Antithetic Noise in Diffusion Models
Authors: Jing Jia
Sifan Liu
Bowen Song
Wei Yuan
Liyue Shen
Guanyang Wang
Venue: ICLR 2026 Poster (2026)
Keywords: diffusion model
initial noise
uncertainty quantification
Primary area: generative models
TL;DR: We find antithetic initial noise yields negatively correlated samples, which enables us to improve sample diversity and construct more accurate estimators.

Abstract: We systematically study antithetic initial noise in diffusion models, discovering that pairing each noise sample with its negation consistently produces strong negative correlation. This universal phenomenon holds across datasets, model architectures, conditional and unconditional sampling, and even other generative models such as VAEs and Normalizing Flows. To explain it, we combine experiments and theory and propose a \textit{symmetry conjecture} that the learned score function is approximately affine antisymmetric (odd symmetry up to a constant shift), supported by empirical evidence.
This negative correlation leads to substantially more reliable uncertainty quantification with up to $90\%$ narrower confidence intervals.  We demonstrate these gains on tasks including estimating pixel-wise statistics and evaluating diffusion inverse solvers.  We also provide extensions with randomized quasi-Monte Carlo noise designs for uncertainty quantification, and explore additional applications of the antithetic noise design to improve image editing and generation diversity. Our framework is training-free, model-agnostic, and adds no runtime overhead. Code is available at https://github.com/jjia131/Antithetic-Noise-in-Diffusion-Models-page.

Published as a conference paper at ICLR 2026
ANTITHETIC NOISE IN DIFFUSION MODELS
Jing Jia'
Sifan L
Bowen Song"
Wei Yuan'
Liyue Shen
Guanyang Wang"
Rutgers University, {jing. jia, wy204, guanyang. wang www.an/gggggrrrre. @rutgers.edi Duke University, si sin. findduukk edu University of Michigan, (bowenbw, liyues] @umi ch. edu
ABSTRACT
We systematically study antithetic initial noise in diffusion models, discovering that pairing each noise sample with its negation consistently produces strong negative correlation. This universal phenomenon holds across datasets, model architec- tures, conditional and unconditional sampling, and even other generative models such VAEs and Normalizing Flows. To explain it, combine experiments and theory and propose a symmetry conjecture that the learned score function is approximately affine antisymmetric (odd symmetry up constant shift), sup- ported by empirical evidence. This negative correlation leads to substantially more reliable uncertainty quantification with up to 90% narrower confidence intervals. We demonstrate these gains on tasks including estimating pixel-wise statistics and evaluating diffusion inverse solvers. We also provide extensions with randomized quasi-Monte Carlo noise designs for uncertainty quantification, and explore ad- ditional applications of the antithetic noise design improve image editing and generation diversity. Our framework is training-free,, model-agnostic, and adds no runtime overhead. Code is available at https: / com/ jjial31/ Ant Antithet: c No se- in-Di 1 F fus on Mode page
INTRODUCTION
Diffusion models have set the state of the art in photorealistic image synthesis. high-fidelity audio,
and video generation (Sohl-Dickstein et al.) 2015 Ho al. 2020; Song et al. 2021b Kong et al.)
2021); they also power applications such as text-to-image generation (Rombach et al. (022). image
editing and restoration (Meng al. 2022). and inverse problem solving (Song et al. 2022
For many pretrained diffusion models, sampling relies on three elements: the network weights, the denoising schedule, and the initial Gaussian noise. Once these are fixed, sampling often deterministic: the sampler transforms the initial noise into an image via successive denoising passes.
Much of the literature improves the first two ingredients and clusters into two strands: (i) architectural and training developments, which improve sample quality and scalability through backbone or objective redesign (e.g., EDM (Karras et al., (2022). Latent Diffusion Models (Rombach et al., (022).
DiT (Peebles & Xie 2023)), (ii) accelerated sampling. which reduces the number of denoising steps while retaining high-quality generation (e.g.. DDIM (Song et al., (021a). Consistency Models (Song
al. 2023). DPM-Solvert+ (Lu et al. 2023 and progressive distillation (Salimans & Ho 2022
However, the third ingredient- the initial Gaussian noise- tooise-has received comparatively little attention.
Prior work has optimized initial noise for generation quality, editing, controllability, Chen et al. 2024a Eyring al., 2024 [Sogg et al. 2025 Wang et al. [0224] Chihaoui et al. (024)). However, most of these efforts are task-specific. A systematic understanding of how noise itself shapes diffusson model outputs still missing.
inverse problem solving (Guo et al. 2024 Qi et al. 2024 Zhou et al. 2024) [Ban et al. 2025|Chhen
Our perspective is orthogonal to prior work. Our central discovery is both simple and universal: pairing every Gaussian noise z with its negation -z-known antithetic sampling (Owen 2013 consistently produces samples that are strongly negatively correlated. This phenomenon holds
"co-corresooning authors

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regardless of architecture, dataset, sampling schedule, and both conditional and unconditional sampling. It further extends to other generative models such as VAEs and Normalizing Flows. We explain this phenomenon through both experiments and theory. This leads to a symmetry conjecture that the score function is approximately affine antisymmetric, providing a new structural insight supported by empirical evidence.
DM(-2,c)
NASO
DM c)
Figure I: Use antithetic noise -2 and (with condition c) to generate visually "opposite" images.
This universal property has direct impact on uncertainty quantification and also enables additional applications:
(i) Sharper uncertainty quantification. Antithetic pairs naturally act control variates, enabling significant variance reduction and thus sharper uncertainty quantification. Our antithetic estimator delivers up to 90% tighter confidence intervals and cuts computation cost by more than 100 times. The efficiency gain immediately leads to huge cost savings in a variety of tasks, including bias detection in generation and diffusion inverse solver evaluation, we demonstrate in later experiments.
(ii) Other applications. Because each antithetic noise pair drives reverse-diffusion trajectories toward distant regions of the data manifold (see Figure []. the paipligs scheme increases diversity "for free" while preserving high image quality, as confirmed by SSIM and LPIPS in our experiments. Moreover, algorithms that rely intermediate sampling or approximation steps also benefit from the improved reliability provided by antithetic noise. As an illustration, we present an image editing example in C.2 and show that our antithetic design serves as a plug-and-plag tool to improve performance at no additional cost.
Building the antithetic pairs, we generalize the idea apply quasi-Monte Carlo (QMC) and randomized QMC (RQMC). The resulting RQMC estimator often delivers further variance reduction. Although QMC has been widely used in computer graphics (Keller [995 Waechter & Keller 2011 quantitative finance (Joy et al., 1996] L L'Ecuyer 2009), and Bayesian inference (Buchholz et al. 2018 Liu & Owen 2021p. this is, to our knowledge, its first application to diffusion models.
In summary, we discover universal property of initial noise, reveal a new symmetry in score networks, and demonstrate conerete benefits in various practical applications. This positions initial noise manipulations as simple, training-free tool for advancing generative modeling.
The remainder of the paper is organized follows. Section 2 defines the problem and outlines our motivation. Section presents our central finding that antithetic noise pairs produce strongly negatively correlated outputs. We offer both theoretical and empirical explanations for this phe- develops nomenon, and present the symmetry conjecture in Section 3.2 Section decvioopde estimators and their confidence intervals via antithetic sampling, and extends the approach to QMC. Section reports experiments the aforementioned applications. Section discusses method and outlines future directions. Appendix C and includee proofs, additional experiments and detailed setups, and supplemental visualizations, respectively.
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2 SETUP, MOTIVATION, AND RELATED WORKS
Unconditional diffusion model: A diffusion model aims to generate samples from an unknown data distribution Po- It first noises data towards a standard Gaussian progressively, then learns to reverse the process so that Gaussian noise can be denoised ste- back into target samples. The forward process simulates a stochastic differential equation (SDE): dx, = [(xt, l)dl + a,dw,, where { (we)(=o denotes the standard Brownian motion, and y((x, l), dg chosen by the users. Let Pr. denote the distribution of x,- Song et al. (2021b) states that if we sample Yr ~ Pr and simulate the probability-flow ordinary differential equation (PF-ODE) backward from time T to 0 as
dy: = (-u(yee)) 1) 2 logp(ye: () dr,
then for every time the marginal distribution of yı coincides with pt. Thus, in idcalized world, one can perfectly sample from po by simulating the PF-ODE (T.
In practice, the score function V log Pt (x, f) is unavailable, and neural network co to approximate it, where denotes its weights. Therefore, one can generate new samples from po by first sampling a Gaussian noise and simulating the PF-ODE (I) through numerical integrator from T to O. For example, DDIM (Song et al. 2021a) has the (discretized) forward process as X X N(Varxi-1o (1 a fork -00.-=0...... -..... 1. and backward sampling process as
(5) (x) is trained
YT ~ N(O, 1), Yr-1 = Va--1 Yc vI at cp (yi)
(y.).
+
Vou
Once 0 is fixed, the randomness in DDIM sampling comes solely from the initial Gaussian noise.
We remark that samples from Po can also be drawn by simulating the backward SDE with randomness each step, as in the DDPM sampler (Ho al. 2020 [Song al. 2021b). Throughout the main text, we focus on deterministic samplers such as DDIM to explain our idea. In Appendix CC.4] we present additional experiments showing that our findings also extend to stochastic samplers like DDPM.
Text-conditioned latent diffusion: In Stable Diffusion and its successors SDXL and SDXL Turbo (Podell et al. 20244] Sauer et al. (024). a pretrained VAE first compresses each image to latent tensor A text encoder embeds the prompt as E R During training, the network eg (2e, c) and learns V., log Pe (2s c). At generation time we draw latent noise 21 ~ [OO, 7) and run the reverse diffusion from t T to 0, yielding denoised latent 2p, which the decoder maps back to pixels. Given a prompt e and an initial Gaussian noise, the sampler produces an image from po c).
(t) receives
Diffusion posterior sampling: Diffusion model can also be used as a prior in inverse problems. Suppose we observe only a partial corrupted measurement yohs A(x) noise from unknown signal x. Diffusion posterior sampling aims to sample from the posterior distribution p(x Yous) x P(yobs x) p(x). where p(x) is given by a pretrained diffusion model. Given initial noise z and observed Vab.. diffusion posterior samplers apply sequence of denoising steps from T to 0, each
step resembling but incorporating Yobs (Chung et al. 2023a Song et al. 2022 2024).
2.1 MOTIVATION
Beyond examining how variations in the initial noise affect the outputs, several technical considera- tions motivate our focus on the antithetic noise pair (2, 22) for diffusion models. Let DM denote certain diffusion model's mapping from an initial noise vector to a generated image sample.
Preserving quality: Since z N(0, I) implies -2 N(0, I), the initial noises z and - 2 share
the same marginal distribution. Consequently, DM(2) á DM(-2) so all per-sample statisties remain unchanged. In other words, negating the initial noise does not degrade generation quality.
Maximal separation in the noise space: High-dimensional Gaussians concentrate the sphere of radius /dimension (Vershynin) 2018), , so any draw and its negation 2 lie at opposite poles that hypersphere. This antipodal pairing represents the maximum possible perturbation noise space, making it natural extreme test of the sampler's behavior.
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(a) CelebA-IIQ
(b) DiT (class 919)
(c) Glow (class 0)
Figure 2: Histograms of standard and centralized Pearson correlation coefficients for CelebA-HQ. and DiT class and Glow in single class. Dashed lines indicate the average.
Measuring (non-)linearity; Recent work has examined how closely score networks approximate linear operators. Empirical studies across various diffusion settings show that these networks behave as locally linear maps; this behavior forms the basis for new controllable sampling schemes (Chen et al. 2024b; Li et al. 2024b, (ong et al., 2025). Correlation is a natural choice to measure linearity: score network were exactly linear, feeding it with noises = and -a would yield correlation of -1 between (2(2) and DM(-2). Thus, the difference between the observed correlation and provides a direct measure of the network's 's departure from linearity.
3 NEGATIVE CORRELATION FROM ANTITHETIC NOISE
3.1
PIXEL-WISE CORRELATIONS: ANTITHETIC vs. INDEPENDENT NOISE
We compare the similarity between paired images generated under two sampling schemes: PN
(positive negative, =) and RR (random random, vs. 22) under three settings: (1)
unconditional diffusion models, (2) class- or prompt-based conditional diffusion models, and (3)
generative models beyond diffusion.
For diffusion models, we evaluate both unconditional and conditional generation using publicly available pre-trained checkpoints. Notably, we include both traditional U-Net architecture and mansformer-based DiT (Peebles & Xie 2023). Implementation details are described Section 5.1 We also test distilled diffusion models (Song el. (023) using consistency distillation checkpoints. For generative models beyond diffusion, we select two representative baselines: an unconditional VAE on MNIST and conditional Glow model (Kingma & Dhariwal 2018) on CIFAR-10. The experimental details for consistency models, VAE and Glow are provided in Appendix C.I
To quantify similarity, we
metrics: the standard Pearson correlation and centralized Pearson correlation. Let I;,1 and 24,2 denote the flattened pixel values of the two images in the i-th generated pair. The standard Pearson correlation is computed directly between T;,1 and 22,2- To correct for lataset-level or class-level bias, we also define a centralized correlation. For a dataset, prompt, or class with K pairs generated, correlation of the i-th pair is defined as the standard Pearson correlation between the centralized images I;, Me and zi22 Me- For each comparison, t-test is conducted to assess statistical significance. In all experiments, the resulting p-values are negligible (< 10-11), which confirms significance; hence, they are omitted from the presentation.
compute the mean N4c = E (Zt,1 + 2222 The centralized
Additional metrics such as the Wasserstein distance presented in Appendix
classes of models. PN pairs consistently show significantly stronger negative correlations than RR pairs, with this contrast also visually evident in their histograms (Figure 2). addition, centralization strengthens the negative correlation, since removes shared patterns. For example, in CelebA-HQ, centralization removes global facial structure, while DiT and Glow, removes class-specific patterns.
Results:
Table T summarizes the statistics
The same behavior appears in both DDPM models and diffusion posterior samplers, which report in Appendix CC.4 These results demonstrate that the negative correlation resulting from antithetic sampling is universal phenomenon across diverse architectures and conditioning schemes.
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Table 1: Correlation results across different models and datasets, shown
(SD). Rows
1-3 are pretrained unconditional diffusion models on different datasets. Rows 4-5 are conditional diffusion models. Rows 6-8 are pretrained consistency models on different datasets. Rows 9-10 are generative models that are not diffusion-based.
Standard Correlation PN
Centralized Correlation RR
Model
Dataset
RR
PN
LSUN-Chureh
-0,62 (0.11)
0.08 (0.17)
-0.77 (0.07)
0.00 (0.17)
Uncon. Model
CelebA-HQ
-0.34 (0.19)
0.25 (0.20)
-0.78 (0.06)
-0.01 (0.20)
CIFAR-10
-0.76 (0.13)
0.05 (0.24)
-0.86 (0.07)
0.00 (0.23)
SD 1.5
LAION-2B(en)
-0.47 (0.05)
0.05 (0.04)
-0.54 (0.08)
0.00 (0.01)
DiT
ImageNet-256
-0.07 (0.27)
0.26 (0.18)
-0.45 (0.08)
0.00 (0.03)
LSUN-Cat
-0.88 (0.06)
0.02 (0.14)
-0.91 (0.05)
0.01 (0.14)
Consistency Model
LSUN-Bedroom
-0.78 (0.07)
0.03 (0.15)
-0.84 (0.06)
0.00 (0.15)
ImageNet-64
-0.71 (0.14)
0.02 (0.19)
-0.75 (0.12)
0.01 (0.19)
VAE Glow
MNIST
0.21 (0.12)
0.42 (0.17)
-0.41 (0.12)
-0.00 (0.24)
CIFAR-10
-0.52 (0.02)
0.08 (0.05)
-0.57 (0.02)
-0.01 (0.02)
3.2 EXPLANATORY EXPERIMENTS: TEMPORAL CORRELATIONS & SYMMETRY CONJECTURE
We aim to explain the strong negative correlation between DM Z and DM Z) for Z N(0, 7). Because DM performs iterative denoising through the score network
(t) c (see (2), we visualize how
these correlations evolve over diffusion time-steps in Figure 3
Throughout the transition from noises to samples, the PN correlation of cg inddiatee by the orange, blue, and green dashed lines in Figure 3 starts 11, stays strongly negative, and only climbs slightly in the final steps. This nearly- -1 correlation is remarkable, as it suggests that the score network cœ learns an approximately affine antisymmetric function. To explain, first state the following results which shows -1 correlation is equivalent to affine antisymmetric, with proof in Appendix B.1
Lemma 1.
Let Z ~ N(O, Ia), suppose a map J Rd + R satisfies must be affine antisymmetric at (0, c) for some c, i.e. f(x) + f(-x) 2c for every x.
= -1, 1, then
Lemmanand Figure leads us to the following conjecture:
Conjecture For each time step I, the score network 6g is approximately affine antisymmetric at (0, c,), i.c.,
(4) (x) + (1) (-x) x 2c;
for some fixed vector c that depends on t.
The conjecture implies that the score network has learned an "almost odd" function up to an additive shift. This is consistent with both theory and observations for large f, where px(x) is close to standard Gaussian, thus the score function is almost linear in Nonetheless, our conjecture far more general: applies to every timestep and can accommodate genuinely nonlinear odd behaviors (for example, functions like sin.r.
Figure 3: Correlation of (solid) and (t)
" (dash) between antithetic (PN) pairs. Step 50 is the initial noise and Step O is the generated image. Shaded bands show +1 std. dev.
This conjecture combined with the DDIM update rule ( immediately explains the negative correlation: The one step iteration of F, in can be written
linear combination be- tween and the output of the score network: Fx(x) = 0, + bee (f) (x). Given the conjecture, F((-x) R axx byc (t) (x) 26,c, - -x((x) 2b, so the one-step DDIM update is affine antisymmetric at (0, b.C.). Thus, beginning with a strongly negatively correlated pair, each DDIM update preserves that strong negative correlation all the way through the final output.
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To (partly) validate conjecture, we perform the following experiment. Using pretrained CIFAR- 10 score network with a 50-step DDIM sampler, we picked time steps t = 1.3,..... 19 (where smaller t is closer to the image and larger is closer to pure noise). For each t, sample N(0, Ia). evaluated the first coordinate of cg (cx) as varied from - to 1 (interpolating from x to x), and We exclude t 2 20, where the curve is very close to straight line.
plotted the result in Figure +
Figure 4 confirms that, at every step the first coordinate of the score net- work is indeed overall affine antisym- metric. Although the curves display nonlinear oscillations at small t, the deviations on either side are approxi- mately mirror images, and for larger t the mapping almost a straight line. The symmetry center o usually lies near zero, but not always- for instance, at t = 11, ~ 0.05 even though the function spans from -0.05 to 0.15. In Appendix C.5 present further vali- dation experiments using alternative co- ordinates, datasets, and a quantitative metric called the antisymmetry score. Despite the distinct function shapes across coordinates, the conjectured symmetry still persists.
t-1l ll
[-13
f-
[-10
Tea
Figure 4: First-coordinate output of the pretrained score network CIFARIO function of the interpolation scalar e for a 50-step DDIM at t = 1,,...., 19.
We further evaluate the antisymmetry conjecture across additional coordinates and datasets. On CIFAR-IO and Church, we assess how closely one-dimensional slices of the network outputs behave like afffin--antiymmmetii functions. To this end, we introduce the affine antisymmetry score, which quantifies the degree of antisymmetry. Across datasets, the resulting antisymmetry scores are consistently high (mean values above 0.99 and even the lower quantiles still near 0.9). These results provide strong empirical support for the conjecture; full experimental details and plots are given in Appendix C.5
Finally, we provide theoretical support for our conjecture in Appendix B.2 B33 and B.4 Ap-
pendix B.2 confirms the conjecture in the large-t regime, Appendix B.3 shows how both the density ratio and the score error converge monotonically as evolves via a Hermite polynomial expansion, and Appendix demonsttaass that all orthogonal symmetries of the data distribution preserved throughout the forward process. These results help explain the behavior underlying the conjecture. The forward process preserves every orthogonal symmetry of the initial distribution, including coor- dinate reflections. Thus, in the ideal case where the data distribution at = 0 is reflection-symment the density remains even and the score remains odd for all / by Proposition 4 Moreover, even if this symmetry holds only approximately for small f, the monotone decay of the density ratio and the score error shows that Pt quickly approaches the Gaussian, so the forward dynamics push the score toward an odd function in a controlled manner.
Both the conjecture and its empirical support could be of independent interest. They imply that the score network learns a function with strong symmetry, even though that symmetry is not explicitly enforced. The conjecture matches the Gaussian linear score approximation in the high-noise regime, which has proven effective (Wang& & Vastola 2024). In the intermediate-to-low noise regime, single Gaussian approximation performs poorly- this is clear in Figure for = 3, 7, 11, where the score is strongly nonlinear- yet our conjectured approximate symmetry still holds.
Besides explaining the source of negative correlation, we believe this result provides new insight into the structure of diffusion models, which are often treated as black-box functions. Leveraging this finding for algorithms and applications is an important direction for future work.
For readers interested in additional theoretical perspective, Appendix B.6 presents discussion that links negative correlation to the FKG inequality, and provides a detailed analysis of the DDIM sampler using this framework.

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4 UNCERTAINTY QUANTIHICATION
In uncertainty quantification for a diffusion model DM, the goal is typically to estimate expectations of the form Ex r) (DM(=))] where 5 is statistic of user's interest. Our goal is to leverage negative correlation to design an estimator with higher accuracy at fixed computation cost inference.
Recent studies have examined epistemic uncertainty and bias in diffusion models, including Berry et al. 2024 2025). Our focus is different: we study aleatoric uncertainty from noise sampling and develop a variancc-reduucion method.
Standard Monte Carlo: To approximate this expectation, the simplest approach is to draw NN independent noises EN ~ N(0, 1), calculate S, = =((M(())) for each i and form the standard Monte Carlo (MC) estimator jN == Σ S:/N by taking their average. A (1 o)
confidence interval, denoted CIN (1 a) is then PN + 21 22-a/2// (2/N), where o% is the sample variance of S1, S2....... SN and 21 is the (1 - a/2) quantile the standard normal. A formal guarantee of the above construction is given in Proposition 5 Appendix B.5
Antithetic Monte Carlo: The observed negative correlation motivates an improved estimator. Let N 2K be even, users can generate K pairs of antithetic noise (21,- -2))...., (=x,- -2 ). Define S 5(DM(,,)). ST = ((D(((2)) and let S. 0.5(5, + + ) be their average. Our Antithetic Monte Carlo (AMC) estimator is f AAMC = Σ S K, and confidence interval CI (1 a) is
AMC + 21-a/2 V 22(6 / where (2 is the sample variance of S,,...,
The intuition is simple: if the pair (S,, .T)) remains negatively correlated, then averaging antithetic pairs can reduce variance by partially canceling out opposing errors-since negative correlation suggests that when one estimate exceeds the true value, the other is likely to fall below it.
The AMC estimator is unbiased, and ( a)) achieves correct coverage as the sample size increases. Let p == Corr(S; S.) We can prove that AMC's standard error and its confidence- interval width are each equal to the Monte Carlo counterparts multiplied by the factor vI+ p. Thus, when p < 0, AMC produces provably lower variance and tighter confidence intervals than MC, with greater gains becomes more negative. See Appendix [B.5 for formal statement and proof.
Since both methods have the same computational cost, the variance reduction from negative correla- tion and the antithetic design yields direct and cost-free improvement.
We can generalize the antithetic noise pair to a collection of K noise variables, constructed so that every pair has the same negative correlation -1/(K - 1). One way generate them is as follows. Draw K independent standard Gaussian vectors (WI,.. ..... wx) and let w be their average. For each i, set = / KK/(K - 1) (w; - w). One can directly check that each 2; is still marginally standard Gaussian, and Corr(zi,, zرر) = -1/(KK - 1) for every * j and every pixel I. In particular, when K- = 2, this construction reduces to our usual antithetic noise pair.
K-antithetic noise:
Quasi-Monte Carlo: The idea of using negatively correlated samples has been widely adopted in Monte Carlo methods (Craiu & Meng (005) and statistical risk estimation (Liu et al.) 2024) for improved performance. This principle of variance reduction can be further extended to QMC methods. As an alternative to Monte Carlo, QMC constructs a deterministic set of N samples that have negative correlation and provide a more balanced coverage of the sample space. QMC samples can also be randomized, resulting in RQMC methods. RQMC maintains the marginal distribution of each sample while preserving the dow-discrepancy properties of QMC. Importantly, repeated randomizations allow for empirical error estimation and confidence intervals (L ECEuee et al. 2023).
Under regularity conditions (Niederreiter 1992), QMC and RQMC can improve the error convergence rate from (N to O for any e > 0, albeit e may absorb a dimensionndeependent factor (log Nyd. For sufficiently smooth functions, RQMC can achieve rates as fast as O(V - 3/2+c) (wwen) 1997b Moreover, the space-fillingg property of QMC may also promote greater sample diversity.
Although the dimension of the Gaussian noise is higher than the typical regime where QMC is expected to be most effective, RQMC still further shrinks our confidence intervals by several-fold compared with standard MC. This hints that the image generator's 's effective dimension is much lower than its ambient one, allowing QMC methods to stay useful. Similar gains arise when QMC methods
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deliberately exploit low-dimensional structure in practical problems (Wang & Fang 2003 Xiao & Wang 2019; Liu & Owen, (023). Developing ways to identify and leverage this structure more systematically is an appealing avenue for future work.
5 EXPERIMENT
Sections 5. and 5.2 present two uncertainty quantification applications: estimating pixel-wise statistics and evaluating diffusion inverse problem solvers. The latter considers two popular algorithms across a range of tasks and datasets. For each task, we apply MC, AMC, and, when applicable, QMC estimators as described in Section For each estimator, we report the 95% confidence interval (CI) width and its relative efficiency. The relative efficiency of new estimator (AMC or QMC) is defined as the squared ratio of the MC CI width to that of the estimator, (CIme/ /Cne/CClww estimator) 2 It reflects how many times more MC samples are required to achieve the same accuracy as our new estimator. Section 5.3 shows that antithetic noise produces more diverse images than independent noise while preserving quality, and Appendix C.2 explores an image editing example.
5.1 PIXEL-WISS WISE STATISTICS
We begin by evaluating four pixel-level statistics. These include the (i) pixel value mean, (ii)
perceived brightness, (iii) contrast, and (iv) the image centroid, with definitions in Appendix C.6
These statisties are actively used in diffusion workflows for diagnosing artifacts and assessing
reliability, with applications in detecting signal leakage (Lin et al., 2024 Everaert et al. 2024).
out-of-distribution detection (Le Bellier & Audebert 2024), identifying artifact-prone regions (Kou
et al. 2024), and improving the reliability of weather prediction (Li et al. 2024a).
Setup: We study both unconditional and conditional diffusion models; details are Appendix
For unconditional diffusion models, we evaluate pre-trained models on CIFAR-10 (Krizhessky et al. 2009), CelebA-HQ (Xia al.) (02T), and LSUN-Chureh (Yu al., 2015). For each dataset, 1,600 image pairs are generated under both PN and RR noise sampling with 50 DDIM steps.
For conditional diffusion models, we evaluate Stable Diffusion 1.5 on 200 prompts from the Pick-a-Pic (Kirstain et al. (023) and DrawBench (Saharia et al. (022) with classifier free-guidance (CFG) scale 3.5, and DiT (Peebles Xie 2023) on 32 ImageNet classes with CFG scale 4.0. For each class or prompt, 100 PN and RR pairs are generated with 20 DDIM steps. We also study the effect of CFG C.4 (Ho & Salimans 2022).
scale on both models the Appendix
Implementation: To ensure fair comparisons under equal sample budget:
For unconditional diffusion models: MC uses 3,200 independent samples, AMC (k = 2) uses 1,600 antithetic pairs, and AMC (k 8) uses 400 independent negative correlated batches. QMC employs a Sobol' point set of size 64 with 50 independent randomizations. Due to the dimensionality limits, QMC is restricted to CIFAR-10.
For conditional diffusion models: for each class or prompt, MC uses 200 independent samples, AMC uses 100 antithetic pairs, and QMC employs 8 Sobol' points with 25 randomizations.
CIs for MC and AMC are constructed as described in Section 4
For QMC, we use a student-l interval to ensure reliable coverage EL'Ecyyer al. (023). with details in Appendix C.6
Across all statistics, both AMC and QMC have much shorter CIs than MC, with relative dramatically reduce cost for uncertainty estimates. AMC and QMC estimators yield comparable results. The performance of QMC depends on how the total budget is allocated between the size of the QMC point set and the number of random replicates, a trade-off we explore in further details in Appendix C.6
Results:
efficiencies ranging from 3.1 to 136. This strongly suggests estimators
5.2
EVALUATING DIFFERENT DIFFUSION INVERSE PROBLEM SOLVERS
Bayesian inverse problems important in medical imaging, remote sensing, and astronomy. Re- cently, many methods have been proposed that leverage diffusion priors to solve inverse problems. For instance, /Zheng et al. (2025) surveys 14 such methods, yet this is still far from exhaustive. As
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Table 2: CI lengths and efficiency (CIMc/CI)² (in parentheses), using MC as baseline.
Dataset
Brightness
Pixel Mean
Contrast
Centroid
MC
2.00
2.04
1.08
0.14
QMC
0.35 (32.05)
0.39 (26.94)
0.22 (23.43)
0.04 (9.65)
CIFARIO
AMC (k- 2)
0.35 (32.66)
0.39 (27.12)
0.23 (22.05)
0.04 (9.73)
AMC (k 8)
0.36 (30.35)
0.39 (27.34)
0.24 (20.37)
0.03 (13.94)
MC
1.77
1.76
0.60
0.60
CelebA
AMC (k 2)
0.26 (47.4) (1)
0.31 (33. 15)
0.19 (10.18)
0.20 (7.82)
AMC (K 8)
0.15 (130.69)
0.16 (115.79)
0.18 (10.90)
0.19 (8.49)
MC
1.66
1.64
1.02
0.82
Church
AMC (k = 2)
0.14 (134.13)
0.16 (103,80)
0,20 (27.16)
0.26 (9.84)
AMC (k -)
0.15 (117.90)
0.16 (107.64)
0.19 (27.81)
0.25 (10.87)
MC
3.86
3,95
2.18
4.03
Stable Diff.
QMC
1.36 (8.07)
1.43 (7.60)
1.03 (4.46)
1.88 (4.54)
AMC (k = 2)
0.90 (18.53)
1.00 (15.53)
0.89 (6.25)
1.62 (6.15)
MC
7.52
7.74
3.32
3.15
DiT
QMC
3.44 (4.78)
3.49 (4.91)
1.82 (3.31)
1.69 (3.46)
AMC (k- (k=
3.13 (5.79)
3.12 (6.16)
1.72 (3.74)
1.74 (3.30)
result, comprehensive benchmarking is costly. Meanwhile, most existing works on new diffusion inverse solvers (DIS) largely ignore uncertainty quantification, though it is critical for fair evaluation. This raises key challenge: how to efficiently evaluate posterior sampling algorithms with reliable estimates of reconstruction quality and uncertainty. Our approach aims efficiently address this gap.
We show that our antithetic estimator can save substantial computational cost for quantifying un- certainty by requiring far fewer samples than standard estimators. We focus on two popular DISs, Diffusion Posterior Sampling (DPS) (Chung et al. 2023a) and Decomposed Diffusion Sampler (DDS) (Chung et al. 2023b), and evaluate them on a range of tasks described below.
5.2.1 HUMAN FACE RECONSTRUCTION
We evaluate DPS across three inverse problems: inpainting, Gaussian deblurring, and super-resolution. Reconstruction error is measured using both PSNR and L, distance relative to the ground truth image. For all tasks, we use 200 human face images from the CelebA-HO dataset. For each corrupted image. we generate 50 DPS reconstructions using both PN and RR noise pairs to compare their estimators. Operator-specifec configurations (e.g., kernel sizes, mask ratios) are provided in Appendix C.6
Results: As shown in Table 3 across all three tasks, AMC achieves substantially shorter confidence intervals than standard MC. This implies AMC yields efficiency gains, ranging from 54% 84% in inpainting, 41% 54% in super-resolution, and 34% 56% in deblurring.
The reconstruction metrics (I,, PSNR) differ by less than 0.2% between the two estimators, showing that they produce consistent results. Meanwhile, our method offers a clear advantage in efficiency.
Table 3:
Comparison of AMC vs. MC across tasks in DPS with efficiency (CIvc/CI)3 in parentheses
L1
PSNR
Task
AMC CI
MC CI
AMC CI
MC CI
Inpainting
0.83 (1.54)
1.03
0.23 (1.84)
0.31
rusou-ieno
1.01 (1.41)
1.20
0.24 (1.54)
0.30
Gaussian Deblur
0.89 (1.34)
1.03
0.23 (1.56)
0.29

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5.2.2 MEDICAL IMAGE RECONSTRUCTION
We use the DDS (Chung et al. 2023b as the reconstruction backbone and apply our antithetic noise initialization to estimate reconstruction confidence intervals. Given a single measurement, we generate 100 reconstruction pairs using 50-step DDS sampling. Reconstruction quality is evaluated against ground truth using L1 distance and PSNR. Dataset details can be found in Appendix [C.64]
Results: Both MC and AMC estimators give consistent estimates, with estimated L1 error of 0.0194 and 0.0195, and PSNR values of 31.48 and 31 .45, respectively. Similarly, we observe that AMC produces much shorter confidence intervals. The CI lengths decrease with relative efficiency of 1.44 for L1 and 1.37 for PSNR. This shows that antithetic initialization consistently reduces estimator variance with no extra cost without degrading reconstruction fidelity in large-scale inverse problems.
Conclusion:
These two experiments show that a single antithetic estimator reduces evaluation costs for diffusion inverse solvers by 34%-84%, saving that becomes especially valuable given the large number of solvers in the literature. This makes large-scale benchmarking far more practical.
5.3 DIVERSITY IMPROVEMENT
Many diffusion tools generate a few images from one prompt to let users select a preferred one. However, these images may look similar with randomly sampled initial noise (Marwood et al.) 2023] Sadat et al. 2024). We show that antithetic noise produces more diverse images than independent noise, without reducing image quality or increasing computational cost. Diversity measured using pairwise SSIM (lower more diverse) and LPIPS (Zhang et al. 2018) (higher more diverse).
We evaluate the diversity metries for both unconditional and conditional diffusion on image pairs generated following the same setup in Section [.1] As shown in Table 4 antithetic noise pairs consistently lead to higher diversity, as indicated by lower SSIM and higher LPIPS scores. Importantly, image quality remains stable for both noise types. We expect our method can be easily integrated into existing diversity optimization techniques to further improve their performance.
Table Average percentage improvement of PN pairs over RR pairs on SSIM and LPIPS. Metric
Unconditional Diffusion
Conditional Diffusion
CIFAR-10
LSUN-Church CelebA-HQ
SD1.5
DIT
SSIM (%)
88.78
45.69 3.54
36.78
28.32 5.78
23.99
LPIPS (%)
6.69
15.14
10.62
6 DISCUSSION AND FUTURE DIRECTIONS
We find that antithetic initial noise yields negatively correlated samples. This is a robust property that generalizes across every model that we have tested. The negative correlation is especially useful for uncertainty quantification, where it brings huge variance reduction and cost savings. This makes approach highly effective for evaluation and benchmarking tasks. Meanwhile, this finding can serve as simple plug-in to improve related tasks, such as diversity improvement and image editing.
limitation of our work is that while the symmetry conjecture is supported by both empirical and theoretical evidence, it remains open in full generality. In addition, our method is not cure-all: the LPIPS diversity improvements in Table 4 are consistent but modest. While we obtain large gains for pixel-wise statisties, the improvements for perceptual or text-alignment metries such MUSIQ and CLIP Score are small, since the strong nonlinearity of transformer decoders attenuates most of the negative correlation present in the diffusion outputs.
Looking forward, future directions include systematically studying the symmetry conjecture; inte- grating antithetic noise with existing coise-optimizztiinn methods; and further leveraging QMC to improve sampling and estimation.
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ACKNOWLEDGEMENT
Guanyang Wang and Jing Jia acknowledge support from the National Science Foundation through grant DMS- 221099 and an Adobe Data Science Research Award. Liyue Shen and Bowen Song acknowledge funding support from the National Science Foundation through grant IIS-2435746, Defense Advanced Research Projects Agency (DARPA) under Contract No. HR001 12520042, Hyundai America Technical Center, Inc. (HATCI), well the University of Michigan MICDE Catalyst Grant Award and MIDAS PODS Grant Award.
ETHICS STATEMENT
This work does not involve human subjects or personally identifiable data collected by the authors. All datasets used are publicly available and distributed under licenses permitting academic research.
Our medical image reconstruction experiments rely on benchmark datasets that are fully de-identified before release (e.g., publicly available MRI data). No new data collection was performed, and no protected health information is included. The experiments are purely methodological, intended to evaluate reconstruction fidelity and uncertainty quantification techniques, and make no clinical claims or diagnostic recommendations.
Our contributions are methodological, focusing on sampling and uncertainty quantification in gen- erative models. While such methods could be applied broadly, including in sensitive domains, we emphasize that our work aims to improve reliability, transparency, and efficiency in evaluation.
REPRODUCIBILIIY STATEMENT
Full experimental details can be found in Section and Appendix C The code is available at https
//gi thub. (ttomm/jjaa1////ae Ant thet i c- No se- in Di fusi Mode 1s page
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Zikai Zhou, Shitong Shao, Lichen Bai, Zhiqiang Xu. Bo Han, and Zeke Xie. Golden noise for diffusion models: learning framework. arXiv preprint arXiv:2411.0950,, 2024.
A
USE OF LLMS
LLMs are used only for writing, editing, or formatting purposes and do not affect the core methodol- ogy, contributions, and originality of the paper.
B PROOFS
B. PROOFS IN SECTION 3.2
Proof of Lemmal 7
It suffices to show Var(J(Z) + /(-2)) = 0. By definition:
(or(
Com(f(-2), Var(f( -1
Therefore
(a(f(()) Var(f(2)) +2
Var(f(2)) + Var(f - (ar(f(Z)) Var(f(-2))
(VVar(f(Z) - VVr(ff
On the other hand, since -Z N(0, 1) given Z N(0, 1), we have Var(f(Z)) Var(f(-Z)) and then Var(f(Z) + f(-Z)) = 0, as claimed.
B.2 THEORETICAL GUARANTEE OF THE SYMMETRY CONJECTURE FOR LARGE
Background: The Uhlltenn-Uhlenbeekk (OU) process. The OU process (X,)220 in IRd solves the SDE
dX, -x, dt + v2dB, Xo ~ po:
where (B,) is standard Brownian motion in IRd. This is Gaussian Markov process with stationary distribution Raa N(0, Ia) and generator
Lf(x) = Af(x) - x Vf(x).
For any initial law Pos let hr denote the law of X,. Then be = PoPe where (Pi)ezo is the OU semigroup defined by
= Xo = z]]
Write the score of μt as (x) = V log,:(r), where pt is Lebesgue density of prs we have Theorem (Score converges to the Gaussian score). Under the setup above,
lim Esullls + xxll3] 0. 1-00
Moreover, the convergence is quantified by
(x, + X+Il3] I(pe (7)) S e-2 I(uo 7a)-
Proof. Recall the definition of relative Fisher information
I(me I ra) = [VVV log(pt///((p))])
Since Vya(x)= = --r, -r, we have:
Ilm 74) Byn (X.) + 1
Moreover, it is well known (e.g. Chapter of Bakry et al. (2013) that I(m 71) S exp(- ( 7a), this concludes our proof.
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diffusion models, the forward noising process is exactly the Omn- stein- Sein-Uhlenbeck semigroup. Theorem Daanniimppi iiplies that the true score converges to the Gaussian score -I. Consequently, mne-dimeniional slices of the score become asymptotically affine antisymmetric, confirming our symmetry conjecture the high-noise limit.
In
The next corollary shows, the 1-step DDIM update has nearly -1 correlation large
Corollary 1 (Correlation of 1-step DDIM). With all the setup the same as above, assuming we have score network e(t) approximating the score function. Let tp == max (Xc()))) 8c (x) Rp, (-X) *r(-X)13²) be the expected (symmetrized) error. Consider one-step
DDIM update Fi(z)) = a + b Co (x) = (Fe.(z),..... F(..()) E R as defined in (2) Let
tt = min Varp. (Fr,(X)), Varp. (Frx(-X)))-
Then
2 lbel
(x + +1| v
/ + V ral)
Vt,
Proof. Theorem 1 shows I(m 72) S exp(-22)
7a). Write c6") = Set and 8t x Do.
so that Ep, (rx)(((()1 (x) S and Ep. (x) = I(mm
7a). Then
Fe(x)) (ae - b,)x l/ (A)(z) + (x)),
-(--))=-Pp Ff(x)) + Ex(x),
with E,(x) : b1 Ax(x) + (-=)) re(x)). By triangle inequality of L (pt) norm,
[E & (V(e= ra) + e "e'VVI'' I(pu ra) + 21)
Using
(Fr (x,)
Corr(Fi,(X) FL,, = --1 (ar((F.(())) (ar (-X))
and Cauchy-Schward with the variance lower bounds gives
((ox((((X).F((x))) +1| s ||Bell
2 lbbll
e 77),
Ur,i
as claimed.
In practice, ut, O, the correlation is close -1 for large 1; its deviation from -1 1 is the order of the neural network approximation error plus an exponentially decaying term Ce
B.3 MONOTONE CONVERGENCE IN
Section B.2 shows that the score converges to the Gaussian score as + oc. now further prove that (1) the score error E,,, (X,) + x and (2) the density ratio Pe/Ta both converge monotonically in t.
B.3.1 DENSITY RATIO CONVERGENCE
Background: Hermite Polynomials in R:: For x E R, the (probabilist's 's) Hermite polynomials (H,(r)) form sequence of polynomials defined by
2
d
x²
Hx(z) (-1)" exp
exp
dx
We summarize their known properties here. The first two can be checked by direct calculation. The latter two can be found on page 105 of Bakry et al. (2013).
Proposition Hermite polynomials (x) nzo satisfy the following
(Recurrence relation) H Hat1 (x) Hn(x) H²(x),,
0.
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(Orthogonality under Gaussian measure) We have:
| H (m)))))) n16mmm
where ?n1 (dx) = (212)_1/2--- /2 d2 denotes the one-dimensional standard Gaussian mea- sure, and 8nm is when n = m and otherwise.
(Completeness) Hermite polynomials (1l (22) an orthogonal basis of the Hilbert space (7) [f f (x) < oo) equipped with inner product (f.g)> fgr(dz))
(Eigenfunction) Let L be differential operator defined as L(f) == f" zz'. Hermite polynomials {H, (z) 1220 are eigenfinctions of L:
L(Hn) -nH
T2 2 0.
Hermite Polynomials in Rd; One can naturally extend univariate Hermite polynomials to the multivariate setting. For x (#1...., za) Rd. let 7/a denote the d-dimensional standard Gaussian measure,
||=|| 2
ra(dr) = (2x) - d/2 exp
dr.
For multi-index a = (ai,.... ad) E Nd, we define the multivariate Hermite polynomial
Ha II H., (2:),
where Ha; is the one-dimensional probabilists' Hermite polynomial. We write |a| := an +... Oa and a! ai!... a.l. Similar to the one-dimensional case, the family (Halaeva forms orthogonal basis of the Hilbert space L (Ta))
[ (x(x)
al 80g.
Monotone convergence of the density ratio. Recall from Section B.2 that the generator of the OU process
Lf(x) = Af(x) Vf(x).
and the corresponding Markov semigroup Pt exp(tL). We show that the multivariate Hermite polynomials are eigenfunctions of L, and thus also of P..
Proposition We have:
I.H, = = Ho.
E No.
P.H, - lalt Ha.
a N2,t200
For every g (12(1a)
Pig(x) = Σ
age t/al H.(
aEN
where on (g, Had Ha)u?
Proof. We can directly calculate
LH. (x) AHa(x) - VHa (x)
Σ (H:) (x- E Dz H' (a.
Σ (Z-, (r,) 12, (x:))
((-a))Ho,(( (x_ (xi)
|a| Ha(x).
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where Ha (x-1) 14 Ha, (zz). The second last equality follows from the fourth property of Proposition
Consequently, the OU semigroup (Pt)e20 satisfies
P.Ha e
-|aa Ha: Ho:
t2 0.
For every E (na), we rewrite 9 according to its Hermite expansion (which is valid according to properties 2-3 Proposition D.
g(z) Σ (z)). aENd
aa Ha)
Therefore
Peg(s)
Σ aENS
tlal Ha(x),
as claimed.
In particular, let (x() = f((x))((2) denote the density at time t of the OU process started from po- It is known that ft = P.fo- Proposition implies
f((x) = Σ
aaa-
f/ai Ha(x) = au +
Σ
a.e
a.e
Hall Ha(x),
|al
aEN
|a|30
where a are the Hermite coefficients of fo. Since Ho = and
ag -ff fo(x) Ho(x) na(dx)= f fo(x) ra(dx) / po(x) 1,
may rewrite
f((x) 1+ Σ a 1/aal Ha())
(3)
|a|>0
We now show fi monotonically converges to 1 in (²²))).
Proposition 3. Let ps filia be as above, and assume fo E (2(())). Then IIfe 11l(2(50)
monotonically decays to at the speed of e
Proof. From we have
f((x) 1 Σ a (a)))
1a|30
Since (lla)aeNs an orthogonal basis of (7r)), obtain
llfe 11(24(0))
Σ
2||a| -11al
a
Since each term the sum decreases monotonically O, the same holds for Ilfe 11]; The overall decay rate is governed by the slowest mode, which corresponds to the smallest |a| = 1, thus the rate of convergence is exp( (f)). Taking square roots yields the desired result.
B.3.2
SCORE ERROR CONVERGENCE
Recall from Section B.2 that Ep. [Xls, X, I(H, na). It follows from equation 5.7.2 in Bakry et al. (013) that
dI(ue I va)) < -2I(14 ra) S 0.
dt
Thus I(me 72) is monotonically decreasing as function of t. Further, Grönwall's inequality implies
I(m 74)) - I[Ho ra)-
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B.4 SYMMETRY PRESERVATION UNDER THE FORWARD PROCESS
This section shows the forward process preserves any orthogonal symmetry in the data distribution. More precisely, let G be a subgroup of the orthogonal group ((d) in IRd (e.g. coordinate sign flips, coordinate permutations, and rotations). We have the following:
Assuming Po is symmetric about I under the action GfG, i.e., for every g E G. we have po( = polg (g (x 11) + p)) for every Then pt satisfies pe(x) pelg (x Mc) + Me) for every n, and (p + g x) = g St (1 + X), where pt = E
Proposition
Proof. Let Xo ~ po.Z ~ N(0, 1a). It is known that X₂ e + e1-e-22 ~ Pr. For any g E G. 9 (X, md) + μr = g (e + v1 e =z - M4) H
= -e (Xo p) + 1) + V1 1 c---22 7 ~ Pt
- μi) + μ has the same distribution Xo (by symmetry) and g Z ~ N(0, Ia). This proves the claim on the symmetry of pt. For the score, since Pe - 9 2) = (1t z), taking derivative on both sides yields:
where the last claim uses that g (X,-
g Cppp + x) = Vpi(He + x).
Since g is an orthogonal matrix, g =g we have:
Pp + g x) = g Vpr(1 + x))
Dividing both sides by Po(p x) proves the claimed result.
Sections B B.2 (large limit), B.3 (monotone convergence), and B.4 (symmetry perservation from t == 0) together provide insights for the conjecture: The forward OU process preserves all orthogonal symmetries of po. including central reflections. Hence, in the idcalized case where Po is reflection symmetric, the density is even and its score is odd. Proposition X ensures that this symmetry is when the symmetry is only approximate near t R: 0, the monotone convergence of p: toward the Gaussian is exponentially fast, so the forward dynamics push 8, toward linear function monotone manner.
preserved for all Furthermore,
B.5 PROOFS IN SECTION
B.5.1 MONTE CARLO ESTIMATOR:
Proposition 5. Denote E,.,No,r(S((MM)) by μ and Var.- by o². Assuming
2² < co, then we have:
MC + and (GMO)2 + 0² almost surely, and (μ CM (1 (1-a)) + a, both as oo.
E[(aMC - 11) VarlaM = o2/N.
In words, the above proposition shows:
(i Correctness: The standard Monte Carlo estimator f converges to the true value, and the coverage probability of CI (1 aa) converges to the nominal level (1 a). (ii) Reliability: The expected squared error of j equals the variance of single sample divided by sample size. The confidence interval has width approximately 202//a/2/ /N.
Proof of Proposition 5 Let S; = ((M())) for i 1,...,N. independently from N(0, I). the random variables S,, S..... are independent and identically dis- tributed with mean y and variance の² < cc.
N. Because the noises z; are drawn
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Consistency. By the strong law of large numbers,
N
PN
Σ
S
2.h.
μ.
N
The sample variance estimator
N 2
N N
N N
(o
S
AN
2
S
(
N
The first term converges B[S almost surely by the law of large numbers, the second term converges to n1² almost surely as shown above. Therefore (a ns. o².
Asymptotic normality and coverage. The classical central limit theorem states that
P N(0.1).
Replacing the unknown by the consistent estimator &MC and applying Slutsky's theorem yields
MC VN
4 1(0. PN
o MC
Hence
P(R E CI a))
VNN 4) S 521-0/2)
a.
Mean-squared error.
Because AN is the average of N iid. variables,
VarlaN
MC
N
Moreover, since MC is unbiased (F(iMM) = 11), its mean-squared error
E [ - nu = E[(2NC E[aNC)) = o²
N
This completes the proof.
B.5.2 ANTITHETIC MONTE CARLO ESTIMATOR:
Proposition 6. Denote ([S(DM(=))) by 14, Var(S(DM(=))) o2, and Cov(St, ST) by p. Assuming
²² <. 00, then we have:
AMC μ and (GAMC)2 (1+ p)o2/2 almost surely N + oc.
ETCAAMC 11)1 VarluAMC (1++p)/1
IP ( CIMC (1)))) +1 +1 N 00.
Proof. Let N = 2K and generate independent antithetic noise pairs 22, for i = I,...,.. K. Recall
St S(DM(2)), ST = -S(DM -2)) 5, 1s + +T),
#= K.
Because the pairs are independent and identically distributed (i.i.d.), the random variables St...., .Sk are i.i.d. with
B[Si] /
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and
Var[Si] Var 2 + +s
(4)
(t) Var [s. + +
(5)
[Var[S;]] + Varls, + 2 C
(6)
4
1 (02 02 +222200²
(7)
1 + P 0².
(8)
2
where we have written Cov(S, .T,) po² 2 as in the statement. Set a K K -i= S, and
let (aAMC)2 2 be the sample variance of Si...., Sk. All three claims in Proposition 6 follow from Proposition
B.6 AN ALTERNATIVE EXPLANATION VIA THE FKG INEQUALITY
We also provide an expository discussion highlighting the FKG connection behind negative correlation. Consider the univariate where scalar Gaussian is fed through sequence of one-dimensional linear maps and ReLU activations. Let F be the resulting composite function. One can show that, regardless of the number of layers the linear coefficients used, Corr F(-z)) S 0. The proof relies on the univariate FKG inequality (Fortuin et al. 1971). well-known result in statistical physics and probability theory.
We generalize this result to higher dimensions via partial monotonicity, under which negative correlation still holds.
We first formally state and prove the claim for the univariate case in Section B.6
Proposition (Univariate case). Let N(0, 1) and let F R R be the output of any one-dimensional feed-forward network obtained by alternating scalar linear maps and ReLU activations:
ho(2) =
he(z) ReLL(w: he-1(2) + be), e= 1=1,.... 1,...,, L,
F(z) h2(2).
For all choices of depths L and coefficients {we, be
Corr (F(2), F(-2)) - 0.
Proof. An important observation that monotonicity is preserved under composition: combining one monotonic function with another produces a function that remains monotonic.
Each scalar linear map x + wyz + by is monotone: it non-decreasing wi 2 and non-increasing if we < O. The ReLU map æ ReLU((z) max (0, æ) is non-decreasing. Hence the final function F is monotone. Without loss of generality, we assume is non-decreasing.
FKG inequality guarantees Cov (ov2--No.)(((()).c..) ((2) 0 provided that f.g are non-decreasing. Therefore, (F()), = Covz (F (F(Z), -F(-)) < 0. Since
Corr(F(2)) F(-z)) Co Covz.NO,,) (F()), F(-2))
VVa((F())) Var(F(-2))
we have Corr(F(z), F(-2)) S..
To generalize the result to higher dimension, define function f Rm - R to be partially monotone if for each coordinate j. holding all other inputs fixed, the map t H f( 2,_ 2, ) is either non-decreasing non-increasing. Mixed monotonic- ity is allowed. For example, f(x, y) = x y is non-decreasing in x and non non-increasing in y, yet still qualifies partially monotone. We have the following result:
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Proposition 8. For diffusion model DM R - I.m and summarr statistics S R B. if the joint map is partially monotone, then Corr(S DM(Z)), S DM s O.
Now we prove the general case:
Proof of Proposition
Let G = So DM. For each coordinate E [d] fix sign
+1, if G is non-decreasing in zj
8 -1, if G is non-increasing in zy.
Write 8 = (81....., E {+1)a and define, for any 2 E Rd,
((2) G za)
Similarly, define
H(=) = G(-2) = G(-8) -dza).
By construction, G coordinate wise non non-decreasing and H is mordinate-wiis non-increasing.
Because each coordinate of ~ N(O, La) symmetric, the random vectors (81 Z......... (dZd) and (Zi...., Za) have the same distribution. Hence
Cov(G(Z),
= Cov (or()
The multivariate FKG inequality (Fortuin et al. 1971) for product Gaussian measures states that, when U and V are foordinate-wiet non-decreasing, (Z)(U(Z), V(Z)) Z 0. Apply it to the pair (G. -H): both components are non-decreasing. so
Cov(G(Z), 2 0 = Cov(O(Z), (()) s
+ G((()()) S
Therefore
0, claimed.
Proposition relies on checking partial-monotnnice If S is partially monotone (including any linear statistic), then the conditions of Proposition { are satisfied by, c.g., Neural Additive Models (Agarwal al. 202 l) and Deep Lattice Networks You al. 2017). Unfortunately, partial-monotonicil is in general hard to verify.
While popular diffusion architectures like DiT and U-Net lack partial monotonicity, include Proposition & as an expository attempt to highlight the FKG connection behind negative correlation.
B.6.1 FKG INEQUALITY FOR DDIM
We first analyze the idcalized DDIM process using the FKG inequality. single step of the idcalized DDIM is
(x) = ax (x))
where s, denotes the score of pt, and a,, C 2 are deterministic coefficients obtained by rearranging
The idealized DDIM trajectory is given by the composition:
DDIM = Fi o F2 ... Fr.
(10)
We present two results that give sufficient conditions under which the local (one-step) DDIM update 99) and the global DDIM procedure (T0) generate negative correlation.
We need the following definition:
Definition 1 (MTP, with curvature bound). Letp: : Rd + (0, oc) be a C probability density. We say that p is multivariate totally positive of order 2 (MTP,) with curvature bound x 2 0 if the second partial derivatives of log p satisfy, for all igd
Bg p(x) for all i + js
and 0g logp(2) 2 - for all i.
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MTP₂ distributions widely studied in statistics and probability Karlin & Rinott (980). In particular, isotropic Gaussian distributions are MTP2.
Proposition 9 (One-step DDIM). With all the notations as above, fix > assume the distribution Pt is 2 with curvature bound Ka, and
at Cehi,
then Ft is coordinatewise nondecreasing: 0n (.) 2 0. for all i,j and all x. Moreover, for any linear statistics we have Corr(S F((Z)), F((-2)) O.
Proof. Differentiate componentwise:
OF (x) = ar ōug + a O. log px().. dæ,
For ( Hj
this is 2 by & 2 0 and the mixed second derivative condition. For j.
a (a) 2 a + ( Nee) 2 0. Ōr;
Let G = S o Fr. Since F; is coordinately non non-decreasing and S is linear, we have that G = S is partially monotone. Thus Corr(S (z)), o F((-2)) 2)) S by the multivariate FKG inequality and Proposition 8
The next result shows that the idealized DDIM trajectory continues generate negative correlation, as long as the curvature bound holds uniformly over all steps.
Proposition (Global DDIM chain). With all the notations as above. Assume the distribution pt is MTP₂ with curvature bound Kit, and
2 CşKa
for every t. Then DDIM, = o F2 F3 ....o o Fr is coordinatewise nondecreasing. Moreover, for any linear statistics S, we have Corr(S DDIM)(Z)), DDIM((-Z) S
Proof. Since each F; is coordinatewise nondecreasing, their composition F1 ... Fr is also coordinatewise nondecreasing. Therefore, the composition S DDIM, is partially monotone, thus Corr(S o DDIM; S DDIMi(-2)) S 0 as claimed.
Remark 1. All these results remain valid if we replace the exact score V log by the neural network 8g, up to a constant rescaling.
Remark 2. We note that the condition a, 2 C2Ky, is satisfied when k, O(1) and the probability flow ODE is discretized with sufficiently small step size. For the deterministic DDIM update under puriance-preserving schedule,
04-1
|at VI ag.
at
-= -=vI- 01-1
C4 =
a1
Thus is of unit order, and at captures the difference la--1 - -re l. In the continuous- -imm limit with step size At and smooth function a(t), a Taylor expansion yields a; 1+ O(At) and c = O(A)): thus 0 2 C,K, holds when At is sufficiently small and stays bounded.
C ADDITIONAL EXPERIMENTS
C.1 CORRELATION EXPERIMENT SETUP OTHER MODELS
C.I.I CONSISTENCY MODEL
We study both unconditional and conditional consistency models using publicly available check- points provided Hugging Face's Diffusers library (von Platen et al.) 2022): openai/diffusers- ed imagenet64 12, openai/diifusers--c__caaa/ 12, and openai/diffusers-cd bedroom256 12.
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For unconditional models, evaluate pre-trate pre-trained models on LSUN-CI and eet al. 2015). For each dataset, 1,600 image pairs are generated under both PN and RR noise sampling with DDIM steps.
Yu
For conditional models, we evaluate pre-trained models on ImageNet on 32 classes (Deng et al. 2009). For each class, 100 PN and RR pairs generated with 1 steps.
C.1.2 GENERATIVE MODELS BEYOND DIFFUSION
A mentioned in Section evaluate correlation VAE and flow-based models. Unlike the diffusion models, which we use public pre-trained checkpoints, both of these models required explicit training before evaluation. Here describe the training setup and generation procedure.
We train the unconditional VAE on MNIST following the publicly available implementation provided Francis 2022). The VAE consists of a simple convolutional encoder-decoder architecture with Gaussian latent prior. After training, generate 1,600 paired samples under PN and RR schemes, respectively.
VAE:
We train the class-conditional Glow model (Kingma & Dhariwal 2018) on CIFAR-I0 using the normflows library. The architecture follows a multiscale normalizing flow design and Glow blocks. After training, we generated 100 and 100 RR pairs per class.
Glow:
C.2 IMAGE EDITING
We apply our antithetic initial noise strategy to the image editing algorithm FlowEdit (Kulikov et al. 2024). which edits images toward target text prompt using pre-trained flow models.
In Algorithm FlowEdit, at each timestep, the algorithm samples Rave random noises Z (0, 1) to create noisy versions of the source image, computes velocity differences between source and target conditions, and averages these directions to drive the editing process.
In the Nav = 2 setting, we replace the two independent random samples with antithetic noises: for each Z ~ NO, I) we also use -Z and average the two velocity updates.
We compare on 76 prompts provided in FlowEdit's 's official GitHub repository. For each prompt, we generate 10 images using both PN and RR. All other parameters follow the repository defaults. We evaluate performance using CLIP(semantic text-image alignment; higher is better) and LPIPS (perceptual distance to the source; lower is better), which jointly measure text adherence and structure preservation.
As result, PN improves the mean CLIP score, winning 56.59% of pairwise comparisons. also reduces LPIPS, winning in 81.58% of all pairwise comparisons.
C.3 IMPLEMENTATION DETAIL
We use pretrained models from Hugging Face's Diffusers library (von Platen et al. 2022): google/ddpmm-curch-2555 google/ddpm--cifr110-2
and google/ddpm-celebahg-53 for uncondi- tional diffusion; Stable Diffusion v1.5 for text-to-image; and the original repository from (Chung ct al. 2023a) for guided generation. Experiments were run on eight NVIDIA L40 GPUs. The most intensive setup setup-Stable Diffusion-taa about five minutes to generate 100 images for single prompt.
C.4 ADDITIONAL EXPERIMENTS ON PIXEL- WISE SIMILARITY
C.4.1 DDPM
Antithetic sampling setup: Unlike DDIM, where the sampling trajectory is deterministic once the initial noise is fixed, DDPM adds random Gaussian noise every timestep. The update function in DDPM is
ai EB (x t)) + a.
(I:
V
ar
VI
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and 1) if 1, else 0. Therefore, in DDPM, antithetic sampling requires not only negating the initial noise but also negating every noise 2, added at each iteration.
Table 5 and Table reppor the standard Pearson correlations and centralized correlation between pixel values image pairs produced using the same pre-trained models under PN and RR noise schemes using DDPM (default 1000 steps) across CI CelebA-HQ, and LSUN-Church. For each dataset, we follow the same setup explained in Section 3 We calculate the pixelwise correction with 1600 antithetic noise pairs and 1600 independent noises.
Consistent with the behavior observed in DDIM, PN pairs in DDPM samples also exhibit negative correlation. Using the standard Pearson correlation, the mean correlation for PN pairs is strongly negative in all datasets-CIFARIA CIFARIO (-0.73), Church (-0.45), and in the more identity-consisten CelebA (-0.18).
The centralized correlation analysis further sharpens this contrast: mean PN correlations are sub- stantially lower (CIFARI0 -.800, CelebA -0667, Church 0.655. These results confirm again that PN noise pairs consistently introduce strong negative dependence, while RR pairs remain close to uncorrelated or weakly positive.
Table 5: DDPM standard Pearson correlation coefficients for PN and RR pairs
Statistic
CIFAR10
CelebA
Church
PN
RR
PN
RR
PN
RR
Mean
-0.73
0.04
-0.18
0.29
-0.45
0.11
Min
-0.95
-0.61
-0.80
-0.50
-0.90
-0.60
25th percentile
-0.81
-0.12
-0.31
0.16
-0.55
-0.02
75th percentile
-0.66
0.20
-0.05
0.44
-0.36
0.23
Max
-0.21
0.80
0.44
0.78
0.07
0.80
Table DDPM centralized Pearson correlation coefficients for PN and RR pairs
Statistic
CIFAR10
CelebA
Church
PN
RR
PN
RR
PN
RR
Mean
-0.80
0.01
-0.67
-0.00
-0.65
-0.00
Min
-0.96
-0.61
-0.89
-0.60
-0.93
-0.56
25th percentile
-0.87
-0.15
-0.73
-0.15
-0.72
-0.12
75th percentile
-0.75
0.16
-0.62
0.15
-0.60
0.11
Max
-0.23
0.76
0.14
0.64
-0.32
0.72
Fa Ohiggnn fanofii Comboloet
Gorl 37 17-kee
Aar of Karoffrre (Ceoebett)
Nard Mestraliee
Prokedaltyy
<038 "029 222 Condacion Doeffdent
-013
Pearson Conditton Confficint
-0.2
-2.3
-229
-0.0
020 Conpeaticn Zoeffieet
(a) CIFARIO
(b) LSUN-
(c) CelebA-HQ
Figure 5: Pearson Correlation histograms for PN and RR pairs across three datasets using DDPM. Dashed lines indicate the mean Pearson correlation for each group.
C.4.2 DIFFUSION POSTERIOR SAMPLING (DPS)
The same pattern persists in the setting of using the diffusion model prior for posterior sampling, too, which has been utilized to solve various inverse problems, such as inpainting, super-resolution, and Gaussian deblurring.
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Since there is ground truth image available in the image restoration task, the standard pixel-wise correlation calculated using the difference between reconstructed images and the corresponding ground truth, and the centralized correlation is caleulated using the same definition described in in Although the overall standard correlation values are shifted up, due to the deterministic conditioning, the posterior nature of sampling PN pairs still shows significantly lower correlations than RR pairs across all tasks.
For the standard Pearson correlation, the mean PN correlations range from 0.20 0.27, while RR correlations consistently lie above 0.50. For the centralized correlation, PN correlations are strongly negative across all tasks (means around -0.72). In contrast, RR pairs remain centered near zero (mean correlations around -0.01 to -.02)..
Table 7: DPS Pearson correlation coefficients for PN and RR pairs
Statistic
Inpainting
Gaussian Deblur
Super-resolution
PN
RR
PN
RR
PN
RR
Mean
0.28
0.57
0.27
0.57
0.20
0.53
Min
-0.14
0.05
-0.22
0.13
-0.34
0.01
25th percentile
0.19
0,52
0.17
0.51
0.10
0,47
75th percentile
0.36
0.63
0.36
0.63
0.30
0.60
Max
0.66
0.83
0.62
0.83
0.64
0.81
Table 8: DPS centralized correlation coefficients for PN and RR pairs
Statistic
Inpainting PN
Gaussian Blur
Super-resolution
RR
PN
RR
PN
RR
Mean
-0.72
-0.02
-0.71
-0.01
-0.72
-0.01
Min
-0.86
-0.43
-0.84
-0.36
-0.87
-0.35
25th percentile
-0.76
-0.07
-0.75
-0.07
-0.76
-0.08
75th percentile
-0.69
0.04
-0.67
0.05
-0.69
0.05
Max
-0.14
0.47
0.01
0.48
-0.14
0.43
voceing PooMi Foomee ech
Tao Dentralled maaaac
Commlttinn Conffcerz
Premson Convaaten Confecirr
Corbaetcn Coetiiceet
(a) Inpainting
Super-rcsolution
(c) Gaussian Deblur
Figure 6: Pearson Correlation histograms for PN and RR pairs across three tasks in DPS. Dashed lines indicate the mean Pearson correlation for each group.
C.4.3 WASSERSTEIN DISTANCE
To complement the correlation-based analyses in the main text, we also evaluate similarity using the Wasserstein distance, measure of distributional discrepancy. It quantifies the minimal "effoort required to transform one probability distribution into another, which means lower Wasserstein values indicate closer alignment between the two distributions, while higher values indicate larger differences.
To calculate Wasserstein distances, treat each generated image pair as sample from distribution under the sampling scheme, PN or RR. As shown in Table 9 PN consistently exhibits larger Wasserstein distances than RR across nearly all models, and the differences statistically significant
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at all 10-10 This implies that antithetic initial noises lead more divergent distributions than random sampling and confirms our results from the correlation analysis.
Table 9: Wasserstein Distance, shown are means (SD) with corresponding t-statistics and p-values.
Model
Wasserstein Distance
t-stats (p)
PN
RR
LSUN-Church
0.16 (0.10)
0.12 (0.07)
12.09,
CIFAR-10
0.19 (0.14)
0.15 (0.09)
1 = 8.30,
CelebA-HQ
0.17 (0.11)
0.12 (0.07)
14.50, p
SD 1.5
0.10 (0.06)
0.09 (0.04)
= =33.111, p= t = 14.17,
DIT
0.19 (0.15)
0.14 (0.12)
VAE
0.07 (0.05)
0.05 (0.03)
f 13.24, t = 7.92, 0
Glow
0.15 (0.09)
0.12 (0.08)
C.4.4 INFLUENCE OF THE CFG SCALE CORRELATION
We extend our pixel-correlation analysis on conditional diffusion models across various Classifier- Free Guidance (CFG) scales. CFG (Ho & Salimans 2022) is technique that strengthens conditioning in diffusion models by interpolating between conditional and unconditional score estimates. The guidance scale controls this interpolation strength, with higher values producing samples more aligned with the conditioning signal such as prompt or class.
We conducted experiments both Stable Diffusion 1.5 (SD1.5) and DiT using CFG scales (,, 9). For each setting, we generated 100 PN and RR noise pairs across 25 prompts/classes, measuring both standard and centralized correlations between the image samples.
Meay Consaton
gourt hoddig Dee lud
Pockay thggel Poshio (ontulzzd
CrC Soare
ChG Scale
(a) Stable Diffuusin
(b)
Figure 7: Average pixel correlation versus CFG scale for SD1.5 and DIT. Both standard and central- ized correlations are shown. Shaded regions indicate standard deviation of mean correlation across 25 prompts/classes
As show in Figure 7 for all CFG scales, negated noise continues to produce strongly negatively correlated samples. At the same time, the standard correlation for both PN and RR grows the CFG scale increases. This can be explained as follows: larger CFG values pull samples more strongly toward the conditioning signal (prompt/class), effectively shrinking the space of plausible outputs. As samples concentrate more tightly around the target distribution, they become more similar to one another. Thus. the correlations of both PN and RR pairs increase, while PN remains much more negative than RR.
C.4.5 PARTIAL NEGATION NOISE VECTORS
We conducted experiments to evaluate how localized antithetic noise influences outputs of uncon- ditional diffusion models trained on LSUN-Church and LSUN-Bedroom For each dataset, generated 200 image pairs by sampling Zı ~ N(0, I) and constructing Z₂ by negating the upper half of Z, while leaving the lower half unchanged.
We calculate the Pearson standard correlation and the centered correlation between corresponding halves to quantify spatial correspondence induced by the antithetic manipulation. As shown in
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Figure 8 the top halves exhibit sharply negative correlations, and the bottom halves are highly positively correlated and visually almost identical (Figure [27] 28). This shows that the negative correlation effect acts locally in noise space and directly affects the corresponding spatial regions in the generated images.
top gaull (Crignna)
Bocomm (Centrakzzal Bottom (eerisl
groonii Dametyy
Top loo) (Centealund))
Probdbillty Densiy
-2.33
-9.25
2.00
23
Coeffkient
Conciation Coctlicient
LSUN-Church
(b)
Figure 8: Average pixel correlation of top-half and bottom- salf across generated image pairs.
C.5 ADDITIONAL EXPERIMENTS ON THE SYMMETRY CONJECTURE
We run additional experiments to validate the conjecture in Section 3.2 Using a pretrained score network and a 50-step DDIM sampler, we evaluated both CIFAR-10 10 and Church. For each dataset, we selected five random coordinates in the C' x H X W tensor (channel, height, width). At every chosen coordinate we examined the network output at time steps 1. ..... 20 (small to the final image, large / is close to pure noise). For each we drew standard Gaussian sample (4) (0, Ia) and computed the value of ég (cx) the selected coordinate. The resulting plots appear in Figures 18
close
To measure how much one dimensioonl function resembles an antisymmetric shape, we introduce the affine antisymmetry score
AS(S) = [_ +0
()))))
where f= f== is the average value of on [-1,
The integral's numerator is the squared average distance between the antithetic mean 0.5 f(-x) 0.5 fxx and the overall mean f, while the denominator is the full variance of f over the interval.
The antisymmetry score well-defined for every non-constant function f: it takes values in the range [0, 1] and represents the fraction of the original variance that eliminated by antithetic averaging. The score attains AS(f) = 1 exactly when the antithetic sum f(x)) + f(-w) constant, that is, when is perfectly aeffin-aantsymmmettne affine-symmtric.
Conversely, AS(f) = if and only if f(-z) f(z) + c for
some constant c, meaning
For each dataset we have 100 (5 coordinates x 20 time steps) scalar functions; the summary statistics of their AS scores are listed in Table TO] Both datasets have very high AS scores: the means exceed 0.99 and the 10% quantiles are above 0.97, indicating that antithetic pairing eliminates nearly all variance in most cases. Even the lowest scores (about 0.77) still remove more than three-quarters of the variance.
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1..00
6
11:11
1.2
Figure 9: CIFARIO: Coordinate (0,21, 21, 0)
1xT
1.19
6
16:11
1417
05
Figure 10: CIFAR10: Coordinate (1, 15)
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14.0
4
1111
1-22
1.1
0.110
Figure 11: CIFARIO: Coordinate (1,
1xT
lx2
1+111
1412
141
140
64
05
2
Figure 12: CIFAR10: Coordinate (1, 29, 15)
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140
187
1.12
1:0
02:0
Figure 13: CIFAR10: Coordinate (2, 0)
4
4
1417
1+D
140
2
33
Figure 14: Church: Coordinate 73, 73, 76)
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R PRREE SEEEE
2.a0 5 :
2 5
0
1.B
1.12
1:22
2.
Figure 15: CIFARIO: Coordinate (0, 117, 192)
4
5 1
2
5 8
Figure 16: Church: Coordinate 54, 244)
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1.22
5 E 2 2 2 12
0
1:-21
2.
12
Figure 17: CIFARIO: Coordinate (0, 56, 219)
3
132
Figure 18: Church: Coordinate (1, 208, 237)
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Table 10: Affine antisymmetry score
Dataset
CIFARIO
Church
Mean
0.9932
0.9909
Min
0.7733
0.7710
1% quantile
0.9104
0.8673
2% quantile
0.9444
0.8768
5% quantile
0.9690
0.9624
10% quantile
0.9865
0.9750
Median
0.9992
0.9995
C.6 ADDITIONAL. EXPERIMENTS FOR UNCERTAINTY QUANTIFICATION
C.6.1 UNCERTAINTY QUANTIFICATION
The image metrics used in uncertainty quantification are used to capture different aspects of pixel intensity, color distribution, and perceived brightness.
Mean pixel value is defined as the average of all pixel intensities across the image (including all channels). Formally, for image / RCx WxW the mean is computed
ΣΣΣ
# HWC
j-
Brightness is calculated using the standard CIE formula: 1129--R+0587---0111 B to produce grayscale value at each pixel, where R, G, B the red, green, and blue color value of the given pixel. It is widely used in video and image compression, and approximates human visual sensitivity to color. The brightness of an image is then the average of the grayscale value across all pixels.
Contrast is computed as the difference in average pixel intensity between the top and bottom halves of an image. Let [0, x x be the normalized image, we define contrast as 100- (/oop Hbottom) where Mrop and Pbottom the average intensities over the top and bottom halves, respectively.
Centroid measures the coordinate of the brightness-weightedd center of mass of the image. For scalar comparison, we focus on the vertical component of the centroid to assess spatial uncertainty. After converting to grayscale M E I2 HW by averaging across channels, we treat the image as 2D intensity distribution and compute the vertical centroid as
Σ
ΣgB Mij.
MN Ej=1
Mij i=
where denotes the row index.
C.6.2 QMC EXPERIMENTS
RQMC confidence interval construction: For the RQMC experiments, we consider R inde- pendent randomization of Sobol' point set size n, with a fixed budget of Rn function evaluations. Denote the ROMC point set in the r-th replicate U,, Uni 1J²). Applying the Gaussian inverse cumulative distribution function --1 1 to each coordinate u, transforms the uniform samples to standard normal samples. Consequently. the estimate in each replicate is given by
(uskhisksn C. [0, 1jd. where
A = Σ S(DM (0--(((~()))) r= 1,.,.., R.
QMC
k=1
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The overall point estimate is their average
OMC PN
1
OMC
Let ( (2 be the sample variance of ju
AMCC
PR
QMC
The Student confidence interval is given
by
QMC R
CIMCC (1 a)
PN
QMC
=
L o/2
VR
where 1-0/1 is the (1 (1-aa22)-uun qquantie of the t-distribution with R 1 degrees of freedom. the estimates fi M,MC are normally distributed, this confidence interval has exact coverage probability 1 a. In general, the validity of Student & confidence interval is justified by CLT. An extensive numerical study by L'Ecuyer et al. (2023) demonstrates that Student intervals achieve the desired coverage empirically.
Exploring different configurations of R and n This experiment aims to understand how different configurations of R. the number of replicates, and n, the size of the QMC point set, affect the CI length of the RQMC method. The total budget of function evaluations is fixed at Rn = 3200, to be consistent with the AMC and MC experiments. We consider the four image metries used Section and one additional image metric, MUSIQ.
For RQMC, each configuration was repeated five times, and the results were averaged to ensure stability. AMC and MC each consist of single run over 3200 images. All experiments are conducted using the CIFAR1O dataset.
The results are shown in Table 11 and underlined values indicate the best CI length among the threc QMC configurations, while bold values indicate the best length across all methods, including MC and AMC. For the brightness and pixel mean metrics, the configuration with point set size = 64 and number of replicates R = 50 reduces CI length the most. In contrast and centroid, the configuration with larger has better CI length. For MUSIQ, more complex metric, changes in CI lengths across configurations are subtle, and the configuration with the largest R has the shortest CI length. While no single configuration consistently advantages, all RQMC and antithetic sampling methods outperform plain MC.
Table 11: Average 95% CI length and relative efficiencies (vs MC baseline). The first three rows are for RQMC methods, with the configuration of R x indicated in the first column.
R x n
Brightness
Mean
Contrast
Centroid
MUSIQ
Cl
Eff.
CI
Eff.
CI
Eff.
CI
Eff.
CI
Eff.
25 x 128
0.37
29.81
0.40
25.71
0.22
24.57
0.04
8.17
0.13
1.11
50 x 64
0.35
32.05
0.39
26.94
0.22
23.43
0.04
7.35
0.13
1.13
200 x 16
0.47
18.38
0.49
17.30
0.29
14.20
0.05
5.84
0.12
1.21
AMC
0.35
32.66
0.39
27.12
0.23
22.05
0.04
6.96
0.13
1.06
MC
2.00
2.04
1.08
0.11
0.13
C.6.3 DPS EXPERIMENT IMPLEMENTATION
We evaluate the confidence interval length reduction benefits of antithetic initial noise across three common image inverse problems: super-resolution, Gaussian deblurring, and inpainting. For each task, the forward measurement operator is applied to the true image, and noisy observations generated using Gaussian noise with = 0.05. We use the official implementation of DPS (Chung al. 2023a) with the following parameters:
Super-resolution uses the resolutionnn operator with an input shape of (1, 3, 256, 256) and an upsampling scale factor of This models the standard bicubic downsampling scenario followed by Gaussian noise corruption.
Gaussian deblurring employs the gaussian blur operator, again with a kernel size of 61 but with intensity set to I, which is intended to test the variance reduction in simpler inverse scenario.
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Inpainting is set up using the inpainting operator with random binary mask applied to the input image. The missing pixel ratio is drawn from a uniform range between 30% and 70%, and the image size is fixed at 256 256. A higher guidance scale (0.5) is used to compensate for the sparsity of observed pixels.
As shown in Table 3 AMC consistently achieves shorter CI lengths than MC across all tasks and metries without sacrificing reconstruction quality, implied by the LI and PSNR metries measure the difference between reconstructed images and ground truth images.
C.6.4 DDS DATASET
Data used in the DDS experiment on uncertainty quantification (Section 4 were obtained from the NYU fastMRI Initiative database (Knoll et al. 2020 Zbonntar al. 2018). listing of NYU fastMRI investigators, subject to updates, can be found at: fastmri. med. nyu edu The primary goal of fastMRI is to test whether machine learning can aid in the reconstruction of medical images.
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D
MORE VISUALIZATIONS
Positive
Negative
Random 1
Random
L
Figure 19: CelebA-HQ Image Generated
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Positive
Negative
Random 1
Random 2
Figure 20: CelebA-HQ Image Generated
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Postziive
Negative
Random
Random
Figure 21: CelebA-HO Image Generated
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Posilive
Neqalive
Random
Ramdem
Figure 22: DiT Class 974; geyser
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Posilive
Neqalive
Random
Ramdem
Figure 23: DiT Class 387: lesser panda, red panda
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Pusilive
Neqalive
Random
Ramdem
Figure 24: DiT Clas 979; valley, vale
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Pusilive
Neqalive
Random
Ramdem
Figure 25: DiT Class 388: giant panda, panda
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Positive
Negative
Random 1
Random 2
Figure 26: Prompt: "most expensive sports car"
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Positive
Negative
Positive
Neqative
Ter
Figure 27: Partial Negation of LSUNN-hurch
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Positive
Negative
Positive
Negative
Figure 28: Partial Negation of LSUN-Bedroom
47


Reviews:
[{"review_id": "g2CZRdpTxF", "reviewer": "Reviewer_cieF", "summary": "The paper discovers a universal negative correlation in outputs from diffusion models when pairing each initial Gaussian noise  with its negation (“antithetic noise”). This holds across datasets, architectures (U-Net, DiT), unconditional/conditional sampling, DDIM / DDPM, and even VAEs / Normalizing Flows. The authors explain the phenomenon with a symmetry conjecture: the learned score function is approximately affine antisymmetric (odd up to a constant), supported by temporal correlation analyses, 1D slices of score outputs, and theory in the high-noise regime via the OU process. Leveraging the strong negative correlation, they propose antithetic Monte Carlo for uncertainty quantification, yielding up to 90% tighter confidence intervals and large efficiency gains, and show complementary benefits from randomized QMC. Applications include estimating pixel-wise statistics, evaluating diffusion inverse solvers (DPS/DDS), and improving diversity, as well as training-free and model-agnostic editing.", "questions": "In addition to the weaknesses outlined in points (i-iii), I present the following questions for the authors to address:\n\n(1) How does the negative correlation behave under varying classifier-free guidance scales and different guidance strategies?\n\n(2) Can the method improve semantics in text-image alignment uncertainty, and object counts/ positions beyond pixel statistics (a qualitative study would suffice)?\n\n(3) How does antithetic pairing interact with distillation methods (e.g., Consistency Models) and very few-step generation (few-step generation would suffice if not applicable to consistency models)?\n\n(4) While results seem promising, I miss a detailed discussion of limitations. Are there scenarios where antithetic pairing reduces diversity or text adherence?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "(i) The paper proposes a simple, training-free, model-agnostic procedure with no runtime overhead\n\n(ii) The paper features broad empirical validation across datasets, samplers (DDIM/DDPM), architectures, and even VAEs/flows\n\n(iii) Antithetic Monte Carlo is well motivated and yields substantial, measurable variance reduction\n\n(iv) Beyond the main paper, the work provides good reproducibility details and extensive appendices", "weaknesses": "(i) While theoretical support for the symmetry conjecture is partial and focused on high-noise regimes, the symmetry conjecture remains unproven in generality. The novelty lies mostly in documenting the phenomenon antithetic variance reduction and its strength in diffusion models.\n\n(ii) Many results use pixel-level correlations and simple statistics, while semantic-level uncertainty and quality metrics (e.g., FID, CLIP alignment) are less explored or could be expanded upon more.\n\n(iii) Conditional settings (e.g., strong CFG guidance, complex prompts) and additional methods (e.g., Consistency Models) are not exhaustively studied.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qQuPB3Al20", "reviewer": "Reviewer_JEpj", "summary": "This paper shows that in the diffusion model sampling, pairing each noise sample with its negation induces a strong negative correlation in the generated outputs.  The authors also note that this effect is universal, observed across datasets and architectures, and even in other generative models such as VAEs and normalizing flows. They claim both empirical and theoretical support for a symmetry conjecture that the learned score function exhibits approximate affine antisymmetry. The resulting negative correlation enables substantially improved uncertainty quantification, significantly reduces the uncertainty, suggested by  90% narrower confidence intervals", "questions": "Please refer to the weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The study of antithetical initial noise in diffusion models appears new to me. \n2. Although the empirical results are not perfectly aligned with the theoretical argument in Lemma 1, the characterization is interesting and could inspire future studies. \n3. The paper is well-written and easy to follow.", "weaknesses": "1. The work would be more complete if the authors could provide some discussion on why the score function admits this affine antisymmetric property.\n2. While the work shows that the found fact could result in new methods to do uncertain quantification with significantly narrower CI, as we deal with the generative models, we can potentially generate an infinite number of samples, which potentially drops the significance of the proposed methods. \n3. I am a bit confused about the results in Table 1. It appears that models and datasets are mixed up.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DaeAbkxwBl", "reviewer": "Reviewer_mwy8", "summary": "This paper investigates the use of antithetic samples in the context of diffusion models. After presenting observational evidence in the form of pixel-wise correlation statistics of generated images across different datasets and models, the authors formulate the conjecture that the learned score network is an odd function up to a constant shift at every fixed time t.\n\nMotivated by the negative correlation of samples generated from antithetic noise samples, the authors propose using antithetic sampling in diffusion models for various tasks such as uncertainty quantification and inverse problems.\n\nFinally, they show that an antithetic Monte Carlo scheme significantly outperforms classical MC in terms of confidence interval width and overall sample diversity.\n\nOverall, this paper reads as an advanced study of an expected statistical phenomenon underpinned by a well-known variance-reduction technique, rather than a fundamentally novel contribution.", "questions": "- The benefits from the conjecture are not stated explicitly, e.g., how could one benefit from this underlying symmetry when designing/training a diffusion model?\n\n- Are there ways to further validate the conjecture in the small-noise regime from a theoretical perspective?\n\n- Could the authors propose hypotheses about the underlying mechanisms (either due to training algorithm/data/architecture) that might lead to the conjectured behaviour?\n\n- Could other control variate schemes be explored?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "- The paper is clearly presented and easy to follow.\n- The underlying principle is simple and elegant and can be implemented easily.\n- The conjecture is simple and relevant to the evidence presented.\n- The experiments are thorough and convincing in terms of the superiority of antithetic vs independent sampling for uncertainty quantification and posterior sampling.", "weaknesses": "- The novelty feels quite limited. Antithetic sampling is a well-established variance-reduction technique, and the fact that fully negatively correlated samples passed into what is essentially a flow map yield back partially negatively correlated outputs feels like expected behaviour, not really a \"universal discovery\".\n\n- The theoretical justification of the conjecture in the high-noise regime, although nicely presented, is fairly trivial since it mostly relies on the well-known fact that the score function of a Gaussian distribution is linear.\n\n- The paper gives very little insight into what might be happening close to the target distribution p_0 (i.e., the data manifold). The omission feels significant, as it is clearly the region where getting the score function right matters most. Figure 3 clearly shows that the strong negative correlation breaks down faster as we approach the data distribution.\n\n- Instead of focusing on the conjecture as a pure observational fact, the paper could benefit from either going deeper in the theoretical justification of the conjecture (in the likes of what is done using the FKG inequality in the appendix), or going deeper in the direction of using control variates to improve the efficiency of statistic computation/uncertainty quantification of diffusion models.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "BUufhz6YjI", "reviewer": "Reviewer_NGgz", "summary": "The paper investigates the effect of sampling noise in generative image models by pairing each noise vector with its negation and analyzing the resulting generated images. The authors find that this antithetic noise approach produces samples that are strongly negatively correlated, revealing an approximate antisymmetry in the learned score function of diffusion models. They verify this symmetry statistically and propose it as a general property of generative models, extending beyond diffusion to VAEs and normalizing flows. Leveraging this finding, they show that antithetic noise can substantially improve uncertainty quantification, yielding tighter and more reliable confidence intervals.", "questions": "1. Did you experiment with partial negation of the noise vector (e.g., negating only the first half or specific components of $z$) to analyze how localized symmetry affects the generated outputs?\n\n2. Can you envision an analogous approach for LLMs, where an equivalent notion of antithetic sampling might reveal symmetry or uncertainty properties in text generation?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1. The paper is thorough both empirically and theoretically, providing strong support for its claims through extensive experiments and proofs.\n\n2. The results are substantial and appear to be novel, as the proposed method generalizes across different classes of generative models for image synthesis.\n\n3. The proposed antithetic noise technique requires no retraining, making it practical and simple to apply.\n\n4. The paper offers a compelling theoretical explanation for the observed symmetry through the antisymmetry of the score function, which deepens understanding of diffusion model behavior.", "weaknesses": "1. Some of the writing could be improved; for instance, the first sentence of the abstract (“We initiate…”) reads awkwardly.\n\n2. The paper would benefit from an evaluation on OOD data, along with quantitative uncertainty metrics such as AUROC or AUPRC, to better demonstrate the method’s reliability in uncertainty estimation.\n\n3. The work discusses uncertainty improvement but does not clearly distinguish between different types of uncertainty (e.g., aleatoric vs. epistemic), which would help clarify the scope and interpretation of the results.\n\n4. Several relevant works on uncertainty in diffusion models are not cited:\n\n[1] Berry, Lucas, Axel Brando, and David Meger. \"Shedding light on large generative networks: Estimating epistemic uncertainty in diffusion models.\" The 40th Conference on Uncertainty in Artificial Intelligence. 2024.\n\n[2] Berry, Lucas, et al. \"Seeing the Unseen: How EMoE Unveils Bias in Text-to-Image Diffusion Models.\" arXiv preprint arXiv:2505.13273 (2025).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)