to make the diffusion process more stable. From an RO viewpoint, this means preparing for the worst- case scenario within a set of plausible perturbations to the function correspondingly. In this context, we introduce two robust update schemes grounded in the RO framework: Sharpness-Aware Sampling (SAS) and Curvature-Awar Sampling (CAS). Both approaches operate by solving min-max max RO problem that seeks to guard against local failures, but they do so with different emphases.
Sharpness-Aware Sampling (SAS). SAS focuses on worst-case spikes in the score function ft. which captures the model's energy landscape time and can be directly derived from RO scheme. We define a function family F, consisting of localized shifts of fc:
F, [us fe,(z)= fe(x + d), 8 M.
where M is a small bounded set (e.g., an bz-ball) of perturbations. For the associated min-max max problem, we minimize the worst-case value of fi within the nearby neighborhood, which hedges against sharp peaks or local inconsistencies that could derail the sampling trajectory. Specifically, we handle the following bilevel optimization problem:
min fe(x 8), where
8 = arg max (f((z, fu F 8'EM
Actually, as in the following, we can generalize the inner maximization to other forms
8 arg max gr (2,
8'M
where gx( 8') defines some type of (undesired) metric to focus on. The details of solving subprob- lems are provided in Appendix B.4
Curvature-Aware Sampling (CAS). In contrast to SAS, which concentrates on worst-case in- creases in function values, CAS focuses on the shape of the score landscape. Specifically, it aims to identify directions where the slope of the function is steepest by maximizing the gradient norm:
Jr,cAs( 8)
[. +
While this form differs from the classical RO objective, it shares the same spirit: guarding against dangerous regions. Large gradient magnitudes indicate steep or unstable directions, so stepping toward them helps uncover problematic curvature. After identifying the high-curvature direction, the sampler can take controlled step to avoid abrupt changes.
7

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EDM-Eular EDM-Heun
RDDS-SAS
RODS CAS
Labeled Normal
Labeled Gmunn Truth
896
176
samee: 155
14
sutric 6
unclerr: 0
Predicted Prediction
Figure 3: AFHQv2 sampling results. Left: Visual comparison across different samplers-EDM (Euler, Heun), and our RODS-SAS. RODS-CAS Hallucinations include misplaced eye, incorrect facial structures,
Right: Confusion matrix at e 0.1 and max ||ô]| 1 over 1,080 samples using the proposed hallucination index H(x). RODS-C CAS detects 87.5% of labeled hallucinations, correcting 10 cases (better), and introduces no degraded cases (worse).
4.3 RODS Algorithm
Therefore, once we detect that the sampling path is entering potentially unstable region, which typically characterized by sharp curvature changes, apply more cautious update to help prevent unrealistic or unstable samples. The full procedure is outlined in Algorithm [ closely follows the deterministic sampling style of EDM [25], but selectively activates robust steps when the local geometry suggests the model may unreliable.
In each iteration, the algorithm first estimates the current score direction and checks whether the score landscape is changing too sharply using the eurvature index H(xx). If the region is flagged as high-risk, it proceeds through three steps: (I) Search look around the current sample to find the most sensitive direction 6; that exposes model uncertainty, obtained by maximizing a function 9r, (z,, 6), which can be tailored to different risk criteria (c.g., SAS or CAS); (2) Descent compute the steepest descent direction d, the identified worst-case point neutralize local instability; and (3) Update move the original sample r; along this safe corrective direction.
5 Experiment
We evaluate our method on three benchmarks: AFHQ-v2 26 FFHQ (26), and 11k-hands [I]
For AFHQ and FFHQv2, we use pre-trained models from EDM 25 For 11k-hands, we train diffusion model using standard frameworks and configurations in EDM. All training and evaluation are conducted on a single NVIDIA A100 GPU (Appendix B.J for dataset and training details).
Due to substantial diffcrences across prior works in methodology, assumptions, and evaluation protocols, there currently standardized benchmark for hallucination detection, generation and correction. We summarize and contrast related methods in Appendix C As a plug-and-play module, RODS reduce to the default EDM-Eule sampler when the detection threshold is high. Therefore, 2DM-Euler serves as our baseline method. In particular, we perform manual hallucination annotation for evaluation (Appendix B.6).
AFHQv2 We generate 1,080 samples from the AFHQv2 dataset using random seeds from 0 to 119, with batch size 9. With the robust region M == [[]8] 1} and the detection threshold = 0.1, our method detects 7 of 8 labeled hallucinations, improves 7 unlabeled cases, and introduces no degraded outputs demonstrating its robustness. successful visual corrections by RODS-CAS, while the right panel summarizes detection and correc- tion performance. Notably, most samples labeled as non-hallucinated by human annotators remain visually unchanged after the robust update (Appendix Figure 13 14). In some cases, however, even these non-hallucinated samples exhibit subtle improvements appearing more semantically coherent after refinement (c.g., top-left in Appendix Figure 8).
shown Figure the left panel illustrates

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EDM Euler EDM-Hicur
RODS-SA SAS
RODS-CAS
Labreie Ground Truth
859
181
better: Satree worde:
29
tamee 24
unclaar:
Predicted Normal
Predicted Hollucination
Figure 4: FFHQ sampling results. Left: Visual results across EDM-Euler, EDM-Heun, and RODS-SAS, RODS-CAS. Hallucinations include distorted eye regions and implausible facial geometry. Right: Confusion matrix at 0.09 and max || = 8 over 1,080 samples using the proposed hallucination index H(z)). RODS-CAS detects 72.5% of labeled hallucinations, corrects cases (better), and introduces no degradation (worse).
EDM-Eular
EDM-Heun
RODS-SAS
RODS-CAS
Labettd Grennd Puth
85
640
0
T
EDM-
EDM-
RODS-
Metric
Euter
Heun
Hall.%
19.4%
18.0%
14.3%
Correction %
N/A
17.7%
(s)
2.71
Figure 1 11k-hands sampling results. Left: Visual results for different samplers. RODS-CAS corrects hallucinations such as extra or missing fingers while preserving anatomical plausibility. Top-Right: Confusion matrix at e = 0.014 and max ||o|| = 30 over 900 samples using the proposed hallucination index Hx)). Bottom- rate (f), new hallucination rate (L), and inference time (in seconds).
Quantitative metrics: hallucination rate (L), correction
FFHQ We then apply the same evaluation procedure to the AFHQ dataset, with M = {1 8} and = 0.09; an ablation study on threshold sensitivity is provided in Appendix B.2 Our method detects 29 of 40 labeled hallucinations and improves 9 of the detected images without introducing any visible worse cases. As shown in Figure 4 the left panel illustrates representative visual corrections, and the right panel reports quantitative results.
l1k-hands To evaluate performance in more challenging setting, test the 11k-hands dataset, where hallucinations are more frequent and severe. We generate 900 samples using seeds 0 to with batch size of 9. To ensure high sensitivity, we adopt a stricter configuration with M = {||o]| 30} and c = 0.014. Under this setting, our method successfully detects 169 out of 1 175 numan-labeled hallucinations. As trade-off for this high sensitivity, the method attempts to revise every suspicious sample.
Based on ablation studies, we found that the key issue lies in the directional error, particularly in low-density, high-uncertainty regions (Appendix B.5). Furthermore, we observe that critical steps (Appendix B.3). those most responsible for hallucinations, tend to occur in the middle of

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Table 1: Comparison of FID, Inception Var, and DreamSim metrics across datasets, each evaluated on 1k generated samples.
FFHQ
AFHQv2
1ik-hands
Dataset
FID 4
Incep
DreamSim
FID
Incep Var
DreamSim
HID
Incep
DreamSim
EDM-Euler
20.48
0.0669
0,407
13.36
0.0539
0.486
14.57
0.0327
0.1427
EDM Heun
19.56
0.0699
0.422
12.66
0.0548
0.488
14.20
0.0332
0.1476
RODS-CAS
22.73
0,0661
0.400
16.08
0.0542
0,481
16.68
0,0318
0.1509
the sampling trajectory. To reduce computational cost while preserving performance, we employ a critical-step truncation strategy (Appendix [B.3]. We compare EDM--uuer, EDM-Heun, and our proposed method RODS-CAS. As shown in Figure RODS-CAS achieves the lowest hallucination rate while correcting the highest proportion of faulty samples, and introduces no new hallucinations.
To provide broader generation quality assessment, we have further computed standard image-quality and diversity metries on 1,000 unconditional samples from FFHQ, AFHQv2, and Ilk-hands. Table ] reports Fréchet Inception Distance (FID) [8] (which quantifies the similarity between the generated and real image distributions in the Inception feature space), Inception Variance 33 (which measures variability in the latent representations), and DreamSim Diversity [17] (which captures the variance of features extracted from batch of generated images). These results show that RODS-CAS preserves diversity while keeping FID comparable. Moreover, in our generation, the main subjects remain sharp and artifact-free.
These results, which spanning multiple datasets and varying task difficulty, demonstrate that our method generalizes effectively: it identifies potentially hallucinated samples at a controllable sensitiv- ity; after applied robust corrections to the identified hallucination, it achieves over 25% correction for true positive depending on the difficulty a given task, and preserves output quality even for false positives.
6 Conclusion
In this work, proposed the equivalence between the diffusion sampling process and the numerical optimization continuation method. Build upon such optimization point-of-view, we introduced RODS, that is to detect and correct hallucinations in generative models. By leveraging geometric information from the loss landscape, our method adaptively adjusts the sampling direction in high-risk regions, significantly improving sample quality without additional training or excessive compute. We validated our approach across three MaansetssAFQv22 FFHO, and 1 Ik-hands. Overall, RODS-CAS delivers equally diverse, comparably high-quality unconditional samples while reducing hallucinations.
While RODS-CAS shows promising improvements over standard samplers in reducing hallucinations and correcting semantic inconsistencies, several aspects remain open for future study. First, curvature-bacs -basd detection provides reliable signals instability, however, more precise localization could further improve correction accuracy. Empirically, we observe that early corrections tend to influence global semantics, while late ones mainly refine local details. This trade-off between early and late interventions offers flexibility, enabling RODS to adapt its correction strength to different scenarios. Second, our RODS algorithm can be directly implemented within the VE-ODE framework, which benefits from the equivalence between VE-based sampling and the continuation method under the usual parameterization. The extension of RODS to other sampling schemes (c.g., VP or latent-space diffusion) conceptually straightforward but remains be validated experimentally. Furthermore, its generalizability to text-to-image or conditional generation tasks should be straightforward but requires a thorough study in future work.
Acknowledgment
This work was supported by the National Institutes of Health (NIH) under award number R01HL159183 and the National Science Foundation (NSF) AI Institute under award number 12085
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A Theoretical Analysis
In this section, we provide theoretical justifications and analytical insights into the proposed method. Specifically, we draw a connection between the diffusion sampling dynamics and the continuation method, and further investigate the relationship between our proposed index and the local geometry of the score landscape, characterized by the Hessian's 's spectral properties.
A.1 Continuation Method and sampling process matching
We formally derive the connection between diffusion sampling and continuation methods, showing that the sampling trajectory implicitly follows an energy descent path under gradually relaxed constraints. This view provides a principled interpretation of diffusion as continuation process. While prior work has explored the link between Score-Matching Langevin Dynamics (SMLD) and our framework generalizes this connection the diffusion sampling.
optimization D
Algorithm 3 Continuation Method
Input: Objective function f(z), regularization term R(x), initial Ao. decay rate a (a 1), stopping criteria
2: Initialize: (0) + initial guess, +- 0
3: while dạ do
4: Start from initial guess (A)),, solve: x (+++1) +- arg min, (f(x) + d R(x))
5: Update: λx+1 + aλ2, + +1
6: end while
7: Output: Final solution (k+1)
Theorem Proof
Proof. Case I: When 6 = TT,,,000 T + oo, the defined function can be simplified as:
wwN(x uis
o
| mu T
[T(2)= log - log
K wiN(x (3)
exp
20g(a0 + T2)
K
w;N(x M4 og)
| H4 T 20 + T2)
- log
W/N(x pi, pn)
exp
1=1 === wzN(x oi)
|| 11-11133 T2 20 T2)
log
wiN(<> (i, exp
log
waN(( n4s oه + T2)
Thus:
min (T((() = min -
(9)
where the resulting pure Gaussian distribution:
K Σ
pr(x) = N(x μr + To),
(10)
Thus, f T, minimizing fr( aligns with minimizing log pr(x), which confirms the behavior of the diffusion process in this case.
Case When t < T, the diffusion process probability distribution form can be expressed in this way:
K waN(x Mi, of + +²),
(x = Σ
(11)
iml
22

---

Then, its log form can be expressed as follow:
- log pc())
log
wiN(a
Mis
oB 1?)
K
v o
1 p 113 +2 20 + (2)
- log
Wi
No pr. (2) exp
+
12
==1
0
K
11 - mil1 t²
log
log
wyN(2 pi, (1 exp
2
%
f2
20/ +t2))
1 log
00
WiN(Wx Mig oō)
1 Mi m1113 12 20/ o +22) }
log
wyN(x pi:
exp
2
0
f2
j= WyN(2 MMj 08)
K
K
w.N(x Mi.
10 - pil13 82 20 + (2) }
log
log Σ
w;N(( Uis og) (
log
exp
2
0
+ 12
Σド= wyN(( Mg of)
1r-Mulll-+ 20
(The last two term is pp)(z) exp
describe the contribution proportion of each Gaussian distribution)
(0 +-2)
1
of
K (.(x)) exp
| 11112 +2 20 + 82)
2 log
log (og) log
0
12
i=1
(w,(x)
w.N(cllu Σド "
E, (x((x)) = =)
o 2 0 + 1
log
+ f((x)
Therefore, minimizing fe(x) over x is equivalent to minimizing -logr(xx) over x.
Case 3: When / is O, corresponding to the original distribution, the second term in the function vanishes. This simplifies our defined function to:
K
fo(x)) = log (z)) - log
Σ
Wz (x)
exp (0)
1=1
loggo(x) log exp (0) log
Σ (x)(()))
(ogg)((z)
In this scenario. the function directly reflects the negative log pooboabbbyiltyy of the original distribution po(x). This demonstrates that, in the absence of the regularization term, the function naturally converges to the original distribution, consistent with the diffusion process at its starting point.
Thus, the minimization of fx()) for varying ł represents gradual smoothing of the optimization landscape, analogous to the continuation method, where progressively subproblems are solved to reach the final target distribution.
Theorem 2 Proof
Proof. The proximal operator is defined as:
((()) 1 11 xe+1 12
prox (5++1) arg min y
(12)
21
Taking the gradient of the objective function with respect to y and setting it to zero gives:
v fe()) + -y- I 0,
(13)
23

---

which implies:
y rt+1 (V
(14)
The diffusion sampling process updates using:
21 21+1 v f2(411)
(15)
Comparing the two expressions, we see that:
St prox (#t+1);
(16)
where the approximation arises from using (((+++11) in place of (z)), which holds when (a)
is sufficiently smooth. Thus, each diffusion sampling step corresponds approximately to solving the proximal operator.
A.2 Understanding of the index
We analyze the theoretical underpinning our proposed index by relating it to the maximum eigenvalue of the local Hessian of the score function. This analysis reveals that the index is inspired the local curvature, offering insights into the stability and reliability of the score field.
To intuitively understand when the learned score function becomes unstable, we look at how rapidly it changes in space. Let v(x)= := v. log (xx be the score function at time L, pointing in the direction of increasing data likelihood. In well-behaved regions, this score field changes smoothly. But in low-density areas, v(z) can shift direction quickly-these are regions where hallucinations are likely to occur.
Ideally, to capture how sharply (xx) bends, we would examine its second derivative, full Hessian tensor V V2 (z). However, this object is large and expensive to compute. Instead, we focus on the magnitude of the score function,
H([|e(=)II) [
Rapid changes this quantity often indicate instability in the local geometry of the score function. To measure how much this score magnitude could change under small spatial shift 6, we apply first-order Taylor expansion:
V + 6) vz|l((>]|| ~ H([v(z)))) 8.
This gives us a way to approximate how sensitive the model's confidence to perturbations input space. The Hessian norm H can be used to compute the maximum change in the gradient due to the perturbation ò. Especially for lo norm, it corresponds to the maximum eigenvalue λ1 A1()):
Hō
||
max |
max [6|==1
|Ho
max VV || =1
+ 6)]l V [=())ll
|
Finally, we measure the sensitivity of to changes in by computing the norm of the gradient difference:
H(x) max V>]lo(< + 0) V=]lv(>)||I
|+||=1
However, directly solving this maximization over is computationally expensive. Instead, we approximate 8 by the direction:
8= arg max ||
|| + 5]
By restricting the search local neighborhood of radius p. this yields the practical form of Eq. (8)
B Ablation study and Visualization
This section provides additional analysis and visualizations to support the main results. We present ablation studies on key components of our method, including the choice of detection threshold and the impact of critical-step truncation. We further compare different strategies in solving ō, report runtime statisties, and outline implementation details that improve efficiency. Finally, we describe the manual labeling protocol used for hallucination identification and provide additional visualizations.

---

B.1 Dataset and Base Model
We evaluate our method on three benchmarks: AFHQ-v2 (26,, FFHQ [26], and 11k-hands [-] AFHQ-v2 contains high-resonn regh-resolution animal face images actoss three domains (cats, dogs, wild), with improved alignment and quality compared to the original AFHQ. FFHQ includes 70,000 high-quality human face images with rich variation in age, ethnicity, and lighting, making it standard benchmark for face synthesis. The k--hands dataset contains 1 11,076 normal hand images (1600x 1200 pixels) from 190 individuals aged 18-75. Each subject was photographed opening both hands, from dorsal and palmar views, against a white background with consistent distance from the camera.
For the AFHQ and FFHQv2 datasets, we adopt the pre-trained models provided by EDM 225 For the I Ilk-hands dataset, we train a diffusion model from scratch using the EDM framework, with input images resized 128 128 and following standard training configurations.
Throughout our experiments, we utilize the Variance Exploding (VE) version of EDM with 40 sampling steps, which strikes a balance between generation quality and computational efficiency.
Training follows the default hyperparameters and architectural settings provided in the official EDM repository.
B.2 Detection Threshold
In current experiments, observed that the choice of is influenced by both the dataset characteristics and the desired sensitivity level. For instance, 1 11k-hands features clean backgrounds and fine-graine fine-graieed hand details, so a smaller helps detect subtle hallucinations in localized regions. In contrast, face datasets (AFHQv2, FFHQ) often involve more complex backgrounds, where slightly larger is preferred avoid overreacting natural variation in curvature.
We further analyze the impact of the detection threshold used in identifying hallucinations. As illustrated in Figure 6 the threshold controls fundamental trade-off: lower detection threshold increases sensitivity and allows the model to detect more hallucinated samples, but at the cost of falsely flagging some normal images suspicious. In this experiment the FFHQ dataset, we observe that a threshold of 0.09 achieves a true positive rate close to 80%, while keeping the false positive rate below 20%.
In practice the threshold can be adapted based on the application scenario. For example, in high-risk domains such medical imaging, a lower threshold may be preferable to ensure higher recall and minimize the chance of missing harmful hallucinations.
In the following ablation study, we address two key concerns related to false positives and efficiency: (I) Does mistakenly detecting normal image as hallucinated degrade its quality after correction? Our experiments suggest that it does not. (2) Does correcting too many samples incur excessive computational overhead? We design targeted correction strategy mitigate this issue.
B.3 Critical steps
As observed Figures 1 and [4] the final differences between generated outputs often arise from few critical steps during the sampling process. This suggests that whether or not hallucination occurs is frequently determined by a small number of specific updates.
To verify and analyze this observation, we visualize, under fixed settings, the timesteps at which corrections are triggered across an entire batch. As shown in Figure 7 critical seep-i.e.., steps where proposed hallucination index exceeds the threshold for at least one image in the batch- tend to concentrate in the middle stage of the trajectory. This pattern is consistent across tasks and datasets, and aligns well with hypothesis that mid-range steps are most influential in determining sample quality.
B.4 Different Strategies in Solving 5
Solving the inner maximization exactly can be costly, especially in high dimensions. A simple and efficient alternative is to approximate the direction using first-order step. If M is an ball of radius p. then:
25

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ROC for thh
fanden
0.2
Troe Posabre Tutta
Pabse Pouitive Rato
Figure 6: ROC curve for hallucination detection the FFHQ dataset. Each point corresponds different detection threshold. Lower thresholds yield higher true positive rates but also increase false positives. For example, threshold 0.09 achieves 800 TPR under 20% FPR. Thresholds can be adjusted depending on the application's 's sensitivity to hallucination risk.
Figure 7: Detection of critical steps. Each Left: batch of generated samples. Each Right: for each timestep, we mark it as "critical" (value I) if the proposed detection index exceeds the threshold for at least one image in the batch. This highlights which steps are most likely to trigger robust corrections.
SAS::
V (x.) [V (x:))l
&,SAS
Since fx, -Vlogr,, (2,), you may flip the sign depending on the implementation. CAS:
V V (x))
CAAS P
ft, (=+)lII
Again, it's common substitute V (z,)
V log Pt. (I:) here well.
Another natural idea is to solve å using multiple iterations of gradient descent. To investigate this, we compare our default single-step update with 5-step and 10-step gradient descent variants. illustrated in Figure g we observe that for the vast majority of cases, the results across all three strategies are nearly Identical suggesinn that additional optimization steps not significantly alter the output. Specifically, only small fraction of images exhibit perceptible differences between
26

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Table 2: Hallucination analysis on the 11k-hands dataset using various samplers. Our method achieves the highest correction rate without introducing new hallucinations, while also maintaining low inference time.
Method
Hall. %4
Correction %
New Hall. %
Time (s)
EDM-Fuler
19.4%
N/A
N/A
2.71
EDM-Heun
18.0%
17.7%
2.5%
5.38
Euler-200
16.9%
18.9%
1.4%
13.76
RODS-CAS 14.3%
26.3%
0.0%
4.82
single-step and multi-step updates (3 out of 184 on AFHQ, and out of 222 FFHQ), as shown on the right-hand side of the figure.
Given the negligible improvement and substantially increased computational cost, we recommend using the single-step update in practical applications.
Origin
Single-step
5 steps
10 steps
Origin
Single-step
5 steps
10 steps
Figure 8: Comparison between single-step and multi-step gradient updates (5 steps, 10 steps). Left: most cases, all update strategies lead to visually identical results. Right: rare examples where multi-step updates differ from the single-step version (3/184 in AFHQ, 4/222 in FFHQ). Given the marginal benefit and higher computational cost, we recommend the single-step strategy.
B.5 Acceleration Strategy and Running Statisties
Building the previous analysis, observe that most hallucination-indueee steps occur in the middle of the sampling trajectory, and that single-step correction is sufficiently effective. For datasets with a high hallueination rate, apply single-step correction to all samples. To improve inference efficiency, we restrict the application of correction sampling to the middle portion of the trajectory specifically, from 10% to 50% of the total sampling steps. This selective strategy preserves the effectiveness of hallucination correction while reducing the risk of quality degradation from excessive perturbations.
We compare EDM-Euler, EDM-Heun, and our proposed method RODS-CAS For fair comparison, also include EDM-Euler with 200 sampling steps. As shown in Figure 9 I0 and Table 2 our method achieves the highest hallucination correction rate without introducing new artifacts, while maintaining low inference time.
This result also explains why simply reducing the step size insufficient. Even with 200 uniform steps, which is five times smaller than the 40-step baseline, the hallucinations persist. In contrast, RODS-CAS adaptively adjusts its update direction in high-curvature regions and eventually removes artifacts. The key issue lies in directional error within low-density, high-uncertainty areas; smaller steps cannot fix wrong direction, while curvature information helps steer sampling toward stable regions.
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EDM-Euler
EDM-Euler200
EDM-Heun
RODS-CAS
EDM-Euler
EDM- Euleer00000
EDM-Heun
RODS-CAS
Figure 9: Examples where our method successfully reduces hallucinations that are not corrected by stronger solvers such as EDM-Euler (200 steps) and EDM-Heun (40 steps).
Euler-40
Euler-200
Heun-40
CAS
Figure 10: Examples where advanced solvers (Euler-200, Heun-40) introduce hallucinations compared to Euler-40, while our method remains stable.
B.6 Labeling Detail
Since there is no universally agreed-upon definition hallucination in generative models, and no established gold standard for evaluation, resort to manual annotation to assess the presence of hallucinations. Given that the identification of visual hallucinations our setting does not require domain-specific expertise, recruited lab members perform the labeling.
For face hallucinations on AFHQv2 and FFHQ, hallucinations are identified and annotated based on human perception, common sense, and visual plausibility. For hand hallucinations on the 1 1Ik-hands
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dataset, each generated sample is manually assigned to one of the following seven categories: (1) normal, (2) extra fingers, (3) missing fingers. (4) incorrect finger structure, (5) abnormal palm, (6) multiple hands, and (7) unrecognizable or implausible. This taxonomy allows for fine-grained analysis of hallucination types and facilitates more targeted evaluation and correction.
To further improve evaluation precision, we conducted pairwise human comparisons on anonymized image pairs (RODS vs. baseline). Unlike binary hallucination labels, these pairwise judgments capture relative image quality, providing more nuanced measure of generation fidelity. The evaluation rubric is as follows:
image is perceived more faithful or realistic, often due to improved correction of visual anomalies (e.g., fixing a misplaced eye) without introducing new defects.
Better:
The
The SODS-generated image is clearly degraded relative to the baseline, typically because it introduces new artifacts or hallucinated elements that were not present in the original.
Worse:
Same:
No human-identifiable difference exists between the baseline and the RODS- generated image.
Unclear: A change occurs, but it does not involve new hallucinations, and it is difficult to judge which version is better (e.g., slight pose changes missing accessories like ring).
All images were labeled independently by two human raters to ensure objectivity. Whenever their judgments diverged, the pair reviewed the cases together and discussed until they reached consensus, producing single, agreed-upon label for each image. This consensus process both minimized individual bias and improved the overall reliability of our ground- -rruh data. To evaluate RODS- CAS's impact relative to the baseline (EDM-EUL -ullrr, annotators first identified hallucinations in the Euler-sampled outputs. They then compared each corrected image against its Euler counterpart, categorizing the result "better," "worse," "same," or "unclear". This side-by-side comparison made easy to see whether our method truly reduced artifacts, or introduced new ones. Examples of annotated hallucinations can be found in Figure [I (faces) and Figure 12 (hands).
Hallucination
Hellucination
Figure 11: Examples of face hallucinations on AFHQv2 (left) and FFHQ (right). Hallucinations are identified and labeled based on human perception, common sense, and visual preferences.
B.7 More Visualization
Figures 3 and 14 show the generation trajectories for the same batch of samples from the AFHQ and FFHQ datasets under different sampling methods. Several interesting observations can be made. For most well-formed samples, our method produces trajectories and outputs that are visually indistinguishable from the baseline. contrast, for samples that eventually exhibit hallucinations, deviations in the trajectory often emerge during the middle stages of sampling, suggesting that few critical steps play a decisive role. more rigorous analysis of this phenomenon provided in earlier sections.
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(c)
Figure 12: Examples of hand hallucination types produced by Euler-40 sampling. From left to right: (a) Extra fingers, (b) Míssing fingers, (c) Incorrect finger structure (correct number but anatomically implausible), (d) Abnormal palms, (c) Multiple hands, and (f) Other distortions or unrecognizable features.
EDM-Euler
EDM-Heun
RODS-SAS
RODS-CAS
Figure 13: Generation trajectories on the AFHQv2 dataset from fixed random seed and batch (batch size = 9). Each row corresponds to a different method, and columns depict intermediate samples Ia from timestep = T (right) to & = 0 (left). Our method selectively modifies poor-quality generations- as correcting the distorted eyes the cat the top-right corner-while preserving images without hallucinations unchanged.
C Literature Review on Hallucination
C.1 Definition of Hallucination
Hallucination generative AI refers to the phenomenon where model produces content that unfounded, unfaithful, or not grounded in the provided input in reality [30 24). In natural language generation, this often means the model outputs nonsensical or factually incorrect statements that were never in the source material 41 [4]. The term has been extended to visual and multimodal
Al: for example, image captioning or language tasks, it denotes instances where the model "hallucinates" objects details that are not actually present in the input image 42]
Hallucination broadly recognized critical problem for trustworthy and safe AI. It undermines users' trust in Al systems and poses serious risks in real-world applications. Models that hallucinate make content unreliable and unpredictable, which is especially problematic high-stakes domains In the healthcare context, the consequences can be life-threatening: for instance, medical system or summarization model that fabricates nonexistent symptom or an incorrect medication instruction could mislead clinicians patients (24)
24 28
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EDM-Euler
EDM-Heun
RODS-SAS
RODS-CAS
Figure 14: Generation trajectories on the FFHQ dataset from a fixed random seed and batch (batch different method, and columns depict intermediate samples Ty from timestep = T (right) to = (left). Our method selectively refines images containing hallucinations such as unnatural artifacts near the mouth, or unrealistic eyebrows and hair-while leaving images without hallucinations unchanged.
size 9). Each row corresponds
C.2 Hallucination in Image Generation
In the context of image generation, a hallucination typically means the model has created visual artifacts or implausible objects in generated images. For example, diffusion models like DALL-E or Stable Diffusion sometimes produce people with anatomically impossible features (such as extra fingers or distorted limbs) or blend background elements in incoherent ways 2 technical standpoint, such hallucinations correspond to the model sampling from outside the true distribution of the training data. other words, the generator is creating images that look superficially plausible but contain details that no real image would have, hence indicating the model has generalized in an incorrect, unrestricted manner.
38 g. From
In scientific and medical domains, the stakes are extremely high because a hallucinated visual detail lead false scientific conclusions diagnostic errors. In medical imaging, diffusion models trained primarily on healthy data may hallucinate away tumors or introduce fictitious lesions, risking dangerous misdiagnoses, as highlighted by [27] denoted structural hallucination. scientific domains such as satellite imagery, models may fabricate false visual features- e.g.. showing floodwater in dry regions-potentiall misleading researchers and policymakers (..) In both cases, hallucinated details not only degrade visual fidelity but also threaten the integrity and safety of downstream decisions, underscoring the urgent need mitigate hallucinations in high-stakes applications (8)
C.3 Reduce Hallucination in Diffusion Model
Although hallucination is not well-defined inherently difficult- remains central concern for deploying generative in high-stakes domains. Recent research has thus focused on developing techniques to address hallucinations during the generation process, particularly in diffusion models, which now dominate image synthesis tasks. Several promising directions have emerged:
its detection and suppression
Trajectory-basse consistency control. Aithal et al. [2] observe that hallucinated samples often exhibit high prediction variance in the reverse diffusion trajectory, signaling off-manifold behavior. They propose simple but effective variance-based metric filter out such aberrant samples. Huang et al. (22) regularizes the sampling trajectory by constraining the norm of the predicted noise to remain aligned with the score direction, ensuring update validity within high-confidence regions.
Local diffusion with OOD region isolation. Kim et al. 27 tackle structural hallucinations in conditional diffusion by partitioning inputs into in-distribution (IND) and out-of-distribution (OOD) regions. An anomaly detector produces a probabilistic OOD mask, and two separate diffusion branches process IND and OOD content, respectively, followed by fusion step.
Early stopping of diffusion decoding. Tivnan et al. [51 propose truncating the sampling process before full convergence to avoid overfitting to noise and hallucinating unnecessary details. Their
31

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results show that early stopping improves structural fidelity without sacrificing perceptual quality. They further introduce a "Hallucination Index" to quantitatively evaluate model faithfulness.
Attention Modulation Adjustment. Oorloff et al. [39 propose Adaptive Attention Modulation (AAM) to reduce hallucinations in diffusion models by dynamically adjusting self-attention behavior during inference. They apply masked perturbations to disrupt early-stage noise that may propagate into hallucinations. AAM directly modulates the internal attention layers, providing a lightweight way to suppress hallucinated artifacts while preserving image quality.
Structure-specific fine-tuning, Several works refine diffusion models in specific hallucination- prone regions. For hand generation, HandRefiner [32] propose to use fine- thee-gained annotations and localized losses to correct anatomical distortions. DreamBooth-sty fine-tuning [4 enable localized hallucination correction with minimal retraining.
In summary, these complementary strategies contribute to growing toolkit for hallucination miti- filtering enables lightweight real-time rejection, local
gation in diffusion models. Trajectory- diffuusin separates difficult regions for targeted reconstruction, early stopping avoids overgeneration, and pruuctue-sppccfii finetuning enables precise corrections in vulnerable subregions. However, hallucination detection and mitigation remain open problem: different methods target different manifestations of hallucination, and the lack of a unified definition or standardized benchmark continues to pose challenges for systematic evaluation and comparison.
D Literature Review on Bridging Optimization and Diffusion Models
This section reviews emerging perspectives that connect diffusion models with optimization. Prior work relates score-based sampling to stochastic gradient methods, uses diffusion as an optimizer in planning and inverse problems, and explores alternative views like Schrödinger bridges. Our approach builds on these by framing diffusion as a continuation method, offering new insights and opportunities for algorithmic improvement.
D.1 Score-Matching Langevin Dynamics Stochastic Gradient Descent
Early work on -aore-Based Generative Models explicitly drew parallels to simulated annealing, introducing annealed Langevin dynamics that uses large noise (high "temperature") initially and decreases it step by step [49]. At fixed noise level, Score-Matching Langevin Dynamics (SMLD), each step essentially a stochastic gradient ascent (SGD) on the log-density of an intermediate distribution (using the learned score V, log (x))) with added Gaussian noise for exploration. This precisely Langevin Monte Carlo, which mirrors gradient descent but injected with noise to sample rather than converge to single mode.
In fact, Langevin sampling can be seen as a close cousin of stochastic gradient descent in optimization: the only difference is the noise term that prevents collapse to mode. Recent research has made this connection explicit. [52] point out the "high connection" between diffusion's 's sampling process and SGD, and import SGD techniques into diffusion. They introduce an adaptive momentum sampler, analogous adding a heavy-ball momentum term to Langevin dynamics.
Similarly, other works have looked at second-order optimization ideas in diffusion: e.g. precondi- tioning the Langevin updates using curvature information. (50) derives an optimal preconditioner for Langevin MCMC based on the Fisher information matrix, effectively a Newton-like method that adapts step sizes the geometry of the target distribution. Such curvature-aware updates amount to using an approximate Hessian to accelerate sampling, paralleling Newton's method in optimization.
D.2 Solving Optimization as Diffusion Sampling
Diffusion models are deeply connected energy- based generative modeling (EBM), and this link further cements the optimization interpretation. An EBM defines an energy function E(x) (or unnormalized log-density); sampling from it typically involves running MCMC (e.g. Langevin
dynamics) essentially gradient descent on E(x) with noise. Score-based diffusion models, instead of specifying single energy, learn a time-dependent score function so (0(x, 1) V, log Pe (x)) for each noise level [47..
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Beyond improving diffusion itself with optimization techniques, a complementary line of research uses diffusion models as optimization solvers for the energy-based objective function or noised objective function. 23 reframes offline reinforcement learning as diffusion-based trajectory generation problem. Diffuser is denoising diffusion probabílistic model that plans by iteratively refining noise into coherent trajectory.
In essence, the diffusion model acts an optimizer in the space of trajectories, finding plans that both resemble the training data distribution and achieve the desired goal (via guidance). Follow-up works have expanded on this concept. For example, [2] decomposes the diffusion planning into two stages: (1) quickly generating feasible rough trajectory (using an autoregressive policy model), and (2) diffusion- based trajectory optimization to refine it for higher quality. (53] introduce safety or constraints into diffusion planners. [35] further advances the idea by leveraging value functions as energy surrogates, enabling efficient trajectory optimization without relying on sampling from the optimal policy.
Overall, in the decision- makinn domain, diffusion models powerful framework solve optimiza- tion problems over sequences of actions by sampling from a progressively refined distribution. The stochastic, continuation-style generation (from noise solution) helps avoid local optima in planning, much like diffusion sampling process.
D.3 Diffusion sampling as continuation method
Our work aligns more closely with the first category but distinctly frames diffusion sampling continuation or homotopy method from numerical optimization. Unlike conventional interpretations based on fixe-noise-leve SMLD, the continuation (homotopy) view of diffusion sampling treats the generation as solving a sequence of gradually changing optimization problems. At = T (maximum noise), the "objective" is trivial (the target distribution is pure Gaussian noise, which the sampler can produce easily). As / decreases, the implicit objective continuously deforms, introducing more data structure; by t 0, it becomes the true data distribution.
Viewing diffusion sampling through the lens of continuation methods not only provides conceptual clarity, but also opens up promising directions for future algorithmic improvements. By treating each diffusion step as a progressively refined subproblem, this perspective aligns with a wide range of techniques in numerical optimization. For example, motivates adaptive step size scheduling, trust- region-basse stability control, and convergence and error analysis. Our proposed robust optimization formulation inspired by this perspective, while other aspects are left for future exploration.
D.4 Other related point of view
Beyond the continuation and optimization viewpoints discussed above, several other perspectives have also been proposed to connect diffusion models with optimization.
One such perspective is the Schrödinger bridge formulation, which seeks the most likely stochastic process (via a controlled drift) that transforms prior distribution into the data distribution over time. This leads to an entropy-regularizzed optimal transport problem, solvable via iterative projections- form of optimization in probability space. Several works reinterpret score-based diffusion Schrödinger bridge problem, thereby establishing a formal connection between diffusion models and stochastic control theory [15].
Another active research direction explores the use of diffusion models for solving inverse problems in imaging and related domains. Diffusion models serve as powerful learned priors, offering structured way to sample from complex data distributions. Recent methods adopt a plug-and-plas approach, where diffusion-basee generation is alternated with steps that enforce data fidelity. For example, Chung et al. 12 propose Diffusion Posterior Sampling (DPS) for general noisy inverse problems, which combines denoising steps with gradient-based corrections based measurement likelihood. At each diffusion timestep, the sample is not only denoised but also guided toward satisfying the observed e.g., using gradients of the log-likelihood V, log p(y -r-rrsemblinng projection onto the data-consistent manifold.
33


Reviews:
[{"review_id": "6xrH6NRy7y", "reviewer": "Reviewer_GnZW", "summary": "The paper proposes RODS, a novel plug-and-play method for mitigating hallucinations in diffusion models, which is an under-explored field of research. The authors interpret diffusion sampling as a continuation method in numerical optimization and integrate robust optimization principles to dynamically detect and correct \"high-risk/unstable\" steps in the sampling process, thereby alleviating hallucinations. RODS considers curvature changes as a proxy for detecting hallucinations, and proposes two updating strategies (SAS and CAS). The authors evaluate RODS on AFHQv2, FFHQ, and Hands datasets, demonstrating promising results for hallucination mitigation.", "questions": "- Is it possible to make \\epsilon not specific to a particular dataset? or to derive epsilon in a data-independent manner? This would help the generalization of the proposed algorithm, especially in cases of high diversity datasets\n- Please include the details of the survey and the qualitative metrics used\n- Discuss the limitations in detail along with the side-effects of corrections such as semantic drift/ over-sanitization\n- As pointed out in the weaknesses and suggestions section, I believe the paper is lacking an evaluation/analysis on the potential trade-off between generative capability and hallucination mitigation.", "limitations": "No", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n- The interpretation of the diffusion process as a continuation method, followed by the adaptation of robust optimization as a mitigation strategy is an elegant and theoretical sound approach. \n- The paper addresses an underexplored research avenue and demonstrates strong hallucination detection rates and mitigation \n- The proposed work does not require retraining (plug-and-play) with a relatively lower overhead in inference time\n- Includes a detailed set of ablations\n- a clear and well-written paper where the proposed algorithm is clearly motivated\n\n**Weaknesses and Suggestions**\n- The proposed method may suppress the generative ability of a diffusion model similar to mode collapse. While this would be acceptable to a certain degree given the hallucinations are mitigated, I believe an analysis of potential trade-offs between hallucination and generation diversity is required. \n- While the use of the sharp curvature as a proxy is justifiable, I would suggest the authors discuss the underlying assumptions behind it as there could be naturally sharp but \"valid\" features\n- The authors use the qualitative analysis results as one of their main metrics but do not present details of the survey. This would be needed in making sure the survey best represents the general community and reproducibility. Further, the use of hand-wavy ratings of terms like  \"better\", \"worse\", \"same\", and \"unclear\" needs to be explicitly defined (i.e., what qualifies for each category)\n- The correction-induced changes by RODS may be unintended, where in certain cases a semantic drift and new subtle hallucinations are introduced by the correction process. For example in Fig. 9: \n    - Row 1 left: Pose and complexion change.\n    - Row 1 right: While the extra finger is removed, the pinky becomes implausibly long.\n    - Row 3 left: Floating hand is not resolved. \n    - Row 3 right: Newly introduced finger has inconsistent tone.\n    - Row 4 right: Missing sleeve and changed hand orientation.\n- The authors do not discuss on limitations, any architectural dependence, applicability to high-diveristy datasets where the notion of hallucination is itself ill-defined. \n- The following additional related work appears to be in the same domain of mitigating hallucinations in unconditional diffusion models: *Oorloff et al.  \"Mitigating Hallucinations in Diffusion Models through Adaptive Attention Modulation\"*", "was_revised": true, "final_justification": "The paper proposes a grounded solution towards mitigating hallucinations in diffusion models, which is an under-explored area. The authors were able to coherently address most of the weaknesses and concerns raised while proposing edits to incorporate them in their final revision. Contingent on the authors including the suggested edits which makes the paper much stronger, I vote towards accepting this work --- raising my score to Accept from Borderline Accept.", "rebuttal": "### **W1 & Q4: Trade‑off between hallucination mitigation and generation diversity**\nWe fully agree that a mitigation technique should *not* come at the cost of collapsing the model’s creative space. Intuitively, **RODS touches only a small subset of trajectories, so most images remain exactly the same**. Visual evidence of this can be found in Appendix B.7 (Figures 13 and 14), while standard diversity metrics further confirm this observation. \n\nBelow we quantify and explain why RODS does *not* suppress diversity. In particular, we assess diversity across 1k samples using: **Inception Variance** [1], which quantifies variability in the latent representations, and **DreamSim Diversity** [2], which measures the variance of features extracted from a batch of generated images (batch size = 64). The experiment results show that RODS-CAS preserves generation diversity on par with the EDM-Euler method while effectively mitigating hallucinations, as detailed in the paper.\n\n| Dataset| FFHQ |   | AFHQ2    |  | 11k-hands | | \n|----------|---------------------|-------------------|----------------------|-------------------|----------------------|-------------------|\n|    | Inception-Var ↑     | DreamSim ↑        | Inception-Var ↑      | DreamSim ↑        |Inception-Var ↑      | DreamSim ↑        |\n| EDM-Euler    | 0.0669              | 0.407             | 0.0539               | 0.486             | 0.0327                    | 0.1427               |\n| EDM-Heun      | 0.0699          | 0.422         | 0.0548               | 0.488         | 0.0332                    | 0.1476               |\n| RODS-CAS | 0.0661              | 0.400             | 0.0542           | 0.481             | 0.0318                    | 0.1509               |\n\n\n[1] Miao, Zichen, et al. \"Training diffusion models towards diverse image generation with reinforcement learning.\" CVPR. 2024.\n[2] Domingo-Enrich, Carles, et al. \"Adjoint matching: Fine-tuning flow and diffusion generative models with memoryless stochastic optimal control.\" arXiv preprint 2024.\n\n---\n\n### **W2: Clarifying the assumptions behind the curvature proxy**\n\nWe appreciate the reviewer’s comment on discussing the assumptions underlying our curvature-based detection. The key hypothesis is that **hallucination correlates with model uncertainty** [1]. As mentioned in our paper, \"in well-behaved regions, this vector field changes smoothly. But in low-density areas, $v(x)$ can shift direction quickly—these are the regions where hallucinations are likely to occur\" (line 467-471). By probing these rapid, spatial fluctuations, i.e., second‑order changes or “curvature”, we obtain a quantitative signal (our curvature index $H(x)$) for potential sampling failures. \n\nIn terms of **the naturally sharp but valid features,** such structures (e.g., object edges) are **unrelated** to the curvature index $H(x)$ used in our method.  While such sharpness in image exhibits high **image gradients** $\\nabla_x I(x)$, this differs fundamentally from the **density-field gradient** $v(x) = \\nabla_x \\log p_t(x)$, which reflects changes in data density.  Our curvature index $H(x)$ measures variations in this score field—not in image intensity.\n\n[1] Sumukh K. Aithal, et al. *Understanding hallucinations in diffusion models through mode interpolation.* Neurips 2024.\n\n---\n\n### **W3 & Q2: Detailing our qualitative survey and rating rubric**\nWe thank the reviewer for pointing this out.\n\n**For hallucination labeling (Appendix B.6) :** \n\n-> *Face images (AFHQv2, FFHQ)* hallucinations are identified and annotated based on human perception, common sense, and visual plausibility. *Hand images (11k-hands)* are each assigned to one of seven categories: normal; extra fingers; missing fingers; incorrect finger structure; abnormal palm; multiple hands; unrecognizable/implausible. \n\n**Current use of “better”/“worse”/\"same\", and \"unclear\".**  \n\nTo make our evaluation protocol more precise, we conducted pairwise comparisons by **human raters on anonymized image pairs** (RODS vs. baseline). Unlike traditional binary hallucination labels, these judgments reflect relative image quality rather than the mere presence or absence of hallucinations. As a result, it provides a more nuanced and reliable assessment of generation fidelity. **_Furthermore, in the interest of transparency and reproducibility we will share our code, data, and labels in Github._** More specifically, the detailed rubric is shown below:\n\n- **Better**: The RODS-generated image is perceived as more faithful or realistic, often due to improved correction of visual anomalies (e.g., fixing a misplaced eye) without introducing new defects.\n- **Worse**: The RODS-generated image is clearly degraded relative to the baseline, typically because it introduces new artifacts or hallucinated elements that were not present in the original.\n- **Same:** No human-identifiable difference exists between the baseline and the RODS-generated image.  \n- **Unclear:** A change occurs, but it does not involve new hallucinations, and it is difficult to judge which version is better (e.g., slight pose changes or missing accessories like a ring).  \n\n---\n\n### **W4 & Q3: Side-effects of corrections**\n\nWe really appreciate the reviewer’s careful analysis and thoughtful comments—these are valuable and nuanced issues that touch on the core difficulty of hallucination correction. Due to the inherently ambiguous definition of hallucination, our correction strategy is not able to make fully “intended” edits. Rather, RODS introduces conservative updates at points identified as high-risk under the base model’s score landscape. \n\n1. **Imperfect correction**: The effectiveness of correction is fundamentally limited by the base model itself. For instance, in cases like Row 3 (right), where the baseline model tends to denoise the fifth finger into blank space, our robust update may reintroduce it in a subtle but imperfect form (e.g., slightly discolored or ill-shaped), as we can only perturb within the model’s generative prior.\n\n2. **Semantic drift**: On one hand, semantic drift is less relevant in our experiments, as there is no external prompt or class label whose meaning could shift. On the other hand, different diffusion timesteps correspond to different levels of abstraction. Among changes, edits made at early stages (high t) can result in semantic drift, while those at later stages (low t) typically affect finer details. Our current method does not constrain which timesteps are eligible for correction, as we found that some large-scale hallucinations require intervention at earlier stages. However, such early-stage perturbations may also affect high-level semantics, such as altering the orientation of a hand, as seen in Row 4 (right). If semantic consistency is a priority, one could optionally restrict updates to later timesteps to avoid undesired high-level changes.\n\n3. **Over-sanitization**: Quantitative diversity remains almost the same: as shown in W1&Q4. Additionally, since most samples are left bit-wise unchanged (Fig 13-14), the model’s creative range is effectively preserved, addressing the over-sanitization concern. We will post all 1k results and the human label for public review.\n\n\nWe acknowledge that achieving more targeted and controllable corrections would require more precise hallucination localization and timestep-aware update scheduling. We view this as a promising direction for future work.\n\n---\n\n### **W5: More discussion on limitation**\n\nThank you for highlighting these important considerations. Here is how we will address them in the revised manuscript:\n\n1. **Architectural dependence (VE formulation)**  \nOur RODS algorithm can be directly implemented within the VE-ODE framework, which benefits from the equivalence between VE-based sampling and the continuation method under the usual parameterization (Section 3). We will clarify this dependency and note that extending RODS to other sampling schemes (e.g., VP or latent-space diffusion) is conceptually straightforward but remains to be validated experimentally.\n\n2. **Dataset scope limitations**  \nCurrently, we have tested RODS on well-established benchmarks (AFHQv2, FFHQ, 11k-hands), where hallucinations can be easily observed without domain expertise. We acknowledge that in highly diverse or specialized domains, i.e., medical imaging, the definition of “hallucination” must be grounded in expert judgment. We are collaborating with domain practitioners to evaluate RODS under these conditions.\n\n---\n\n### **W6: Relevant work suggestion**\n\nThank you for pointing out this relevant work. While it takes a different approach (adaptive attention modulation) compared to our trajectory-level geometric regularization, we agree that it addresses a related goal. As it was posted on arXiv in February 2025 and had no available code at the time of our submission, a direct experimental comparison was not feasible. We will gladly include a citation and discuss its relation to our work in the revised section.\n\n---\n\n\n### **Q1: Selection of epsilon**\n\nIn our current experiments, we observed that the choice of $\\epsilon$ is influenced by both the **dataset characteristics** and the **desired sensitivity level**. For instance, 11k-hands features clean backgrounds and fine-grained hand details, so a smaller $\\epsilon$ helps detect subtle hallucinations in localized regions. In contrast, face datasets (AFHQv2, FFHQ) often involve more complex backgrounds, where a slightly larger $\\epsilon$ is preferred to avoid overreacting to natural variation in curvature. More discussion about the epsilon selection has been presented in Section 4.1 (line 205-207) and Appendix B.2. Given a desired sensitivity level, we believe it is possible to automatically choose $\\epsilon$ (there are similar approaches in traditional literature), but we consider it as a future research direction."}, {"review_id": "kUu14BBj5k", "reviewer": "Reviewer_jSsH", "summary": "This paper tackles the hallucination problem in diffusion models by viewing diffusion sampling as a optimization procedure, which frames each sampling steps as a proximal sub-problem on a gradually sharpening potential landscape. \nBased on this, the authors introduce Robust Optimization-inspired Diffusion Sampler, RODS, a novel curvature-based sampler detects sudden changes in score vector field, which can auto-adjust sampling directions and thus improve robustness and remove hallucination. \nImportantly, RODS does not introduce any modification to the pretrained diffusion. \nExperiments on AFHQ-v2 (cats/dogs/wild), FFHQ (faces) and 11k-hands show that RODS-CAS detects ≥ 70 % of annotated hallucinations and corrects > 25 % of faulty samples.", "questions": "- Why using curvature metric to detect hallucination?\nI don't see why fluctuation in score magnitude can infer hallucination. Is it possible that in some region, score magnitude can be a false alarm? Maybe some region of the image has this type of score behavior. \n- Can you show some image generation quality evaluations? For example, FID, NLL. Just want to see that this sampler doesn't tread hallucination to generation quality. \n- How you make sure that the human labeled data is trustable? \n- For hyperparameter epsilon and rho, can you provide a principled way to choose?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n- Paper is well structured and easy to follow, also algorithms and theorems / proofs are self-contained: \nNice derivations bridge Euler updates to the proximal optimization, and novel link between robust optimization and diffusion sampling. Unlike the previous prior temporal-variance filters, the Curvature-driven risk detector focus more on the shape of the score landscape, \n- Comprehensive experiments across three datasets with manual hallucination annotation. \n- Nearly no extra cost: since the method is plug-and-play, there is no change to the model. \n\nWeaknesses: \n- Although promising experimental results, the authors only provide hallucination related metrics. I think it will be more reasonable to also show some image quality evaluations, e.g. NLL, FID -- just want to make sure that while solving the hallucination issues, the image quality and density estimate are also guaranteed to be good. \n- It's nice to use curvature index to reflect fluctuation in score magnitude, however, making connection between curvature and hallucinate is not rigorously illustrated. To me this fluctuation in score magnitude just tells me how rapid the score changes in this area, and maybe I should adjust my step size to be smaller to avoid introducing more error. These may be just correlation not causality.", "was_revised": true, "final_justification": "My main questions and concerns are answered. Therefore I recommend to accept the paper.", "rebuttal": "### **W1&Q2: Evaluation on image quality**\nThank you for encouraging a broader quality assessment. We have therefore computed standard image-quality and diversity metrics on 1,000 unconditional samples from FFHQ, AFHQv2, and 11k-hands. The table below reports FID, Inception Variance [1] (which quantifies variability in the latent representations), and DreamSim Diversity [2] (which measures the variance of features extracted from a batch of generated images, and our batch size is 64). Because RODS adjusts only the sampling trajectory and never alters the trained score network, its negative log-likelihood (NLL) is identical to that of the baseline sampler and is omitted for brevity. \n\n| Dataset  | FFHQ                |        |         | AFHQ2               |        |         | 11k-hands        |        |         |\n|----------|---------------------|--------|---------|----------------------|--------|---------|-------------------------|--------|---------|\n|          | FID ↓               | Inception-Var ↑ | DreamSim ↑ | FID ↓               | Inception-Var ↑ | DreamSim ↑ | FID ↓               | Inception-Var ↑ | DreamSim ↑ |\n| EDM-Euler    | 20.48              | 0.0669 | 0.407   | 13.36              | 0.0539 | 0.486   | 14.57          | 0.0327                    | 0.1427| \n| EDM-Heun      | 19.56          | 0.0699 | 0.422 | 12.66          | 0.0548 | 0.488 | 14.20          | 0.0332                    | 0.1476               |\n| RODS-CAS      | 22.73              | 0.0661 | 0.400   | 16.08              | 0.0542 | 0.481   | 16.68          | 0.0318                    | 0.1509               |\n\nThese results show that RODS-CAS preserves diversity while keeping FID comparable. The modest FID rise is largely explained by background smoothing: our curvature test sometimes removes ambiguous, high-frequency textures in the complex backgrounds, which the FID metric interprets as loss of detail. However, such effect could be analogous to portrait-mode photography, where a deliberately blurred background can worse a texture-based metric (FID) without reducing perceived quality. In our generation,  the main subjects remain sharp and artifact-free, which is crucial as it is exactly the goal of our correction. **_This is further supported by the 11k-hands experiment, where our method improves hand confidence (HAND-CONF) from 0.9216 (EDM-Euler) and 0.9227 (EDM-Heun) to 0.9347 (RODS-CAS)._**  HAND-CONF is computed by running a pre-trained hand detector [3] and calculating the confidence scores of the detections. This metric is widely used to assess the anatomical plausibility of generated hands. Because hands are structurally well-defined and challenging to synthesize, this benchmark provides a sensitive and interpretable proxy for visual realism. Overall, **_RODS-CAS delivers equally diverse, comparably high-quality unconditional samples while reducing hallucinations._**\n\nAdditional conditional metrics such as CLIP-score are not applicable here, as we work in the unconditional setting. We will incorporate this table and the accompanying discussion into the revised manuscript to present a more complete picture of image quality alongside hallucination reduction.\n\n\n[1] Miao, Zichen, et al. \"Training diffusion models towards diverse image generation with reinforcement learning.\" CVPR. 2024.\n[2] Domingo-Enrich, Carles, et al. \"Adjoint matching: Fine-tuning flow and diffusion generative models with memoryless stochastic optimal control.\" arXiv preprint 2024.\n[3] Zhang, Fan, et al. \"Mediapipe hands: On-device real-time hand tracking.\" arXiv preprint 2020.\n\n---\n\n### **W2&Q1: Connection between curvature, uncertainty, and hallucination**\nThank you for asking us to clarify the connection between curvature, uncertainty, and hallucination. The key hypothesis is that **hallucination correlates with model uncertainty** [1], which can be quantitatively captured by the **“rapid fluctuations (curvature)”** of the density field. Experimentally, we showed that the hallucination is not caused by various step size (Figures 9–10), but rather a directional error within low-density, high-uncertainty regions: merely changing the step-size of the sampler does not help if the direction itself is unreliable.\n\n\n**Curvature is a proxy for model uncertainty**\n\n- **Score field.**  We denote the learned score by $v(x) = \\nabla_x \\log P_t(x)$. \n- **Curvature index.**  As stated in §4.1 (Eq. 8), we define our curvature index is:  \n\n\n$$\nH(x) = \\|\\nabla_x \\|v(x + \\delta)\\| - \\nabla_x \\|v(x)\\|\\|, \n\\quad \n\\delta = \\arg \\max_{\\|\\delta\\| = \\rho} \\|v(x + \\delta)\\|,\n$$  \n\n\nBecause $H(x)$ compares the gradient of $\\|v\\|$ at two nearby points, it measures **second-order variation** of the density field, i.e., how smoothly the score field changes in space. This reflects the key idea that **hallucination correlates with model uncertainty** [1].  By probing these rapid, spatial fluctuations, i.e., second‑order changes or “curvature”, we obtain a quantitative signal for potential sampling failures. In Appendix A.2, we further describe the conceptual foundation in detail:  \n\n> “In well-behaved regions, this vector field changes smoothly. But in low-density areas, $v(x)$ can shift direction quickly—these are the regions where hallucinations are likely to occur.” (line 467–471)\n\n> “Rapid changes in this quantity often indicate instability in the local geometry of the score function.” (line 475–476)\n\n\n**Why “just shrink the step” is insufficient**  \n\nIntuitively, shrinking the step size is a natural approach. We evaluated a naive fix, using a much smaller step size by running an EDM-Euler sampler with 200 uniform steps (Figures 9–10). Although each step is five times shorter than the 40-step baseline, hallucinations still appear. In contrast, in tons of our experiments, we often observed that RODS-CAS actively changes the update direction whenever the curvature index is high and succeeds in removing most of these artifacts. This comparison shows that the problem lies in directional error within low-density, high-uncertainty regions: merely changing the step-size of the sampler does not help if the direction itself is unreliable. Curvature information is therefore essential for steering the trajectory away from such regions.\n\n[1] Sumukh K. Aithal, Pratyush Maini, Zachary Lipton, and J. Zico Kolter. *Understanding hallucinations in diffusion models through mode interpolation.* Neurips 2024.\n\n\n---\n\n### **Q3: Human Evaluation Trustworthiness**\n\nAs shown in Appendix B.6, our rating rubric is detailed. \n\n> *Face images (AFHQv2, FFHQ)* hallucinations are identified and annotated based on human perception, common sense, and visual plausibility. *Hand images (11k-hands)* are each assigned to one of seven categories: normal; extra fingers; missing fingers; incorrect finger structure; abnormal palm; multiple hands; unrecognizable/implausible.  \n\nAll images were labeled independently by two human raters to ensure objectivity. Whenever their judgments diverged, the pair reviewed the cases together and discussed until they reached a consensus, producing a single, agreed-upon label for each image. This consensus process both minimized individual bias and improved the overall reliability of our ground-truth data.\n\nTo evaluate RODS-CAS’s impact relative to the baseline (EDM-Euler), annotators first identified hallucinations in the Euler-sampled outputs. They then compared each corrected image against its Euler counterpart, categorizing the result as “better,” “worse,” “same,\" or \"unclear\". This side-by-side comparison made it easy to see whether our method truly reduced artifacts, or introduced new ones. We will further update our Appendix B.6 with more details.\n\nFinally, in the interest of transparency and reproducibility, we will make our complete annotation guidelines, raw labels, and final consensus labels publicly available. This will allow other researchers to verify our results and build on our work with full confidence in the human-evaluation process.\n\n---\n\n### **Q4: Selection of Hyperparameter**\n\nThank you for the question. Our paper has provided simple, data-driven principles for both hyper-parameters, detailed as follows. We will study the automated hyperparameter selection in the future. \n\n\n**$\\epsilon$ — detection threshold (Section 4.1, Appendix B.2)**\n\n* **Role:** controls sensitivity; lower ε is more sensitive to the score changing, i.e., uncertain steps.  \n* **Guideline (line 205–207):** “In high-stakes domains such as medical imaging, a lower $\\epsilon$ may be preferred to prioritise sensitivity.”\n\n**$\\rho$ — perturbation radius (Section 4.1)**\n\n* **Role:** sets the neighbourhood size over which we probe the score field.  \n* **Choice:** match $\\rho$ to the characteristic feature scale of the dataset. If the surrounding region is trustworthy (e.g., low visual complexity), a larger $\\rho$ is safe; if not, a smaller $\\rho$ focuses detection on truly uncertain zones."}, {"review_id": "eq09M5mqaj", "reviewer": "Reviewer_ZPMe", "summary": "The paper views the reverse diffusion process as a continuation scheme, where each timestep solves an increasingly ill-conditioned optimization subproblem. Building on this perspective, the authors integrate a curvature-based detector into standard samplers to identify regions where the score may be unreliable, and apply a worst-case-aware correction to keep the trajectory stable, without requiring retraining. This approach reduces hallucinations, as validated by experimental results.", "questions": "1. Is there any theoretical justification for why sharp curvature changes indicate high-risk regions during sampling?\n2. Is there any analysis supporting the idea that minimizing the worst-case outcome (as in RODS) can give overall more robust sampling? Under what scenarios is this true?\n3. Just to confirm my understanding: is the core idea behind Algorithm 2 to minimize the objective shown after line 181, when $\\mathcal{F}_t$ is in the form of the expression after line 219?\n4. The experiment results focus on hallucination. Is there any evaluation of overall generation quality, e.g., FID or similar metrics, to ensure RODS doesn’t degrade sample quality?", "limitations": "yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "### Strengths\n1. The paper introduces a novel mathematical perspective on diffusion sampling by framing it as a continuation method.\n2. The work is fairly complete: combining theoretical insight with a practical algorithm that is both lightweight and easy to integrate. The proposed method demonstrates meaningful empirical improvements across several datasets.\n\n### Weaknesses\nTo me, the main value of the paper is in the theoretical viewpoint, while the experiments mostly serve to support and illustrate that idea. I do wish the paper leaned a bit more into the theory: right now, the mathematical part is quite straightforward and mainly used to justify the optimization interpretation. But it doesn’t really explain why or when RODS should reduce hallucinations. It would be more convincing if there were some analysis showing, for example, that under certain conditions on the score function or the data distribution, RODS would provably outperform standard samplers. That kind of result would help bridge the gap between the conceptual contribution and the algorithmic claims.", "was_revised": true, "final_justification": "I thank the authors for their detailed response that addressed my questions and concerns. I would like to increase my recommendation score to 4.", "rebuttal": "### **W: Explain why / when RODS reduce hallucinations**\nThank you for pointing this out. Below we summarize the key take‑aways and point to the detailed explanations we have now added (see the two responses to the questions below).\n\n1) Sharp local curvature ⇒ high score uncertainty ⇒ higher hallucination risk (**Q1**)\n\n2) Our look‑ahead worst‑case update provably lowers the maximum one‑step error in exactly those high‑risk zones while (i) score locally Lipschitz, (ii) score error bounded by $\\rho$, (iii) step stays inside the ball. (**Q2**)\n\n---\n\n### **Q1: Connection between Curvature, Uncertainty, and Hallucination**\n\nThank you for asking us to clarify the connection between curvature, uncertainty, and hallucination. The key idea is that **hallucination correlates with model uncertainty** [1], which can be quantitatively captured by the **“rapid fluctuations (curvature)”** of the density field. \n\n\n**Curvature is a proxy for model uncertainty**\n\n- **Score field.**  We denote the learned score by $v(x) = \\nabla_x \\log P_t(x)$. \n- **Curvature index.**  As stated in §4.1 (Eq. 8), we define our curvature index is:  \n\n$$\nH(x) = \\|\\nabla_x \\|v(x + \\delta)\\| - \\nabla_x \\|v(x)\\|\\|, \n\\quad \n\\delta = \\arg \\max_{\\|\\delta\\| = \\rho} \\|v(x + \\delta)\\|,\n$$  \n\nBecause $H(x)$ compares the gradient of $\\|v\\|$ at two nearby points, it measures **second-order variation** of the density field, i.e., how smoothly the score field changes in space. This reflects the key idea that **hallucination correlates with model uncertainty** [1].  By probing these rapid, spatial fluctuations, i.e., second‑order changes or “curvature”, we obtain a quantitative signal for potential sampling failures. In Appendix A.2, we further describe the conceptual foundation in detail:  \n\n> “In well-behaved regions, this vector field changes smoothly. But in low-density areas, $v(x)$ can shift direction quickly—these are the regions where hallucinations are likely to occur.” (line 467–471)\n\n> “Rapid changes in this quantity often indicate instability in the local geometry of the score function.” (line 475–476)\n\n\n---\n\n### **Q2: Connection between optimization against the worst-case scenario and robustness**\n\nThis follows a common principle in robust optimization: by guarding against the **worst-case scenario**, we ensure **robust and reliable performance** across all admissible uncertainties. If loss can be reduced even under the most adverse local perturbation, then all other nearby cases are guaranteed to be no worse. This approach is valid when the score function is locally smooth, the uncertainty is bounded, and the step size remains small. Below, we provide a detailed analysis to explain why it enhances robustness and under what conditions the guarantee holds.\n\n1. **What the algorithm actually does**\n\n| Step | What we do | Why we do it |\n|------|------------|--------------|\n| **Search** | Look around the current sample $x$ inside a small radius and pick the neighbour $\\hat{x}$ where the score network is least reliable. ‑ **SAS** : choose the point with the largest score magnitude (most likely to overshoot). ‑ **CAS** : choose the point whose score changes fastest (quickest direction flip). | Identify the spot most prone to error. |\n| **Descent at $\\hat{x}$** | Compute the steepest descent direction $d$ *at that worst point*. | Neutralise the largest local error. |\n| **Update the $x$** | Move the original sample a small step along the safe direction just computed. | Apply the error‑proof direction to the true state we care about. |\n\n**Key idea** : we *peek* at the place where the model is most uncertain, borrow a direction that corrects that uncertainty, and use it to update the current sample.\n\n\n2. **Why this look‑ahead step is robust - from robust optimization perspective**\n \n$$\n\\min_x f_t(x + \\delta),\\\n\\text{ where } \\\n\\delta = \\arg\\max_{\\substack{\\delta' \\in \\mathcal M}}\n\\hat f_t(x, \\delta').\n$$\n\n* **Inner maximization ( $\\delta(x)$ )**  \n  Chooses the perturbation inside the admissible set $\\mathcal M$ that makes the surrogate loss $\\hat f_t$ worst.  \n  ➜ *Find the scenario where the current score estimate is least trustworthy.*\n\n* **Outer minimization**  \n  Once that worst case is fixed, we ask: *“What move of $x$ will reduce that worst loss the most?”*  \n  The answer is to step along the **negative gradient at the worst point** (our $d$).  This follows standard convex duality: the steepest descent direction minimises the first‑order approximation of $f_t$ under that fixed worst perturbation.\n\n* **Why this is safe**  \n  Because the score field is Lipschitz‑continuous, every other perturbation in $\\mathcal M$ produces a loss that is **no worse** than the one at $\\delta(x)$.  Therefore, a direction that decreases the loss at the worst perturbation will decrease (or at least not increase) the loss for *all* admissible perturbations, i.e., the surrounding uncertainties.  \n\n\n3. **Conditions for the guarantee**\n- **Smooth nearby score** – The score network is locally Lipschitz (smooth in a small neighborhood).\n- **Error is bounded** – We assume the network’s score is never “too far off” from the true score inside the small ball we inspect.  \n- **Step stays small** – The time step $\\Delta t$ is small enough for the second‑order Taylor expansion to remain valid.\n\n\nUnder these three assumptions the look‑ahead worst‑case update provably minimises the one‑step worst‑case loss and, as shown in Appendix A, yields more robust sampling in practice.\n\n---\n\n### **Q3: Clarification on Algorithm 2**\nThanks for asking. The standard diffusion sampler solves  $\\min_x\\ f_t(x).$\nRODS instead solves the following **bilevel** problem (introduced right after line 232), which is the core of Algorithm 2:  \n$$\n\\min_x f_t(x + \\delta),\\\n\\text{ where } \\\n\\delta = \\arg\\max_{\\substack{\\delta' \\in \\mathcal M}}\n\\hat f_t(x, \\delta').\n$$\n\nTwo cases we addressed in the paper are:\n\n1. **Sharpness‑Aware Sampling (SAS)** (lines 219–223):  $\\hat f_{t,\\delta'}(x,\\delta') = f_t\\bigl(x + \\delta'\\bigr).$\n2. **Curvature‑Aware Sampling (CAS)** (line 224):  $\\hat f_{t,\\delta'}(x,\\delta') = \\bigl ||\\nabla f_t(x + \\delta')\\bigr ||.$\n\nAlgorithm 2 simply plugs in the chosen $\\hat f_{t,\\delta'}$ once the curvature detector $H(x_t)\\geq\\epsilon$ signals a “high‑risk” step, and then takes the corresponding robust update.\n\n---\n\n### **Q4: Evaluation on image quality**\n\nThank you for encouraging a broader quality assessment. We have therefore computed standard image-quality and diversity metrics on 1,000 unconditional samples from FFHQ, AFHQv2, and 11k-hands. The table below reports FID, Inception Variance [1] (which quantifies variability in the latent representations), and DreamSim Diversity [2] (which measures the variance of features extracted from a batch of generated images, and our batch size is 64). \n\n\n| Dataset  | FFHQ                |        |         | AFHQ2               |        |         | 11k-hands        |        |         |\n|----------|---------------------|--------|---------|----------------------|--------|---------|-------------------------|--------|---------|\n|          | FID ↓               | Inception-Var ↑ | DreamSim ↑ | FID ↓               | Inception-Var ↑ | DreamSim ↑ | FID ↓               | Inception-Var ↑ | DreamSim ↑ |\n| EDM-Euler    | 20.48              | 0.0669 | 0.407   | 13.36              | 0.0539 | 0.486   | 14.57          | 0.0327                    | 0.1427| \n| EDM-Heun      | 19.56          | 0.0699 | 0.422 | 12.66          | 0.0548 | 0.488 | 14.20          | 0.0332                    | 0.1476               |\n| RODS-CAS      | 22.73              | 0.0661 | 0.400   | 16.08              | 0.0542 | 0.481   | 16.68          | 0.0318                    | 0.1509               |\n\nThese results show that RODS-CAS preserves diversity while keeping FID comparable. The modest FID rise is largely explained by background smoothing: our curvature test sometimes removes ambiguous, high-frequency textures in the complex backgrounds, which the FID metric interprets as loss of detail. However, such effect could be analogous to portrait-mode photography, where a deliberately blurred background can worse a texture-based metric (FID) without reducing perceived quality. In our generation,  the main subjects remain sharp and artifact-free, which is crucial as it is exactly the goal of our correction. **_This is further supported by the 11k-hands experiment, where our method improves hand confidence (HAND-CONF) from 0.9216 (EDM-Euler) and 0.9227 (EDM-Heun) to 0.9347 (RODS-CAS)._**  HAND-CONF is computed by running a pre-trained hand detector [3] and calculating the confidence scores of the detections. This metric is widely used to assess the anatomical plausibility of generated hands. Because hands are structurally well-defined and challenging to synthesize, this benchmark provides a sensitive and interpretable proxy for visual realism. Overall, **_RODS-CAS delivers equally diverse, comparably high-quality unconditional samples while reducing hallucinations._**\n\n[1] Miao, Zichen, et al. \"Training diffusion models towards diverse image generation with reinforcement learning.\" CVPR. 2024.\n[2] Domingo-Enrich, Carles, et al. \"Adjoint matching: Fine-tuning flow and diffusion generative models with memoryless stochastic optimal control.\" arXiv preprint 2024.\n[3] Zhang, Fan, et al. \"Mediapipe hands: On-device real-time hand tracking.\" arXiv preprint 2020."}, {"review_id": "OOavHQMhmj", "reviewer": "Reviewer_UPcW", "summary": "The authors introduce an inference-time method (RODS) to reduce the number of hallucinations in diffusion models. In particular, if the curvature index $\\mathcal H(x)$ of a particular sample at a particular diffusion time exceeds some hyperparameterized value $\\epsilon$, RODS uses a guidance-like term $\\delta = \\arg\\max_{||\\delta|| = \\rho} ||v(x+\\delta)||$ to guide the sample towards stability. The method is validated over AFHQv2, FFHQ, and 11k-hands and shows good performance.", "questions": "See weaknesses. Would love to see further comparison with other methods.\n\nAdditionally, qualitatively examining Figure 4, RODS-CAS clearly achieves fewer hallucinations, but it also results in the background becoming blurrier. Is there an explanation for this phenomenon?", "limitations": "No real discussion of limits of current method on current task, only discussion of how it was not yet applied to another task (conditional generation).", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1. Paper is clear and well written.\n2. Good performance on reducing hallucinations in AFHQv2, FFHQ, and 11k-hands.\n3. Method is relatively efficient in inference time.\n\nWeaknesses:\n1. Similar idea using norms to dynamically adjust diffusion sample present in other paper [1].\n2. Not enough evaluations of other hallucination reducing methods, e.g. [2], [3]\n\n[1] Constrained Diffusion with Trust Sampling.\n[2] Mitigating Hallucinations in Diffusion Models through Adaptive Attention Modulation.\n[3] Understanding Hallucinations in Diffusion Models through Mode Interpolation.", "was_revised": true, "final_justification": "My main notes were regarding other works, to which they have provided satisfactory explanations / plans to address, as well as the  Overall the method is novel and achieves fewer hallucinations across multiple datasets. Therefore, I am increasing my rating to accept.", "rebuttal": "### **W&Q1: Related works and Comparison**\nThank you for pointing out these related works. We will include citations and a brief discussion of [1]–[3] in the revised version.\n\nWhile [1] (\"Constrained Diffusion with Trust Sampling\") uses the norm of the predicted noise, aligned with the score direction, as a constraint for update validity, our approach instead focuses on the norm of local changes in the score field to guide updates. Given the differing objectives and mechanisms, a direct comparison is not applicable.\n\nRegarding [2] (“Mitigating Hallucinations through Adaptive Attention Modulation”), it adopts a different strategy—modulating attention during sampling—while our approach focuses on trajectory-level geometric regularization. As the paper was posted on arXiv in February 2025 and no code was available at the time of our submission, a direct experimental comparison was not feasible.\n\nAs for [3] (“Understanding Hallucinations through Mode Interpolation”), we do include a comparison in Section 4.1 and Appendix C.3. Their method detects hallucinations via **temporal instability** along the denoising trajectory, whereas our approach targets **spatial-domain instability** and further enables **correction**. Additionally, their released code (for now) only covered 1D&2D Gaussian and SIMPLE SHAPES dataset, which could not be naturally transferred to our datasets. We plan to provide a fair experimental comparison once their full implementation becomes available.\n\nWe appreciate the reviewer’s emphasis on thorough comparison. As noted in Section 5, due to substantial differences across prior works in methodology, assumptions, and evaluation protocols, there is currently no standardized benchmark for hallucination detection and correction. We are committed to releasing our code and annotated data to facilitate future research and establish a common basis for meaningful comparison across methods.\n\n---\n\n### **Q2: Evaluation on background blurry problem**\n\nThank you for pointing out the background blurrier phenomenon. This occurs because it is hard to distinguish between **high-frequency textures and hallucinations**. In our generation, the main subjects remain sharp and artifact-free, which is precisely the goal of our correction. The slight background smoothing does not compromise the diversity of the generation or the sharpness of the primary objects. \n\nTo validate such idea, we have further computed standard image-quality and diversity metrics on 1,000 unconditional samples from FFHQ, AFHQv2, and 11k-hands. The table below reports FID, Inception Variance [1] (which quantifies variability in the latent representations), and DreamSim Diversity [2] (which measures the variance of features extracted from a batch of generated images, and our batch size is 64).\n\n\n| Dataset  | FFHQ                |        |         | AFHQ2               |        |         | 11k-hands        |        |         |\n|----------|---------------------|--------|---------|----------------------|--------|---------|-------------------------|--------|---------|\n|          | FID ↓               | Inception-Var ↑ | DreamSim ↑ | FID ↓               | Inception-Var ↑ | DreamSim ↑ | FID ↓               | Inception-Var ↑ | DreamSim ↑ |\n| EDM-Euler    | 20.48              | 0.0669 | 0.407   | 13.36              | 0.0539 | 0.486   | 14.57          | 0.0327                    | 0.1427| \n| EDM-Heun      | 19.56          | 0.0699 | 0.422 | 12.66          | 0.0548 | 0.488 | 14.20          | 0.0332                    | 0.1476               |\n| RODS-CAS      | 22.73              | 0.0661 | 0.400   | 16.08              | 0.0542 | 0.481   | 16.68          | 0.0318                    | 0.1509               |\n\nThese results show that RODS-CAS preserves diversity while keeping FID comparable. In our generation, the main subjects remain sharp and artifact-free. **_This is further supported by the 11k-hands experiment, where our method improves hand confidence (HAND-CONF) from 0.9216 (EDM-Euler) and 0.9227 (EDM-Heun) to 0.9347 (RODS-CAS)._**  HAND-CONF is computed by running a pre-trained hand detector [3] and calculating the confidence scores of the detections. This metric is widely used to assess the anatomical plausibility of generated hands. Because hands are structurally well-defined and challenging to synthesize, this benchmark provides a sensitive and interpretable proxy for visual realism. \n\n\n[1] Miao, Zichen, et al. \"Training diffusion models towards diverse image generation with reinforcement learning.\" CVPR. 2024.\n[2] Domingo-Enrich, Carles, et al. \"Adjoint matching: Fine-tuning flow and diffusion generative models with memoryless stochastic optimal control.\" arXiv preprint 2024.\n[3] Zhang, Fan, et al. \"Mediapipe hands: On-device real-time hand tracking.\" arXiv preprint 2020.\n\n---\n\n### **L: Limitation on current dataset**\n\nThank you for raising this up. In the unconditional generation scenario, we have tested RODS on well-established benchmarks (AFHQv2, FFHQ, 11k-hands), where hallucinations can be easily measured without domain expertise. We acknowledge that in highly diverse or specialized domains, such as medical imaging, the definition of “hallucination” must be grounded in expert judgment. We are collaborating with domain practitioners to evaluate RODS under these conditions."}]

Author rebuttal:

Meta-review: * Summary
The paper tackles the problem of reducing hallucinations in generative models in a novel manner. 
They recast the problem of hallucinations as an outlier and discontinuity selection problem, which can be solved with tools and methods from optimization theory and robust optimization. Practically, this is done by looking at the stochastic process of diffusion, recasting the dynamical equation in a dynamical programming style optimization procedure, and showing it's equivalent to robust optimization. Then show the methodology on several well-known datasets.

* Strengths/ reasons for acceptance 
The paper is very well written and clear; the authors state all of their claims in a very crisp manner. They also justify it with the analytical definitions and some theoretical bounds. The empirical results on several different data sets also support the authors' claim. The code for the paper will be released and can be thoroughly tested.
From a scientific point of view, the paper is excellent, casting the problem into one from optimization theory, the results are even stronger since they can be regularly verified and have a massive body of literature with full proof results. 
Authors also made an above-and-beyond effort to rebut and answer all of the questions the reviewers had and then some. 
With all of the above, I think this is a very strong paper, and I would suggest accepting it

* Weakness 
The first weak point of the paper is the lack of standardization of benchmarks in the field, which makes it slightly hard to compare and judge the results with other SOT. 
The second issue is in how hallucinations are defined in the paper. It's a bit too loose, but then again, it's not just here; it's a general issue with the field.
The third weak point is indeed how the curvature of the loss function relates to hallucination and the optimization process, while the authors addressed this as part of the reviewers' GnZW questions, it's still a bit lacking in the justification part. 

All of the above, though, do not reduce the novelty and significance of the paper, and I would suggest a strong acceptance
Decision: Accept (poster)