Moment- and Power-Spectrum-Based Gaussianity Regularization for Text-to-Image Models
Authors: Jisung Hwang
Jaihoon Kim
Minhyuk Sung
Venue: NeurIPS 2025 poster ()
Keywords: Normal Distribution
Gaussian Distribution
Text-to-Image Models
Primary area: deep_learning
TL;DR: We propose a novel regularization loss that enforces standard Gaussianity, encouraging samples to align with a standard Gaussian distribution.

Abstract: We propose a novel regularization loss that enforces standard Gaussianity, encouraging samples to align with a standard Gaussian distribution. This facilitates a range of downstream tasks involving optimization in the latent space of text-to-image models.
We treat elements of a high-dimensional sample as one-dimensional standard Gaussian variables and define a composite loss that combines moment-based regularization in the spatial domain with power spectrum-based regularization in the spectral domain. Since the expected values of moments and power spectrum distributions are analytically known, the loss promotes conformity to these properties. To ensure permutation invariance, the losses are applied to randomly permuted inputs. Notably, existing Gaussianity-based regularizations fall within our unified framework: some correspond to moment losses of specific orders, while the previous covariance-matching loss is equivalent to our spectral loss but incurs higher time complexity due to its spatial-domain computation. 
We showcase the application of our regularization in generative modeling for test-time reward alignment with a text-to-image model, specifically to enhance aesthetics and text alignment. Our regularization outperforms previous Gaussianity regularization, effectively prevents reward hacking and accelerates convergence.

Moment- and Power-Spectrum-Based Gaussianity Regularization for Text-to-lmage Models
Jisung Hwang
Jaihoon Kim
Minhyuk Sung
KAIST
(4011hjs,jh27kkt mhsung) tkaist Ckaist.ac ac.kr
Abstract
We propose a novel regularization loss that enforces standard Gaussianity, encour- aging samples align with standard Gaussian distribution. This facilitates range of downstream tasks involving optimization in the latent space of text-to-image models. We treat elements of a high-dimensional sample one-dimensional stan- dard Gaussian variables and define a composite loss that combines moment-baa based regularization the spatial domain with power spectrum bbased regularization in the spectral domain. Since the expected values of moments and power spec- trum distributions are analytically known, the loss promotes conformity to these properties. To ensure permutation invariance, the losses are applied to randomly permuted inputs. Notably, existing Gaussianity-baase
regularizations fall within our unified framework: some correspond to moment losses of specific orders, while the previous covariance-matching loss equivalent to our spectral loss but incurs higher time complexity due to its spatial-domain computation. We showcase the application of our regularization in generative modeling for test-time reward alignment with text-to-image model, specifically enhance aesthetics and text alignment. Our regularization outperforms previous Gaussianity regularization, effectively prevents reward hacking and accelerates convergence.
Introduction
The Gaussian distribution plays a central role in numerous machine learning applications, widely used model measurement noise, uncertainty, and the mean of independent samples. Its mathematical simplicity and analytical tractability have made it default modeling choice in many contexts. In particular, within generative modeling, the standard Gaussian is commonly used the latent distribution mapped to complex data distributions.
Given the pervasive presence of Gaussianity not only in machine learning but also across the broader sciences, quantifying how closely a data point follows-or deviates from a Gaussian distribution,
ie., measuring Gaussianity, has become fundamental technique [1 32 9 22 13
30 16
10 3 In generative modeling in particular, measuring Gaussianity can facilitate optimization by identifying the latent Gaussian variable that best maps desired data point, thereby enabling more precise and controllable generation. For example, in the context of widely used text-to-image models (20] II 6 23 25), prior work has shown that Gaussianity- regularization can improve
downstream tasks such as aesthetic image generation 333 and text-image alignment [1]].
In this work, study the structural properties of the standard Gaussian distribution and propose novel regularization loss that unifies various existing approaches under a unified theoretical framework. Leveraging the identity covariance of the standard Gaussian, treat high- dimenmiieiii sample as a collection of i.i.d. scalar standard Gaussian variables-that is, as samples drawn from a one-dimensional standard Gaussian. With specific ordering, we analyze the distribution of these scalar values in both the one-dimensional spatial and spectral domains. In the spatial domain, we utilize the moments of the samples-the expected values of their powers-and define moment-based
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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regularization loss. Notably, previously proposed losses such as norm-based [27 12], kurtosis- based 77. and <L-divergence-basee [18] regularization can be interpreted as, or shown to be asymptotically equivalent to, specific instances of this moment-based formulation.
Spatial-domain regularization alone is often insufficient in generative modeling, it can leave residual patterns that lead to mappings to unrealistic data points (Figure [ (c)). To address this, we further analyze the samples in the spectral domain and introduce an additional regularization loss based on the fact that the empirical power spectrum of i.i.d. Gaussian samples follows a chi- square distribution. This corresponds spatial domain, aligning with prior work [33]. However, by operating in the spectral domain, we reduce the computational complexity from O(D²) O( D). thereby eliminating the need for dimension wise sampling.
fitting the sample covariance to the identity matrix in the
Crucially, because both spatial and spectral regularization should hold regardless of the ordering of elements in high-ddmeeniioal Gaussian sample, we enforce permutation invariance by applying our losses to randomly permuted versions of the input.
As applications Gaussianity regularization, inspired by Eyring et al. [12] we showcase test-time reward alignments using one-step text-to-image model [20], specifically for enhancing image aesthetics and text alignment. Reward alignment refers to the task of optimizing the latent sample of generative model so that the resulting output maximizes given reward function, such as aesthetic quality or textual alignment. However, directly optimizing the latent can lead to overfitting to the reward signal-known realism). Our Gaussianity regularization effectively mitigates this issue by encouraging the optimized latents to remain close to the original Gaussian prior. Across both aesthetic and text alignment tasks, our method outperforms existing regularization approaches, achieving the highest scores across all metrics while preventing reward hacking and accelerating convergence.
reward whicchh often degrades output quality (e.g., reduced image
2 Related Work
2.1 Gaussianity Testing and Regularization
The Gaussian distribution is fundamental component in statisties, and numerous methods have been developed to assess whether set of samples conforms to it. Classical tests such as K-S, A-D, and CvM N1
measure discrepancies between empirical and theoretical cumulative distribution functions. Other approaches assess Gaussianity using order statistics (31) or quantile alignment 30) These methods, however, are based on non-differentiable functions and are therefore limited in their applicability to optimization-base techniques in machine learning.
32
2
g [2
There have also been differentiable approaches applied as regularization across range of machine learning applications. A classical example is the KL divergence used in variational autoencoders (VAEs) [8] measure the difference between the approximate posterior and a standard Gaussian prior though this operates on distributional parameters rather than directly on samples. Chmiel et al. introduced kurtosis-baat based regularizer that penalizes heavy-tailed behavior in model weights to improve training stability. More recently, norm-based regularization (27 [2] 4 has been used to constrain latent vectors lie hyperspherical shell, matching the expected norm of unit Gaussian. However, these methods only address marginal statistics and do not account for inter-component dependencies. We propose unified approach that incorporates these approaches as components, with the relationships detailed in Section 3.1.3
Probability- Regullii-- Noise Optimization (PRNO) 33 is notable example that aligns the empirical covariance matrix of latent samples with the identity, thereby capturing inter-component dependencies. While effective, this method incurs quadratic memory and time complexity, making it less scalable in high-dimensional settings. In our unified framework, we propose a more efficient regularization that achieves the same objective but in the spectral domain, with details provided in Section 3 3.2.2 and Appendix C
2.2 Reward Alignment via Latent Noise Optimization
Reward alignment refers to the process of steering generative models to produce outputs that maximize a given reward function. While previous approaches using direct fine-tuning 26 g reinforcement learning [5] 37 have been widely studied, these methods are computationally expensive and require retraining the model for each new poarnng significant scalability challenges.

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(a) Neither Domain Matched
(b) Spectral Domain Matched
Spatial CDF
Spectral CDF
Lateni
Sampled Imugee
Spotial CDF
Spectral CDF
Latent
Sampled Image
Spatial Domain Matched
(d) Both Domains Matched
D1
0.004
0.455
0.004
0.002
Spatial CDF
Spectral CDF
Latent
Sampled Image
Spatial CDF
Spectral CDF
Latent
Sampled Image
Figure I: Effect Spatial and Spectral Distribution. Each block presents the spatial and spectral CDFs, a latent visualization, and the generated image. The numeric value in each CDF plot quantifies the deviation from the ideal Gaussian distribution (lower is better). All images are generated using FLUX [20] with the prompt "A car." When both spatial and spectral properties matched (d), the output clean and realistic; mismatches either domain lead to visible degradation in quality.
In contrast, noise optimization offers an efficient alternative by directly optimizing the initial Gaussian noise at inference time, improving the outputs of pretrained generative models without additional training. Due to its effectiveness, noise optimization
been widely applied in various domains, including images (27 12 4 33,, motion [1 38), and music 29
Recently, ReNO [12 demonstrated that noise optimization can be effectively applied to one-step generative models, offering superior reward alignment compared to multi-step models thanks to the much clearer expectation of the final output from the latent sample. However, ReNO builds on previous Gaussianity regularization method [27], which covers only part of our unified framework. As result, the optimized latent samples often deviate from the standard Gaussian prior, leading to degraded image quality. We show that our unified regularization more effectively preserves Gaussianity, maintaining high image quality while maximizing the reward.
Regularization for Standard Gaussianity
We propose a regularization term that captures the structural propertics of the standard Gaussian distribution and unifies several existing methods. Let (.1,....>)) with i..dd. components, I; ~ (O0,, 1). We specifically view x as an ordered sequence of samples, where any random permutation should yield statistically equivalent vector. Thus, must exhibit both the correct marginal distribution and no inter-component dependencies. A valid regularization must enforce both aspects of i.i.d. Gaussianity.
TD) RDD be latent vector
With a specific ordering of the sample sequence, our method enforces Gaussianity in both the spatial and spectral domains. In the spatial domain, match moments of the elements in generalizing several existing methods such as KL-divergence--aat norm-based (27 regularization. In the spectral domain, match the empirical power spectrum, leveraging the fact that the empirical power spectrum of i.i.d. standard Gaussian samples follows a chi-squared distribution.
[18), kurtosis-based [7], and
observe that this dual-domain regularization is especially crucial for the latent samples of generative models, which assume an i.i.d. standard Gaussian prior. As shown in Figure enforcing Gaussianity in only one domain is insufficient-spatian and spectral properties are complementary and must both be satisfied to replicate the behavior of true Gaussian samples. By jointly regularizing both domains, our method yields more faithful latent representations and improves generative performance.
In the following subsections, we formalize the spatial and spectral regularization terms in Section 3.] and 3.2 respectively, and present our final loss formulation Section 3.3

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3.1
Regularization in Spatial Domain
In this subsection, we introduce a regularization term based on the moment properties of the standard Gaussian distribution in the spatial domain. We show that this approach unifies several existing regularization methods as special cases of moment matching the spatial domain.
3.1.1 Moment Conditions of Standard Gaussianity
To formally justify our approach, we begin by recalling a classical result that uniquely characterizes the standard Gaussian distribution via its moments.
Theorem 1. Let X be real-valued random variable. Suppose that for all integers 2 0. the n-th moment of X, defined as [[ satisfies
n odd,
E E
(2k)
if 2 is even.
2
Then X follows the standard Gaussian distribution, i.e., X ~N(0,1)
Proof. We consider the Moment- Generattn Function (MGF) of X:
X"
22 (2k)
00
2
Elexx]
EE
EX
- (2)
n
(2k
2*
2
Since this matches the MGF of the standard Gaussian distribution (O, 1), and the MGF uniquely determines the distribution, we conclude that X ~ N0,11,
3.1.2 Moment-Based Regularization Loss
Motivated by Theorem [ we define the following moment- based loss term to enforce the n-th moment condition of the standard Gaussian distribution:
1/n
Cn
Σ
D
pn
1/n
where μ4, denotes the theoretical n-th moment of the standard Gaussian distribution, specified in Theorem T This loss penalizes differences between the empirical n-th moment and its target value, encouraging each latent component match the desired marginal distribution.
The computation Cn scales linearly with the latent dimension D, resulting in both time and memory complexity of ODD). This makes the loss highly efficient and suitable for application to high-dimensional latent spaces commonly used modern generative models.
3.1.3 Connection to Existing Regularization Terms
The moment-based regularization introduced above is designed to reproduce the marginal statisti- cal properties of a standard Gaussian distribution in the spatial domain. Several widely adopted regularization methods can be viewed as specific instances constrained approximations of this principle. Below, we revisit three representative methods-KL. divergence [88, kurtosis [7], and norm-based 22 12 losses and interpret them through the lens of spatial-domain moment matching.
KL Regularization Loss [8]] The Kullback-Leite (KL) divergence is widely used in the VAE framework to align the latent distribution with a standard Gaussian prior. Assuming that the empirical distribution E RDD is approximately Gaussian with empirical mean Px and variance of. the KI. divergence from the standard Gaussian is given by:
1 (v2 + of -og LKI. 2
log 1).
This loss penalizes mismatches the first and second moments the latent distribution, encouraging both the empirical mean and variance to match those of (0, 1). As such, minimizing the moment

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losses L1 and L2 effectively minimizes LKL. The KL loss thus serves as a compact surrogate for enforcing low-order moment alignment in the spatial domain.
Kurtosis Regularization Loss (7) Another approach focuses on matching the fourth central moment (kurtosis), which controls the tail behavior of the distribution. While this loss is often used to improve robustness against quantization noise in neural networks, it also serves as a valid constraint for Gaussianity enforcement. For standard Gaussian distribution, the kurtosis exactly The kurtosis regularization loss penalizes deviation from this value:
D
2
D Σ D
D 2 - 1x)
Lkan =
Fj ux
Px
Ji
ox = D
(5)
0x
This loss encourages the latent distribution match the Nourth-order structure of the standard Gaus- sian. When normalization by empirical mean and variance is omitted, minimizing Lkurt becomes equivalent minimizing the fourth-order moment loss L1. Even with normalization, jointly minimiz- ing L1, C2. and L4 naturally reduces the kurtosis loss. Thus, Caun can be interpreted as constrained variant of higher-order moment regularization the spatial domain.
Norm Regularization Loss [27 12]. A common approach is to penalize deviations in the & norm of the latent vector x E RD, assuming that each x; is independently drawn from a standard Gaussian distribution. In this case, the norm ||x|22 follows a chi distribution XD. whose probability density function is given by:
Pnor (r)
D-1 e E r
(6)
2 2 -
Maximizing the likelihood of this distribution leads to the norm-based loss:
||x l1 (D log ||x||2 +
Luora (x) - log Promm [|xx|2]
77)
where is a constant independent x. This loss is minimized when the squared norm satisfies |xx1 =
D - 1, which corresponds to an average squared component magnitude of 1 ΣD x2 =1 D. As D > oc, this converges to the second moment of the standard Gaussian. Thus, Lnom implicitly enforces the second moment condition in the spatial domain and is asymptotically equivalent to our moment loss L2.
In summary, many existing regularization terms can be reinterpreted as moment- matthntg strategies in the spatial domain. Our framework generalizes these ideas by explicitly formulating and unifying them through the lens of empirical moment alignment.
3.2 Regularization in Spectral Domain
As shown in Figure (c), even a latent vector has the correct marginal distribution in the spatial domain, spectral mismatch can degrade generative performance. This reflects a failure to fully reproduce the i.i.d. nature of the Gaussian prior. Hence, in this subsection, we propose regularization term enforce the spectral-domain properties of the standard Gaussian.
While the moment- based loss (Equation 3 in the previous section invariant under permutations, spectral regularization is sensitive to the ordering of vector elements. The discrete Fourier transform (DFT) treats the latent vector as a structured signal, and structural dependencies -uchh as spatial correlations appear in its frequency components. We focus the power spectrum, defined as PA = E[ I), which captures how signal energy is distributed across frequencies.
We show that for i.i.d. Gaussian vectors, the normalized DFT magnitudes | disl/yD follow X2/ 22/22 distribution for most k. Based on this, we design a loss that aligns the empirical power spectrum with its expected distribution. We also relate our method to the covariance-based regularization in PRNO (33] highlighting the efficiency and theoretical grounding of our approach.
3.2.1 Spectral Distribution of Standard Gaussianity
We begin by analyzing the statistical distribution of the square root of the empirical power spectrum, defined as the magnitude of the DFT coefficients; [&z], where & DFT(x). When the latent vector
5

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X E IRD consists of i.i.d. standard Gaussian samples, this distribution exhibits well-defined behavior that forms the theoretical basis for our spectral regularization.
Lemma 1. Let RR be random vector with i.i.d. elements 24 ~ (0, 1), and let & DFT(x).
Assume D even. Then,
X2/V2
if k (0, otherwise.
(8)
X1
The proof provided Appendix B This lemma shows that when the latent vector consists of i.i.d. standard Gaussian samples, the magnitudes of its discrete Fourier transform (DFT) coefficients follow specific scaled chi distributions. For most frequency indices k & D/2), |&xl/VD x2/v2, while for l 0 and & D/2, |& X1.
In practice, the latent dimension D is large (e-g.. 65,536 FLUX 20 ) so the bulk of the spectrum follows the X2/ /2 distribution. Since this property is preserved under permutation, we apply our spectral loss after randomly shuffling the latent vector to remove any ordering bias.
3.2.2 Power-Spectrum-Base Regularization Loss
As shown in Section 3.2.1 the magnitudes of the normalized DFT coefficients of i.i.d. standard Gaus- sian samples whiih correspond to the square roots of the empirical power follow a X2/ 2 distribution for most frequency indices. A naïve approach, inspired by norm-based regularization 27), is construct spectral loss by applying the negative log-likelihood of this distribution to the empirical power spectrum. Specifically, we define the loss as follows:
2
Cspectral all
Σ
log
VD
D
where the probability density function of X2/ 2 is given by f(r) = 2r e-r² for 2 0.
The X2/ v2/v2 distribution inherently exhibits high vari- ance, as illustrated in Figure 2 However, minimizing fispccraa_nii drives all spectral components toward its
Probability Density
X2lv2 2
E16X22/2 2]
peak (i.e., = 1/- 1/)2), concentrating the spectrum a single value. This contradicts the natural variability of the underlying distribution and fails to faithfully re- produce its spread. As a result, this loss formulation overly sharpens the empirical power spectrum distri- bution, ultimately distorting the marginal distribution of the latent vector and degrading its Gaussianity.
Hpower
0.0 0.5
1.0
2.0
Figure 2: Distribution of X2 / and mean of 16 independent X2/v2 samples.
To mitigate this issue, we adopt an alternative strat- egy that preserves variance while promoting spectral alignment: instead of applying the loss to individual coefficients, we compute over the mean of randomly sampled frequency subsets. As shown in Figure the distribution of the sample mean over 16 i.i.d. X2 2 variables exhibits bell-shaped curve with significantly reduced variance, while still reflecting the underlying distribution.
Based on this observation, we define our power spectrum regularization loss as follows:
+ VD
power
/pooer
(10)
B
B
BEB
KEB
where B denotes the set of batches, and B represents the indices within each batch. In our experiments, we set the batch size to B 16, and the target mean to /1power 0.875, which approximates the expected value of X2/ 22.
This batched averaging preserves natural variation across the spectrum and avoids collapsing the distri- bution. It also reduces the influence of outlier frequencies ent distributions. Notably, this regularization achieves the same objective as PRNO B3 minimizing
and h: D/2, which follow differ-

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INITIAL LATENT
Checkerboard
KL 18
(Equation 4)
JOK iters.. 11.2 sec.
Spatial CDF
Spectral CDF
Laleni
Sampled Imugee
Spatial CDF
Spectral CDF
Latent
Sampled Image
Kurtosis
(Equation 5
IOK iters., 16.1
Norm 27
(Equation 7
10K iters.. 10.4 sec.
1,708
Spatial CDF
Spectral CDF
Latent
Sampled Image
Spatial CDF
Spectral CDF
Latent
Sampled Image
PRNO
100 iters., sec.
C (Equation
100 iters., 0.26 sec.
0.332
0.305
0.003
0.000
Spatial CDF
Spectral CDF
Latent
Sampled Image
Spatial CDF
Spectral
Latent
Sampled Image
Figure 3: Effectiveness of Regularization Losses in Guiding Latents. A latent vector initialized with checkerboard pattern is optimized using various regularization losses. Each block follows the format of Figure Images are generated by FLUX 20 with the prompt "A guitar".
deviation from identity covariance- but enforces it in the spectral domain in a more efficient and effective manner. detailed discussion of this connection is provided in Appendix
Computational Perspective. Calculating Cpower is highly efficient due to the use of the Fast Fourier Transform (FFT) algorithm. Unlike the naive discrete Fourier transform, which has computational complexity of (D2)) the FFT reduces this to O D log D). Furthermore, FFT operations are inherently parallelizable and benefit significantly from GPU acceleration. Given that FFT core component in many scientific computing libraries, it is already highly optimized in most modern frameworks. As result, we observed no significant increase in runtime compared to simpler O(D))basee based regularization methods.
3.3 Our Gaussianity Regularization Loss
Table 1: Comparison of regularization methods for standard Gaussianity.
By combining the two loss components the moment regularization loss (Equation 3) and the power spectrum regularization loss (Equa-
Time Complexity
Connection Complexity with Our Loss
Memory
Method
tion 10 we define our final regularization loss as:
KL
18
O(D)
O(D)
Li La
Kurtosis
O(D)
O(D)
LA
Norm
27
O(D)
O(D)
Le
Lx(0,)) = Σ Ln Xpowerfquwer. nEK
(11)
PRNO
O(Dk) O(Dk)
L1, Epwwer
Ours
O(Dlog
O(D)
where denotes the set of moments used for matching. our experiments, we set {1,2), as enforcing the first and second moments, together with the power spectrum regularization Lpowers is empirically sufficient to approximate the standard Gaussian distribution. We Apower 25.0.
3.4 Toy Experiment Using an Image Generative Model
We conduct toy experiment using the image generative model FLUX 20 to evaluate how different regularization terms guide latent vector toward standard Gaussian distribution when optimized from a highly structured initialization. The initial latent is set to a checkerboard pattern, and the results 77. and Norm reduee deviation in the spatial domain but retain strong checkerboard artifacts even after 10K optimization iterations. PRNO 33 improves alignment in both spatial and spectral domains, yet the latent still exhibits visible structure and the sampled image shows unnatural textures. In contrast, method
are shown in Figure 3 Spatial-only methods- KL [8] Kurtosis

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Acsthetic Image Generation
Text- Text-Aligned Aligned Image Generation
0.24
0.30
0.700
Image Reword
07
0.V1
0.005
0.45
No
0.00
0.40
PRMO
0.30
Oum
0.75
Orn
10
06.0
0242
5.390
MUS5
0.23
6:48
GUSS
Prckscore
Figure 4: Quantitative Results for Aesthetic Image Generation Text-Aligned Image Generation. Curves show performance at 100-iteration intervals, with dot sizes indicating progress from 100 to 500 iterations. Points closer to the upper right represent better trade-offs between the given reward (X-axis) and held-out reward (y-axis). Our method reaches the highest reward with equal iterations and consistently yields better trade-offs on HPSv2 35 and ImageReward [36].
effectively matches distributions in both domains, producing clean, high-quality output from noise-like latent-while requiring approximately 50x less time than PRNO.
4 Applications: Reward Alignment in Text-to-Image Generative Models
Inspired by the previous work, ReNO [12), we present two applications of reward alignment in a one- step text-to-image generative model: aesthetic image generation and text-aligned image generation.
Baselines. In all experiments, we use FLUX 20 the base generative model, which a one-step text-to-image model. We compare our regularization method against KL [18), Kurtosis [7], ReNO (Norm) [2], and PRNO 33
Additionally, we report two reference baselines: one without any optimization (No Opt.) and one without regularization (No Reg.).
Implementation Details. We initialize the latent vector from the prior distribution (a unit Gaussian) and perform optimization for 500 iterations using Nesterov momentum with a coefficient of 0.9 and gradient clipping set to 0.01. The generated images are evaluated every 100 iterations. The learning rate is set to 0.1 for aesthetic score (28) and 1.0 for PickScore [19]. We set the regularization coefficient to 2.0 for all regularization methods. The regularization gradient normalized and scaled to match the magnitude of the reward gradient, ensuring balanced contributions during optimization. All experiments were conducted on an NVIDIA A6000 GPU with 48GB VRAM, taking approximately minutes per 100 optimization iterations.
4.1 Aesthetic Image Generation
We use aesthetic score (28). which measures the visual appeal of an image, as the given reward- the objective used optimize the latent. Evaluation is conducted on the animal prompts from DDPO [5] We report the given reward along with held-out rewards-Immgell ImageReward (36) and HPSv2 (35 are not used during optimization and serve to assess image quality and text alignment.
whhch
Results. We present quantitative and qualitative results in Figure and Figure 5 respectively. Figure[Jshow curves the baseline aesthetic score plotted against the held-out rewards. The size of each dot indicates the number of optimization iterations, shown in intervals of 100 iterations.
Notably, optimizing the latent solely based on the given reward-withou any regularization--eadl to phenomenon known reward hacking, where the model exploits flaws in the reward function to achieve higher scores without improving. or even degrading. the actual image quality. This is evidenced by the steady decline of the blue curve in Figure 4 which indicates that the image quasiry-masurred by the held-out rewards such as HPSv2 (35] and ImageReward [36] deteriorates as the optimization progresses, despite the given reward increasing. The corresponding visual degradation also apparent the cartoon-like artifacts observed in the generated samples (column No Reg. in Figure While previous works introduce regularization terms that enforce Gaussianity, these methods fall short as they fail to capture both the spatial and spectral properties of unit Gaussian, leading to suboptimal results (see KL, Kurtosis, and ReNO columns in Figure 5) In contrast, our regularization is robust to reward hacking and consistently achieves the highest scores across all metrics-outperrfeeee outperforming ReNO [2 and PRNO
33 at every optimization iteration.
8

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No Opt.
No Reg.
KL
18
Kurtosis
ReNO
12
PRNO 33
Ours
"Ilama"
Aesthetic Image Generation
"camel"
"An oval serving dish and square bowl."
Text-Aligned Image Generation
"The designer carefully selected fabrics and colors for the upcoming fashion line."
Figure 5: Qualitative Results for Aesthetic and Text-Aligned Image Generation. Our method generates images with higher aesthetic quality and better prompt alignment compared baselines [18 33 and not regularized optimization.
4.2 Text-Aligned Image Generation
For text-aligned image generation, image-text alignment and perceptual image quality. This setup is a special case because the given reward is closely aligned with the objective of Gaussianity regularization which also improves the perceptual quality of the generated images in text-to-image generative models [20]. We evaluate on 60 prompts sampled from T21-CompBench++ [15). comprising 10 prompts from each of the six categories: 3D spatial, complex, non-spatial, shape, spatial, and texture.
PickScore 19] the given reward, which measures both
Results. Quantitative and qualitative results are presented Figure and and Figure respectively. shown in Figure methods incorporating Gaussianity regularization initially exhibit strong positive scaling trend, reflecting the close alignment between the PickScore reward and the regularization objective. However, these regularization methods-incluse spatial-domain constraints [8 [2], and quickly plateau in performance-aa limitation further evidenced by the suboptimal generation quality shown in Figure By explicitly guiding latent vectors to stay close to the unit Gaussian manifold, our method promotes stable gradient flow property also noted in prior work (24 14 21. As result, it achieves higher rewards throughout optimization and outperforms baselines with fewer updates.
the No Reg. case-rely solely
Conclusion
We introduced unified regularization framework for enforcing standard Gaussianity. Unlike prior approaches that focus solely on marginal statistics or covariance matching, our method captures both spatial and spectral properties, improving conformity to unit Gaussian distribution while remaining computationally efficient. We validated the effectiveness of our approach on reward alignment tasks in text-to-image generative models, including aesthetic and text-aligned image generation. Our method consistently outperforms existing Gaussianity regularization techniques by mitigating reward hacking and accelerating convergence. These results underscore the importance of structured Gaussianity enforcement and open up directions for broader applications.

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Limitation and Societal Impacts. While our loss effectively guides latent vectors toward Gaussian- ity during optimization and serves well as regularization objective, we observe that its value alone does not reliably indicate how closely given latent matches a true standard Gaussian distribution. That is, low loss value does not necessarily guarantee full Gaussianity. Developing an efficient and principled metric for directly evaluating Gaussianity remains an open direction for future research. In addition, as our method builds on pretrained generative models, which may have been trained on uncurated datasets, it inherits the inherent biases and artifacts of the underlying base model. This may lead to the unintended generation of biased or undesirable visual content.
Acknowledgements
This work was supported by the NRF of Korea (RS-20023-000009223)) ITPP grants (RS-2022- 11220594, RS-2023-00227592, RS-2024-00399817, RS-2025-25441311, RS-2025-25443318 RS- 2025-026531 and the Technology Innovation Program (RS-2025-023173266, all funded by the Korean government (MSIT and MOTIE), well as by the DRB-KAIST SketchTheFuture Research Center.
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Appendix
Stroke-based Image Generation
Inpui
image
image
Input
Gen. imuge!
imuge
Reward Alignment given Lightness
0 iter.
100 iters.
500 iters.
iter.
100 iters.
500 iters.
Reward Alignment given Darkness
0 iter.
100 iters.
500 iters.
iter.
100 iters.
500 iters.
Figure A.I: Showcase of Our Regularization Term in Other Applications. We demonstrate the versatility of our regularization three additional tasks: stroke-based image generation (top) and reward aligmment under lightness (middle) and darkness (bottom). In the first row, images are generated from coarse stroke imputs, where our regularization helps produce semantically faithful and visually coherent outputs. In the second and third rows, we apply reward-guine guided optimization to adjust image lightness or darkness over time, showing smooth and stable progression from the initial sample (0 iters.) to reward-aligned outputs (500 iters.).
A Other Applications
To demonstrate the broader applicability of our proposed regularization, present three additional applications, as shown in Figure A. stroke-based image generation (top), and reward alignment based on lightness and darkness objectives (middle and bottom, respectively).
In the stroke-based image generation task, the input a coarse solor-stroke map that encodes rough spatial layout and scene structure. We begin by performing image inversion using the FLUX-dev (0) model to obtain an initial latent representation. We then optimize this latent by minimizing the distance to the inverted latent while applying our regularization loss, which encourages the optimized latent to remain close to both the standard Gaussian manifold and the original inversion. The final image is generated from the optimized latent. This optimization step helps preserve semantic fidelity to the stroke input while enabling the generation of photorealistic images. As shown in the first row of Figure A.I our method produces semantically aligned and visually coherent results that maintain the structure and framing of the original strokes.
In the second and third rows, we present reward- reward-aligned image generation for lightness and darkness. For the lightness task, the reward is defined as the mean of pixel values-encon encouainn brighter outputs-whibe for the darkness task, we use the negative of this mean to promote darker results. We apply our regularization with these rewards during latent optimization. In both cases, our method effectively guides the latent toward the desired visual attribute without sacrificing image quality.
19

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B Proof for Lemmal
Lemma A.1. Let x RD be random vector whose elements are i.i.d. samples from the unit Gaussian distribution, i.e,, ~ (N),,) 1). Let & DFT(x) E CD denote its discrete Fourier transform. Then:
For all k & {0, D/2] (assuming D is even), the magnitude ( follows a scaled chi distribution with 2 degrees of freedom:
X2
(12)
VD
V2
For k = 0 and k = D/2, where Sy. E R, the magnitude follows scaled chi distribution with degree of freedom:
VD
XI.
(13)
Proof. Let = (wo; 11,
LD-1)
ERD with i..dd In ~N(0,) 1). The DFT of x is defined as
D-1
In
e 2 k= 0, +
D 1
(14)
7=0
This is linear transformation with complex coefficients of unit magnitude. For & (0. D/23, by. complex with real and imaginary parts
D-1
2
B((x)= = Σ
In cos
(15)
D
n=0
D
3(*))
Σ
sin
(16)
D
#=0
which are independent Gaussian variables with variances
2xrkn
Var(HR(fx) = Σ cos?
(17)
D
2
22rkn
Var(S(2x)) = Σ sin?
(18)
(D
2
Thus, (((x), ((2, and the magnitude satisfies
[xx|
DD22 X2.
(19)
which implies
h
X2
(20)
VD
V2
For E (0, D/2) (when /) is even), the Fourier coefficients are real:
D
20
Tn
(21)
D-1
In
(-1"
(22)
==0
Each is sum of D i.i.d. standard Gaussian variables, so &x N(0, D) and hence
|
VD.XI
X1-
(23)
DD
20

---

C Connection to Probability-Regulariz Noise Optimization (PRNO)
In this section, we show that the power spectrum regularization Lpower 10
a practical surrogate for the likelihood-style regularization used in Probabilit-RRggularized Noise Optimization (PRNO) 33
Intuitively, the regularization Ppwwr in the spectral domain spreads energy evenly across frequencies and suppresses global offsets. Theorem A.1 formalizes this by proving that minimizing Lpower drives the first and second moments to approximately and 1, respectively, and renders the latent vector rotation-ison isotropic (i.e.. with no preferred direction). Then, Theorem A.2 shows that these properties align with the objectives of PRNO by pushing the latent vector toward maximal likelihood in broad range of cases.
We first introduce the notation and setup, and then present the two theorems with their proofs.
Setup and notation. Let D mk PB > 10,000 be even, with the spatial block size fixed to (as the official PRNO implementation code (33) and the frequency block size set B = 16 (these constants are used in all our experiments). Let IT E RDxD be a uniformly random permutation matrix and let DDD denote the DFT matrix. For any x RR, define the permuted signal and its DFT by
x(T)= F
(24)
Frequency blocks. For Bp = {pB,
(p + 1)B 1} with selector Rs, E {0,1) define
x|0 (1I) = Rs, x(II) Ca.
(25)
Then, Lpower (T0) can be written as follow:
1 P-1
Cpower (Ilx) = Σ
| (II) ||
Hpower
(26)
DD
p=0
Spatial blocks. For K.: (ik, .... (i 1)k 1} with selector Rx, E (0, 10,1) define
i (II) = Rx, (Ix) E
(27)
Define the block mean and block covariance
m-1
m S(T) m
xII)=
(T).
(II) ( (TI) (I))
(28)
m
Diagnostics:
Mi (IIx) = ||x(1|]
M2(IIx) = ||S(II) 14 11
(29)
For M 2 0 define
(M( = min 2exp( 12 M²) }}
(30)
((M)= = (VT + M - /k/m
(31)
+²
Pa(M) min { 2 eexp( ( 1).
(32)
PRNO 33 maximizes En [log P1 (M, (IIx))) and En [log P2 (Ma ([(x)))
Theorem A.1. Assume that x minimizes Crower (Ix))] Then
D 1-1
| of =
#x
D
Σ 1=
x 20
0,
D
u R0 1
(33)
and x is -otation-isotropic..
21

---

Proof of Theorem A.1
Recall
P-1
power (Ix))
1
s (1I)
Hpower B
(34)
D
To analyze the minimizer of En Powerr (IIx)] we begin by investigating the expectation of the magnitude of each DFT coefficient.
Let x E R and write x - Px1 + r with Σ/= Tj = 0 and ト Σー 1 xx Let la be
non-DChnon-Nyquist DFT row of F, so fyl = 1, E fj = O. and Σド |fz| = D.
Since Σ, fg =0
vD II)) = vD
(fx. IIr).
(35)
Define oj : ://VDDDss so that E0 Cj 0 and
D-1 =
ley1 2 = 1. Then
D 1
Y =
(fe. 1l) = Σ
(36)
VD
1=
Because IT is uniform and the population mean of [ry} is 0,
En[y] 0.
(37)
Using the known covariances of sampling without replacement, Var(rug)) Cov(TTT Tm))) 1 for i #
a and
Then, we obtain
D-1 Varm(Y) = 2
D-1
0x
D EaE = (1+ 1 ox = D-1 2f (1+D_)
1el D
G.
(38)
0=
Y is sum of many small, mean-zero terms sampled without replacement; by Hoeffding's combina- torial CLT (or Lindeberg-ty type arguments),
D-1 Y = Σ =o CN(0, D-1 oz {
(39)
1-0
If Z ~ C. then ||| = La 2 σ. Hence,
ET[[ ~ Vā D
En
(fe, (fe,
2 1 D - 0x.
(40)
For the DC DFT row,
1
fo 1,
(fo, Ix)
VD (1,x) VD 1x-
(41)
VD
Thus,
1
En
(fo,
mx]||
=VD 11xl-
(42)
VD
For the Nyquist DFT row fn/2 we take f, = 1111,, c, (-JDD=- R and
D-1
(ED/2) 11r)
Σ
(TT(()) E R.
(43)
1=0
22

---

As before,
Varn (1) (Eo/2. lrr))
1
En
Ir)) M1)
0,
ox
(44)
D
D-1
By imite-population (permutation) CLT in the real case,
1
(fD/2. IT)) 2 N N(1, D-1 12)
(45)
VD
and therefore, using E|
v 2/ a for Z ~ N(0,002)
1
2 D
D
E
(fD 2fD lxx
~
V ox.
(46)
We analyze y,, J
|
(1))l1
Ppower B, which constitutes Spower (Ixx).
The expectation over all permutations is
(x+1)B-1
En[Yp= =
Σ
En
(fx. lx)|
Mpower
(47)
k=pB
DC-including block (p- 0).
VT
D
Em = -B
+ VDlus - Ppower B.
(48)
2
D-1
Nyquist-including block (p- P.
En[Vp/2] (B - T
D
2
D
D-1
0x
+
D
Mpower B.
(49)
Ordinary (non-DChon-Nyquuis) blocks.
VT
D
En
B
2
D-1
0x
Ppower B.
(50)
By convexity of | |. for each block mean absolute deviation equals absolute bias + nonnegative fluctuation. Using the block-wise biases En (p). we obtain
P-1
En[ Cpower (Txx)] = D
Σ P=0
En Y + +(x),
(51)
where +xx) 2 is the fluctuation term
To minimize Enj Spower (Ix))., we first consider minimizing
[Y [n]] + |nn[ +
[En[Y]]]
(52)
p U
The only term containing #x is En (Yol Thus, setting
V
D
|ux|
VD MMpower B- (B-
xx
(53)
D
minimizes the |m [Y]] to 0.
23

---

The rest terms are only dependent to 0x
Eu(Yr/2l] +
Σ
En
(B-1)
VT
D
+
2
D
0x
Mpower B.
2
D
D-
D
2#0.2
VT
D
(P-
B
0x
Mpower
D-1
(54)
This is minimized when
D-B B
Ox
Poower
(55)
BD 1)
vR
D
D
V
Given that B 16, ppower 0.875, and D 10a, 0x 0.987, and |xx 0.009. Therefore,
Ux Ri 0,
xx R with error of roughly 0.01.
(56)
Next, let us consider the fluctuation term. For easier notations, set g(t) == It Mpooer B and define. for each block p.
(p(1x;x) == VD |
hx(x) = (1,xxx Ep(1l:x) = Ep(1x;x) - 1xp(x)) (57)
By Jensen's inequality,
Enl)(2p)] (()((x) + Ap(x), and hence
Ap(x) = En((((p(()()))) + + - -((b 20, (58)
P-1
P-
Em| Soower (Ix)) = D
Σ p=0
(lpp(x)) + f(x),
x(x) = 1 ツ=ク
Σ
(x)
(59)
Let be the uniform probability measure on O(D), independent of x, and let
Xiso
Rx,
R ~ U,
(60)
be an isotropized representative. Define the rotation-averaged block-mean profile
(x(x) En hp (Ptx Er 17 Rx)] hp (xisn),
p 0, ... P 1.
(61)
Applying Jensen the scalar convex map g after exposing both sources of randomness (R. II) yields. for each p.
(Z, Rx))] 2 9 (pT Rx)) = (())(x)).
(62)
Using Equation 588 for Rx and taking expectation in R.
En.ulg(Z, Z2 Rx))] En| (((p)((((x))))
En (Rx)) 2 ((T((xx) (Rx)) (63)
where also used Jensen y obtain E (Rxx)) (x),(x)). On the other hand, evaluating Equation (58) at Xiso and using Equation (11.
Enfg(Z,, (II; xiso))] 9 (x) + (X
(64)
By convexity and permutation/DFT invariance, the mapping x + Ax(x) is convex in the (unordered) block-magne -aggnitge profile, so averaging rotations cannot increase it; hence
(xioo) < (Rx),,
p p= P 1.
(65)
Summing the last two displays over p and dividing by D gives
EulCpowes (Ix s ER-D Cpower (IIR))..
(66)
This establishes that the isotropic law has expected loss larger than the rotation-average of the loss for x.
24

---

Theorem A.2. Assume that x minimizes Enf Crower (Ixx)]. Then,
Pr (M) =1) 2 0.883,
Pr P2 =1) 2 0.999.
(67)
Consequently, for the majority permutations IT, pi
pz (M2((T)))) 1.
Proof of Theurem A.2
We first show for pi (TTxx) and then show for p2(xx([Ixx).
Proof of P1.
Recall
(T) E MI(IIx) = ||x(1]]
(68)
m
Because II is uniform permutation and the spatial blocks are formed by reshaping into blocks of size k, each coordinate of x(1I) is the average of m distinct entries sampled without replacement from the D-point population {x, DD-1 Hence, for each coordinate a E (1,...,k},
{x));=o!
[n[x((1)a)
Px-
(69)
For the variance, write X(II). = Ei X,, where XI...., Xm are the m draws without replace- ment from {xy}. Then
m
Var X
Var(X,) + Σ Cor(X,,X,)
(70)
m
itj
A direct counting argument for sampling without replacement gives Var(X; o² and, for + j. Cov(X, X,) = / 1). Substituting.
m² mo + -mmm - (10-0-
oZ 0
D - m
Var(X)
(71)
D-1
Therefore,
1 D 2
Varn (x(11)a)
m D xx
(72)
For the cross cross-covariance of two distinct coordinates a * of x(1I), use the same decomposition and the same counting covariance for distinct draws:
Covm (x(I),,, x(I)))
Σ Cov(X ,x()))
(73)
m
i,=1
m²
m²
DD D
o
)====
=-
m D
1 1
o2
(74)
Collecting the coordinates,
2
Covn (x(II)
m D-1 ( - m) Is -(11 It))
(75)
Since D m > 7500 > 1, the covariance can be approximated as
(x(I)) m D D m)4 L..
(76)
At the Em [powwr) minimizer, fxx 2 0 and of ~ 1. Therefore, Equation 69 75
become
1 D Ix-
[II)((I) = 0, Covn(x([II) m D
(77)
25

---

The minimizer is also rotation
rotation-isotropic isotropic. Under isotropy and m >> 1, multivariate combinatorial CLT (sampling without replacement) implies that
x(II ~ M(0,
m(D 1) lx).
D- m
(78)
In particular,
(I(Ix)² = ||( d D - m
m(D - 1) Xi-
(79)
From Equation 79
Pr log - m
M 2) = Pr(x 2k log
D D-1)
(80)
2k
D - m
For k and m D/4 we have D f b). and the xi CDF is x z/2). Therefore the probability admits the exact closed form
-
=
Pr Pr log 2 m (Txx) 20) = 1 2- 1+ P
16 11
11
log2)
(81)
g
3
D
which is increasing in D and, for every D) 10 exceeds
1 1-2
16/3 (1
16 3
log 2) > 1 1- 2 4 (1-10))
1
16
3
(1-
10
log 2
~
0.883.
(82)
Combining with Equation 11) gives
Pr 11
log nh M1 (1lx)² 20) 2 0.883.
(83)
By the definition p1 (M) = min (2xp( 2x M 1}.
log 2 21 M 2 0} { log (Mi)) = =0} [PP((M)) =13.
(84)
Therefore,
Pr (p( =1) 2 0.883.
(85)
I1
Proof of P2-
Recall
S(I): =
x i (1I) (11)
E lxk M2(Ixx) = |[S(1]) - 1x11.
(86)
Fix particular block and coordinates # b. Then,
En (x [IT) x (IT) D(D 1) 344
Σ xjxx = D(D D(D-1) -1)
x
(87)
j=1
Using ED Σp-13 x,= Dux and Ep-133 x3 D(122 + o2), we rewrite Equation
87 as
12 D + +
02
En
x (1) (II)
= 11
(88)
D(D 1)
D-1
Likewise, for a b (a single position),
1
En
x
(89)
D
Since all blocks are identically distributed under the permutation, Equation 88 block i.
889
hold for every
26

---

Let SIII E B denote the block sample covariance
S(
1
Σ IT] ( (I))
(90)
m
Taking permutation expectation and using linearity,
En [S(IT)
1
Σ En
[x
(
(TT)
[ x(1) (m) (m)],
(91)
so it suffices compute the expectation for single block, already encoded by Equation (88
89
Collecting the diagonal and off-diagonal entries:
o --1
(En(S(II)) = = + o2
(En(S(II)),,. ab 2he
(a + b).
(92)
In matrix form,
d
En = 11 11 + +
D-1 (1I -In).
(93)
Let
W =
E R
u span
u' = (uE (v, = =0,
(94)
V
and the associated orthogonal projectors
p== unl
11,
Pi : P.
(95)
We the identities
11 kP,
I = P1, 11 - I (k- 1)P Pl,
(96)
to expand En (S(I)) along P @ PI. Subtract Jg to focus on the deviation:
of
[n( - I = 2 11 + +
(11 -In) Ik.
(97)
D
Insert the projector decompositions:
2x
= + P1) - D -
((k: 1)P - -P)) - PL).
(98)
Group the P and P1 coefficients separately:
A 2= - 1-
k: ---6
xx P+
0xx 1 1+ D-1
1 2
P1
(99)
D-1
Thus A is diagonal in the orthogonal decomposition @ 4 with eigenvalues
k 1
\1 = oz 1+ ky
xx
(100)
1 02
t 0% - -
+
(101)
D 1
Since is diagonal on 1/ @ L
[S(()) Is. 11| = || max (Ayl: [di l}.
(102)
At the minimizer of En En[[pooee [Ix))], Theorem A.1 implies μx ~ 0, of x 1. Applying these to
Equation (100) (TOT) gives
h: 1 k- D-1* 1'
Pail 0 1 1 k
(103)
D
27

---

Trl = 0x 1 D
=
(104)
D-1
For k: and > 104 we have k < D, so both right-hand sides are of order 1/D and therefore negligible at the fluctuation scales considered below. Consequently,
[[][)( Is l2 = max{ Ayp [rrl}
(105)
Hence, at the minimizer, (IIx governed by fluctuations around Ig rather than by deterministic
bias.
To obtain an exact i.i.d. model for the fluctuations, let
g (o, g = (g g (m ())
g(t E IR*
(106)
so g( i... NO0,, Tt). Enforce q = by projecting onto the sphere:
c:= VD
cg.
(g
llol22
(107)
Thus
SIII)
Σ
x0)
=²²
m Σ 00) ( (t)
m S Wisharts (m, Ig).
(108)
Hence the only dependence among the x is the common scalar By x² concentration,
D
1 = 1 + 1/(D-113).
(109)
Using the operator-nowe norm triangle inequality,
[III)) 1+1l = ||c2s, Ixll le2 - 1| |S, 1xll.
(110)
Since | Amax (8,) concentrates (1+ with 7 := k/m, the first term is Oy neg-
ligible compared to the spectral-edge fluctuation scale m 2/3 that drives the second term. Therefore, up vanishing radial error, M2(IIx) is governed by the Wishart shape fluctuations of Sg.
Let Amax denote the maximum eigenvalue S, and set 7 = k/m. The (orthogonal) Tracy-Widom edge law (with & fixed, m - xi) states
Amax - (1+ 72
4 &, E~ TWT.
(111)
(1 /7)1/3 2/3
and the upper tail satisfies Pr(s > 1) I exp( exp(-21'0 2 3/22)
Expanding the square root at the edge
(1 + 7T2 gives
- vi) + 2/) 1/3
-
m
4/3 4/3 ( (1+
(112)
4
Because ||, T&ll
Amax 1, Equation 112
controls the surrogate of the shape fluctuation.
From Equation 110 and (09)..
M2(IIx) |I Iall
||S, Ixll op().
(113)
Since æ + (VI + - v is 1-Lipschitz a neighborhood of the edge, we may transfer
Equation (12) to S(II):
46 (M2((Ix)) = (V1 (I1k)))( S (1+ 2/3
m 4/3 (E+)2 (2+)2 (1+o(1)), (114)
4
with the same & ~~ T,1, The o(1) is uniform for fixed k and
28

---

The clipping event (log 2 4 (M22) 2 0} fails only if
4 log 2
(Mb((Ix))
(115)
m
By Equation 114). this implies
16 16log
1/2
{ > fm = ( + 1/32/3
m 1/6
(116)
Using Pr(f > t) 1 exp( 2///2) yields
Pr log2 n. (I2 <0) x exp
16
(log 2)3/1
m
(117)
3
(1+ + //)///2 k1/4
For k 4, k2/4 = 22, with D/4 we have y 16/D and v 4/VD 0.04, hence
1/2 z VI.0I. Writing (1+ /3)
16 (log 3/4
Cmin
v2 v1.04 2.80,
(118)
and using m2 = 1VD 50 for D > 10a Equation
gives
Pr log - T (M2 2 0) 2 1 1-exp(- - mmin 2 1 exp(-140)
(119)
By the definition Pz(M) == min { 22xpp( - 10--TTC0 (MM)). }}
(M2) { (lgg (M2(((( 2 0} = flogm (M2) =0] = -fpa(Mb) = 13. (120)
Therefore,
Pr (2(2) =1) 2 1 exp(-140) > 0.999. I1
(121)
D Ablation Studies
We present results for text-aligned image generation with varying Apower and IC
Apwer
0
5
10
25
50
100
PickScoret
0.2481
0.2516
0.2513
0.2548
0.2521
0.2533
HPSv21
0.2968
0.2974
0.3032
0.3028
0.3060
0.3042
Table A. Quantitative results with varying Apower.
We observe that performance remains stable across a wide range Apower values and consistently improves relative to the case Apower 0. These results suggest that our loss is not sensitive this parameter. We chose 25.0 considering the trade-off.
K
0
(1)
(1,2)
(1,2,3,4)
(1....,6)
(L....,8)
PickScore?
0.2484
0,2500
0.2548
0.2510
0.2521
0.2525
HPSv21
0.2933
0.2962
0.3028
0.3012
0.2999
0.2984
Table A.2: Quantitative results with varying K..
The performance with {1,2} shows clear improvement over both 0 and (1). However, we did not observe further gains when extending to higher-order moment terms. The first two terms were sufficient to surpass baseline performance.
29

---

E Full Quantitative Results
We present the full quantitative results for the two main applications. Table A.3 reports results for aesthetic image generation, while Table A.4e results for text-aligned image generation.
Given
100 iterations
200
Revations
300 iterations
400 izeriations
500 iterations
Aesthetic
Score
Aest.
7
Image
HPSV2T
Aest..
Image
(t HPSV2?
Score
Aest.
Image Reward
HPSV2f
Aest. c Image
HPSV21
Aest. Image ?
HPSV21
No Opt.
5,9880
0.8299
0.2958
55.9880
0.8299
0.2958
5,9880
0.8299
0.2958
3,9880
0.8299
0.2958
5,9880
0,8299
0.2958
No Reg.
6.3861
0.3988
0.2726
7,0175
0.4076
0.2679
7.4236
0,3418
0.2645
7.8216
0.3514
0.2639
8.1079
0,3460
0.2641
KL. 18
6,3499
0.7132
0.2863
6.7753
0.5287
0.2794
7.3469
0.4812
0.2749
7.8422
0.4815
0,2749
8.1393
0.4225
0.2735
Kurtosis
6.5597
0.7932
0.2930
6.9711
0.7446
0.2900
7,4069
0.7424
0.2897
7.7676
0.7329
0.2914
8.0477
0,8041
0.2920
RcNO
6.5455
0.8331
0.2937
7.0600
0.7545
0.2915
7.4974
0.7913
0.2918
7.8621
0.7493
0.2908
8,1508
0.7749
0.2883
PRNO
6,5860
0.8235
0.2913
6.9030
0.7689
0.2869
7.2650
0.7203
0.2830
7.7159
0.6493
0.2836
8.0760
0.6375
0.2833
Ours
6,6110
0.8798 0.2964
7.1356
0.7538 0.2922
7.6589 0.7837 0.2923
8.1061 0.8243
0.2934
8,4435
0.8397 0.2927
Table A.3: Quantitative Results Aesthetic Image Generation. We report the aesthetic score
values used as given reward during inference-time optimization. To assess generalization across reward metries, we also include ImageReward and I/PSv2 held-out rewards Bold values indicate the best performance in each metric at each optimization iteration.
Given
700 iteraiinss
200 Reeations
300 iterations
400 iterrations
500 iterations
Pickscore
Pick
Reward
HPSM21
HPSv27 Reward
Pick
Reward
HPSv2f
Pick
Reward
Image
HPSV21
Pick
Rewir!
HPSV21
No Opt.
0.2246
0.4825
0.2752
0.2246
0.4825
0.2752
0.2246
0.4825
0.2752
0.22246
0.4825
0.2752
0.2244
0.4825
0.2752
No Rog.
0.2327
0.4911
0.2795
0.2374
0.5139
0.2825
0,2412
0.5193
0.2868
0.2430
0.5238
0,2868
0.2445
0,5198
0.2876
KL [8
0.339
0.2382
0.3750
0.2806
0.2416
0.2874
0.2834
0.24-10
0.2464
0.4127
0.2876
Kurlosis
7
0.2327
0.4587
0.2751
0.2377
0.5405
0.2793
0.2407
0.5405
0.2823
0.2440
0.5322
0,2853
0.2459
0,5315
0.2863
ReNO
0.2342
0,4615
0.2819
0.2388
0.2871
0.2436
0.5245
0.2913
0.2461
0.2485
0,5343
0.2945
PRNO
33
0.2339
0.534]
0.2785
0.2408
0.5548
0,2843
0,2449
0.5647
0.2876
0.2481
0.5495
0.2896
0.2514
0.6061
0.2936
Ours
0.2360
0.5514 0.2862
0.2422 0.5900
0.2931
0.2480
0.5824 0.2985
0.2514 0.5625 0,3001
0.2548
0.5897 0.3028
Table A.4: Quantitative Results on Text-Aligned Image Generation. We report the Pickscore
values used as given reward during inference timee optimization. To assess generalization across
reward metrics, we also include ImageReward and I/PSv2 as held-out rewards Bold values indicate the best performance each metric at each optimization iteration.
30


Reviews:
[{"review_id": "at41h88KQE", "reviewer": "Reviewer_Thdj", "summary": "The Gaussian distribution plays a central role in optimizing diffusion-based models. In particular, when optimizing diffusion for reward alignment, it is important to incorporate a regularization technique to minimize reward hacking and other negative effects of optimization. This paper investigates existing regularization techniques using the structural properties of the standard Gaussian distribution and proposes a novel regularization method that incorporates information from both spatial and spectral domains. The paper performs experiments on reward alignment in a one-step image generation model and text-aligned image generation. The results show that, compared to baselines such as no regularization, KL divergence, kurtosis, ReNO, and PRNO, the proposed regularization technique outperforms them all.", "questions": "- How sensitive is \\mathcal{K} in the joint objective?\n- Did you experiment with moments higher than 2?\n- How sensitive are the experiments to \\lambda_\\text{power}?\n- Is the joint objective just a combination of KL and PRNO? If so, can you elaborate on why regularization is needed in both the spectral and spatial domains simultaneously?\n- Could you explain how spectral and spatial domain regularization work together?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\nWhen performing reward alignment, it is crucial to regularize models to prevent reward hacking and degradation in performance. This paper introduces a novel, simplified formulation of alignment, synthesizing existing work by demonstrating that most current approaches are optimizing a subset of moments of a Gaussian distribution that have a general form, as presented. Not only is the paper well-written, but it also clarifies how the proposed idea relates to previous methods, making the unification clear. \n\nWeakness:\n\nThe paper appears to lack clear ablation studies demonstrating how sensitive the $\\mathcal{K}$ or $\\lambda_\\text{power}$ parameters are in practice. For example, presenting ablation experiments with different \\mathcal{K} and \\lambda_\\text{power} values would offer valuable insight into which factor has the greatest impact. Additionally, since the observation indicates that KL matches the first two moments, it's important to clarify whether the primary contribution of the experiments is in adding the constraint within the spectrum domain, given \\mathcal{K} is set to either 1 or 2.", "was_revised": true, "final_justification": "The author addressed my concerns about specific conceptual details in the paper.", "rebuttal": "Thank you for your thoughtful review and valuable comments. We address the concerns in detail below.\n\n---\n***Q:*** *How sensitive is $\\mathcal{K}$ in the joint objective?*; *Did you experiment with moments higher than 2?*\n\n***A:*** We report additional results with different $\\mathcal{K}$ values in text-aligned image generation:\n\n| $\\mathcal{K}$ | $\\emptyset$| \\{1\\} | \\{1,2\\} | \\{1,2,3,4\\} | \\{1,...,6\\} | \\{1,...,8\\} |\n|---|---|---|---|---|---|---|\n| **PickScore**$\\uparrow$ | 0.2484 | 0.2500 | 0.2548 | 0.2510 | 0.2521 | 0.2525 |\n| **HPSv2**$\\uparrow$ | 0.2933 | 0.2962 | 0.3028 | 0.3012 | 0.2999 | 0.2984 |\n\nThe performance with \\{1,2\\} is clearly improved compared to $\\emptyset$ and \\{1\\}. However, we did not observe further improvements when extending to higher-order moment losses. The first two terms were sufficient to surpass baseline performance, as our regularization loss was designed not only to incorporate moment-based losses but also to include a power-spectrum-based loss that effectively suppresses residual spatial structure. (**Section 3.2**) Additionally, we provide a unified framework that integrates regularization losses which had previously been used independently. (**Section 3.1**)\n\n---\n***Q:*** *How sensitive are the experiments to $\\lambda_\\text{power}$?*\n\n***A:*** We present the following results on text-aligned image generation with varying $\\lambda_{\\text{power}}$:\n\n| $\\lambda_{\\text{power}}$| 0 | 5 | 10 | 25 | 50 | 100 |\n|---|---|---|---|---|---|---|\n| **PickScore**$\\uparrow$ | 0.2481 | 0.2516 | 0.2513 | 0.2548 | 0.2521 | 0.2533 |\n| **HPSv2**$\\uparrow$ | 0.2968 | 0.2974 | 0.3032 | 0.3028 | 0.3060 | 0.3042 |\n\nWe observe that the performance remains stable across a wide range of $\\lambda_{\\text{power}}$ values, and is consistently better than the case with $\\lambda_{\\text{power}} = 0$. These results indicate that the method is **not sensitive** to this parameter.\n\n---\n***Q:*** *whether the primary contribution of the experiments is in adding the constraint within the spectrum domain*\n\n***A:*** We would like to clarify that our contributions extend beyond simply introducing a spectral-domain constraint.\n\n- **Moment-based loss (Section 3.1):** Unifies independently developed regularization terms such as **KL divergence**, **norm**, and **kurtosis** losses within a single framework.\n\n- **Power-spectrum-based loss (Section 3.2):** Defined via the **DFT** in the spectral domain, it regularizes DFT coefficients to match their theoretical distribution. Unlike **PRNO**, which removes spatial correlation by aligning the empirical covariance matrix with the identity and incurs quadratic complexity, our method uses **FFT** for efficient computation with $O(D \\log D)$ complexity and operates via a fundamentally different mechanism.\n\n- **Dual-domain regularization (Section 3.3):** We apply spatial and spectral losses **jointly**, leveraging their **complementary strengths**—the spatial loss ensures marginal distribution alignment, while the spectral loss enforces spatial neutrality. (**L116**)\n\nTherefore, we emphasize that our contribution is not limited to the addition of a spectral loss, but lies in the integration and synergy of both domains within a coherent and efficient framework.\n\n---\n***Q:*** *Is the joint objective just a combination of KL and PRNO?*\n\n***A:*** Our loss is not simply a combination of KL and PRNO. We would like to clarify two key points:\n\n- The primary distinction lies between **PRNO** and our **power-spectrum-based loss**. PRNO suppresses spatial correlation by encouraging the empirical covariance matrix to approximate the identity matrix (**L80**), which incurs **quadratic complexity**. In contrast, our method leverages the **FFT** to compute **DFT coefficients** and regularizes them to match the theoretical distribution. This approach results in a significantly lower complexity ($O(D \\log D)$) (**L48**). As shown in **Figure 3**, PRNO takes **14.1 seconds** for 100 iterations, whereas our method requires only **0.26 seconds**, achieving over **50× speed-up**—while more effectively matching the Gaussian distribution.\n\n- We also emphasize the **complementary roles** of spatial and spectral regularization, as discussed in **Section 3** and illustrated in **Figure 1**. While both KL and PRNO do not address regularization in the spectral domain, our formulation incorporates **spectral-domain regularization** to more effectively enforce Gaussianity. Importantly, our regularization loss shows **superior performance** compared to these baselines in the experiments.\n\n---\n***Q:*** *Why regularization is needed in both the spectral and spatial domains simultaneously*; *How spectral and spatial domain regularization work together*\n\n***A:*** Moment-based losses **permutation-invariant** and therefore do not account for the **ordering** of elements in a sequence. (**L185**) For example, consider a set of 10K Gaussian samples that have been sorted. Although this sequence does not resemble a natural realization of random Gaussian samples—since such perfect ordering is highly improbable—the moment-based loss cannot distinguish it from a genuinely random Gaussian sample because it only matches marginal statistics.\n\nTo address this, we propose a power-spectrum-based loss, which is **permutation-variant** and sensitive to spatial correlations. (**L43**, **L186**) By analyzing the DFT coefficients, it captures deviations from white Gaussian noise and penalizes residual spatial structure. In **Lemma 1**, we provide a theoretical characterization of the expected distribution of the power spectrum under the assumption of independently sampled Gaussian noise in the spatial domain. This formal foundation justifies the use of spectral loss as a principled way to enforce spatial neutrality.\n\nHowever, the **spectral loss alone** does not ensure alignment with the target **marginal distribution**, as it focuses only on spatial structure. Therefore, **both losses must be used together**:\n\n- The **moment-based loss** ensures the correct marginal distribution.\n\n- The **spectral loss** enforces the absence of spatial correlation.\n\nThese two losses play **complementary roles** in fully regularizing toward Gaussianity (**L116**). To further support this, we provide visual evidence in **Figure 1**, which illustrates why regularization in both domains is necessary.\n\n---"}, {"review_id": "1iu2yyz2nN", "reviewer": "Reviewer_LvLi", "summary": "- Authors propose a spatial and spectral regularization loss that enforces gaussianity in latents used for reward alignment in text to image models. \n- In the spatial domain, authors treat a N dimensional gaussian as a collection of 1-D iid gaussian values and utilize the moments of the samples to define a moment-based regularization loss. Authors show that the commonly used KL-divergence, Kurtosis and Norm regularizations are special cases of this spatial domain gaussianity regularization loss.\n- Authors observed that while the latent can have correct marginals in the spatial domain but there can be a mismatch in the spectral domain. To correct this, authors propose another regularization loss in the spectral domain to enforce gaussianity. This is done by a loss that aligns the empirical power spectrum with its expected distribution i.e. $\\chi_2$ distribution.\n- Quantitative results on Aesthetic image generation and text-Aligned image generation show the effectiveness of the proposed regularizations in reward alignment.", "questions": "- In L314 authors claim that their method achieves higher rewards and outperforms baselines with fewer updates. Please provide experimental evidence to support this statement.\n- Authors only focus on text to image models for their downstream task. I would recommend rewriting the paper to focus directly on that and propose the two regularizations instead of the way it is written right now for me to improve my ratings. In its current form, I find the paper very hard to follow.", "limitations": "yes.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 1, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n- The unification of existing methods under the spatial regularization proposed in this work is very informative and useful to the community. \n- Qualitative and Quantitative results show the effectiveness of the proposed method.\n\n**Weaknesses**\n- The introduction and abstract was very hard to read. After reading the abstract, as a reader, I wasn't sure what was the application of the current method. Authors talk briefly about their method in the abstract but what is not clear to me was \"Why is any of this necessary?\". \nAuthors claim their method is useful for a range of downstream tasks but only provide results for reward alignment in text to image models. Then, a better way to organize the paper and the abstract would be to first state that for reward alignment, latents are optimized, and it is important to keep them close to the gaussian manifold. The paper would read very well if it is rewritten as a latent regularization loss for reward alignment instead of its current form. \n- Authors claim gaussianity is important and propose a spatial regularization term but end up using 1st and 2nd moments only in the loss. Why not enforce more terms since matching more moments implies getting much closer to the gaussian manifold.", "was_revised": true, "final_justification": "Authors have addressed my concerns about using higher order moments in the matching loss. Authors also said they would make changes to the text which is hard to verify before camera ready. I trust the authors to make the necessary chanes to make the paper more readable and vote to increase my score.", "rebuttal": "Thank you for taking the time to carefully review our work and provide valuable feedback. We address the concerns below.\n\n---\n***Q:*** *After reading the abstract, as a reader, I wasn't sure what was the application of the current method.*; *The paper would read very well if it is rewritten as a latent regularization loss for reward alignment instead of its current form.*\n\n***A:*** We appreciate your comment regarding the clarity of our paper. Our intention was to introduce the proposed regularization loss in a broader context, as Gaussianity is a fundamental concept with wide applicability across various domains. However, we understand that this broader framing may have made it difficult to identify the specific focus of our work. In response to your helpful suggestion, we will revise the paper to more clearly position it as a latent regularization method for reward alignment.\n\n---\n***Q:*** *Why not enforce more terms since matching more moments implies getting much closer to the gaussian manifold.*\n\n***A:*** We report additional results using different $\\mathcal{K}$ values in text-aligned image generation:\n\n| $\\mathcal{K}$ | $\\emptyset$| \\{1\\} | \\{1,2\\} | \\{1,2,3,4\\} | \\{1,...,6\\} | \\{1,...,8\\} |\n|---|---|---|---|---|---|---|\n| **PickScore**$\\uparrow$ | 0.2484 | 0.2500 | 0.2548 | 0.2510 | 0.2521 | 0.2525 |\n| **HPSv2**$\\uparrow$ | 0.2933 | 0.2962 | 0.3028 | 0.3012 | 0.2999 | 0.2984 |\n\nThe performance with \\{1,2\\} shows a clear performance improvement over both $\\emptyset$ and \\{1\\}. However, we did not observe further improvements when extending to higher-order moment losses. The first two terms were sufficient to surpass baseline performance, as our regularization loss was designed not only to incorporate moment-based losses but also to include a power-spectrum-based loss that effectively suppresses residual spatial structure. (**L42**, **L186**) Additionally, we provide a unified framework that integrates regularization losses which had previously been used independently. (**Section 3.1**)\n\n---\n***Q:*** *In L314 authors claim that their method achieves higher rewards and outperforms baselines with fewer updates. Please provide experimental evidence to support this statement.*\n\n***A:*** We acknowledge that this expression may be misleading. Our intended meaning is that the proposed regularization helps to reach *comparable* reward with *fewer* optimization steps. We will revise the sentence to improve clarity as follows:\n\n“*As a result, it achieves higher rewards over baselines for the same iterations.*”\n\n---"}, {"review_id": "ZQR1YNBQ5r", "reviewer": "Reviewer_p3aM", "summary": "This paper proposes a novel regularization loss function to enforce standard Gaussianity in the latent space of text-to-image generative models. The method combines moment-based regularization in the spatial domain with power spectrum-based regularization in the spectral domain, treating high-dimensional samples as collections of i.i.d. standard Gaussian variables. The authors demonstrate that several existing Gaussianity regularization methods (KL divergence, kurtosis, norm-based regularization) can be viewed as special cases of their unified framework. In the spectral domain, they leverage the fact that the power spectrum of i.i.d. Gaussian samples follows a chi-square distribution to design corresponding regularization terms. The method shows superior performance in test-time reward alignment tasks for text-to-image models, effectively preventing reward hacking and accelerating convergence.", "questions": "+ How were key parameters like λpower=25.0 chosen? Can you provide parameter sensitivity analysis or adaptive parameter selection methods?\n\n+ Given the acknowledged limitation that loss function values cannot reliably indicate Gaussianity, could you design better Gaussianity evaluation metrics? This is important for practical applications.\n\n+ How does the method perform on other generative models (e.g., Stable Diffusion, DALL-E)?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n+ The paper provides a unified theoretical framework that encompasses multiple existing Gaussianity regularization methods under the moment-matching paradigm, with Theorem 1 providing a rigorous characterization of standard Gaussian distributions via moment conditions.\n\n+ The combination of spatial and spectral domain regularization is innovative, particularly the use of DFT coefficient distributions to design spectral domain regularization.\n\n+ Compared to PRNO's O(D²) complexity, the method achieves O(D log D) time complexity through FFT, significantly improving efficiency.\n\n+ The method demonstrates superior performance on both aesthetic image generation and text-aligned image generation tasks, with detailed ablation studies.\n\nWeaknesses:\n\n+ While the paper provides theoretical foundations for moment conditions and spectral properties, it lacks deeper theoretical analysis of why dual-domain regularization is necessary.\n\n+ Key parameters like λpower=25.0 lack theoretical justification or systematic hyperparameter analysis.\n\n+ The authors acknowledge that their loss function value alone cannot reliably indicate how closely a latent vector matches a true standard Gaussian distribution, which is a significant limitation.\n\n+ Experiments are primarily conducted on the FLUX model, with insufficient validation of generalization to other generative models.", "was_revised": true, "final_justification": "After reviewing the rebuttal and discussions, I am inclined to support this paper. Given its originality and relevance, I believe this paper could be a strong candidate for acceptance at NeurIPS.", "rebuttal": "Thank you for taking the time to review our work and provide thoughtful and constructive feedback. We organize our responses into four categories:\n\n---\n***Q:*** *While the paper provides theoretical foundations for moment conditions and spectral properties, it lacks deeper theoretical analysis of why dual-domain regularization is necessary.*\n\n***A:*** Our core rationale for dual-domain regularization is rooted in the complementary limitations and strengths of moment-based and power-spectrum-based losses—an insight supported theoretically.\n\nMoment-based losses are **permutation-invariant** and therefore do not account for the **ordering** of elements in a sequence. (**L185**) For instance, consider a sorted sequence of 10K samples drawn from a standard Gaussian distribution. Although this sequence does not resemble a natural realization of random Gaussian samples—since such perfect ordering is highly improbable—the moment-based loss cannot distinguish it from a genuinely random Gaussian sample because it only matches marginal statistics. This limitation is theoretical, not merely empirical.\n\nTo address this, we propose a power-spectrum-based loss, which is **permutation-variant** and sensitive to spatial correlations. (**L43**, **L186**) By analyzing the DFT coefficients, it captures deviations from white Gaussian noise and penalizes residual spatial structure. In **Lemma 1**, we provide a theoretical characterization of the expected distribution of the power spectrum under the assumption of independently sampled Gaussian noise in the spatial domain. This formal foundation justifies the use of spectral loss as a principled way to enforce spatial neutrality.\n\nHowever, the **spectral loss alone** does not ensure alignment with the target **marginal distribution**, as it focuses only on spatial structure. Therefore, **both losses must be used together**:\n\n- The **moment-based loss** ensures the correct marginal distribution.\n\n- The **spectral loss** enforces the absence of spatial correlation.\n\nThese two losses play **complementary roles** in fully regularizing toward Gaussianity (**L116**). To further support this, we provide visual evidence in **Figure 1**, which illustrates why regularization in both domains is necessary.\n\nIf you have a specific theoretical direction or additional depth in mind, we would greatly appreciate further guidance and are open to incorporating it into the revision.\n\n\n---\n***Q:*** *How were key parameters like λpower=25.0 chosen?*\n\n***A:*** We present results on text-aligned image generation using different values of $\\lambda_{\\text{power}}$:\n\n| $\\lambda_{\\text{power}}$| 0 | 5 | 10 | 25 | 50 | 100 |\n|---|---|---|---|---|---|---|\n| **PickScore**$\\uparrow$ | 0.2481 | 0.2516 | 0.2513 | 0.2548 | 0.2521 | 0.2533 |\n| **HPSv2**$\\uparrow$ | 0.2968 | 0.2974 | 0.3032 | 0.3028 | 0.3060 | 0.3042 |\n\nWe observe that performance remains stable across a wide range of $\\lambda_{\\text{power}}$ values and consistently improves over the case with $\\lambda_{\\text{power}} = 0$. These results suggest that our loss is not sensitive to this parameter. We chose 25.0 considering the trade-off.\n\n---\n***Q:*** *Loss function values cannot reliably indicate Gaussianity, could you design better Gaussianity evaluation metrics?*\n\n***A:*** We would like to clarify the intent of our statement in L327. The key point of L327 is that while our loss is effective for regularization—optimizing it guides the latent toward the Gaussian manifold (**Figure 3**) and prevents deviation—its absolute value is not meant to serve as a global Gaussianity score for comparing arbitrarily distant points. As the title \"**Gaussianity Regularization**\" suggests, our focus is on **designing an effective regularization**, not a general-purpose metric. This distinction does not diminish its utility, as our experimental results consistently support its effectiveness in practice.\n\n---\n***Q:*** *How does the method perform on other generative models?*\n\n***A:*** We evaluated our method using the Stable Diffusion XL-Turbo (SDXL-Turbo) model for text-aligned image generation:\n\n| SDXL Turbo | ReNO | PRNO | Ours |\n|---|---|---|---|\n| **PickScore**$\\uparrow$ | 0.2564 | 0.2595 | **0.2628** |\n| **ImageReward**$\\uparrow$ | 0.6735 | 0.6430 | **0.7063** |\n\nOur method achieves **improved performance** over the two strongest baselines in both PickScore and ImageReward metrics, demonstrating its effectiveness beyond the originally tested model, FLUX. We will include this result in the revised version of the paper.\n\n---"}, {"review_id": "lvZDgML3WG", "reviewer": "Reviewer_mYPt", "summary": "1, This paper proposes a unified regularization framework for latent optimization in text-to-image generative models. Authors combine moment-based regularization in the spatial domain with power-spectrum-based regularization in the frequency domain to better enforce standard Gaussianity of latent variables.\n\n2. This approach is motivated by the limitations of existing regularizers (e.g., KL, norm, or kurtosis penalties), which may fail to eliminate residual structures or correlations in the latent space. By encouraging the latent to match both the marginal moment statistics and the expected power spectrum of white Gaussian noise, the method prevents reward hacking and improves optimization stability during test-time reward alignment. \n\n3, Experiments on aesthetic and text-alignment tasks with a one-step text-to-image model show that the proposed method achieves higher reward objectives while maintaining better image quality compared to existing approaches.", "questions": "Would the method generalize to other text-to-image model except FLUX, multi-step diffusion models or text-to-video models? It would be good to see the discussion.\n\nIs there more quantitative / qualitative results that support the conclusion? Right now, the experiments are very toy.", "limitations": "1.it would be good to expand on how latent regularization might propagate biases from the pre-trained generator.\n\n2. Only on a single text-to-image model (FLUX) with limited prompts are mainly evaluated.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strength: \n\n1. The method is well-motivated and theoretically grounded, and very good written.\n\n2. Reward hacking is well known issue and important topic.\n\nWeakness:\n\n1. This paper is mainly evaluated only on a single text-to-image model (FLUX) with limited prompts. Broader evaluation on other architectures or diffusion models would strengthen the paper.\n\n2. It would help if authors can discuss more regard whether matching higher-order moments could yield further improvements or not.\n\n3. It assumes that optimizing the latent alone suffices to align rewards, but i didnt see it explore interactions with jointly optimizing text or conditioning variables, which might be relevant in practice.", "was_revised": true, "final_justification": "The author has addressed my concerns.", "rebuttal": "We sincerely appreciate your time and thoughtful feedback. We are especially grateful for your recognition that our method is well-motivated, theoretically grounded, and addresses the critical challenge of reward hacking. We also value your observation regarding the complementary roles of spatial and spectral domain regularization in enforcing Gaussianity.\n\nWe would like to take this opportunity to re-emphasize our contributions:\n\n- **Moment-based loss (Section 3.1):** Unifies independently developed regularization terms such as **KL divergence**, **norm**, and **kurtosis** losses within a single framework.\n\n- **Power-spectrum-based loss (Section 3.2):** Defined via the **DFT** in the spectral domain, it regularizes DFT coefficients to match their theoretical distribution. Unlike **PRNO**, which removes spatial correlation by aligning the empirical covariance matrix with the identity and incurs quadratic complexity, our method uses **FFT** for efficient computation with $O(D \\log D)$ complexity and operates via a fundamentally different mechanism.\n\n- **Dual-domain regularization (Section 3.3):** We apply spatial and spectral losses **jointly**, leveraging their **complementary strengths**—the spatial loss ensures marginal distribution alignment, while the spectral loss enforces spatial neutrality. (**L116**)\n\nBelow, we address the specific questions and concerns raised:\n\n---\n***Q:*** *This paper is mainly evaluated only on a single text-to-image model (FLUX) with limited prompts. Broader evaluation on other architectures or diffusion models would strengthen the paper*\n\n***A:*** To assess the generality of our approach, we additionally evaluated our method using the Stable Diffusion XL-Turbo (SDXL-Turbo) model for text-aligned image generation:\n\n| SDXL Turbo | ReNO | PRNO | Ours |\n|---|---|---|---|\n| **PickScore**$\\uparrow$ | 0.2564 | 0.2595 | **0.2628** |\n| **ImageReward**$\\uparrow$ | 0.6735 | 0.6430 | **0.7063** |\n\nOur method **outperforms** the two strongest baselines on PickScore and ImageReward, demonstrating its effectiveness beyond the original FLUX model. We will include these results in the revised version of the paper.\n\n---\n***Q:*** *It would help if authors can discuss more regard whether matching higher-order moments could yield further improvements or not.*\n\n***A:*** We report additional results using different $\\mathcal{K}$ values in text-aligned image generation:\n\n| $\\mathcal{K}$ | $\\emptyset$| \\{1\\} | \\{1,2\\} | \\{1,2,3,4\\} | \\{1,...,6\\} | \\{1,...,8\\} |\n|---|---|---|---|---|---|---|\n| **PickScore**$\\uparrow$ | 0.2484 | 0.2500 | 0.2548 | 0.2510 | 0.2521 | 0.2525 |\n| **HPSv2**$\\uparrow$ | 0.2933 | 0.2962 | 0.3028 | 0.3012 | 0.2999 | 0.2984 |\n\nThe performance with \\{1,2\\} shows a clear performance improvement over both $\\emptyset$ and \\{1\\}. However, we did not observe further improvements when extending to higher-order moment losses. The first two terms were sufficient to surpass baseline performance, as our regularization loss was designed not only to incorporate moment-based losses but also to include a power-spectrum-based loss that effectively suppresses residual spatial structure. (**L187**) Additionally, we provide a unified framework that integrates regularization losses which had previously been used independently.\n\n---\n***Q:*** *It assumes that optimizing the latent alone suffices to align rewards, but I didn't see it explore interactions with jointly optimizing text or conditioning variables, which might be relevant in practice.*\n\n***A:*** While we could not find prior work exploring joint optimization of conditional variables in test-time reward alignment for text-to-image models, we investigated this direction by updating both the text embedding and the latent variable during optimization. The results are summarized below:\n\n| No Reg. | No Emb. Opt. | Emb. Opt.| // | Ours | No Emb. Opt. | Emb. Opt.|\n|---|---|---|---|---|---|---|\n| **PickScore**$\\uparrow$ | 0.2445 | 0.2436 | // | **PickScore**$\\uparrow$ | 0.2548 | 0.2526 |\n| **HPSv2**$\\uparrow$ | 0.2876 | 0.2802 | // | **HPSv2**$\\uparrow$ | 0.3028 | 0.2912 |\n\n**Reg.** denotes regularization, and **Emb. Opt.** denotes text embedding optimization.\n\nWe did not observe any performance improvement from text embedding joint optimization.\n\n---\nAdditionally, we would greatly appreciate it if you could kindly elaborate the following point you raised as a limitation:\n\n“*it would be good to expand on how latent regularization might propagate biases from the pre-trained generator.*”\n\nWe are eager to understand your concern more deeply so that we may address it properly in the revised version.\n\n---"}]

Author rebuttal:

Meta-review: The paper introduces a novel regularization technique for text-to-image models that ensures Gaussianity in the latent space. This technique aims to prevent issues like "reward hacking" during reward alignment optimization. The method uses a dual-domain regularization approach, combining:
- Moment-based loss (spatial domain): This unifies existing regularization methods (KL divergence, norm, kurtosis) into a single framework to ensure correct marginal distribution.
- Power-spectrum-based loss (spectral domain): Defined via the Discrete Fourier Transform (DFT), this loss is sensitive to spatial correlations, enforcing spatial neutrality and penalizing residual spatial structure. It uses Fast Fourier Transform (FFT) for significantly faster computation compared to prior methods like PRNO. The paper claims that these two losses play complementary roles: the moment-based loss ensures the correct marginal distribution, while the spectral loss enforces the absence of spatial correlation.

(b) Strengths:
- The paper proposes a novel regularization method and provides a unified theoretical framework that encompasses existing Gaussianity regularization methods under a moment-matching paradigm.
- It tackles the crucial issue of reward hacking and performance degradation during reward alignment in diffusion models.
- The spectral domain regularization achieves significantly improved computational efficiency (over 50x speed-up compared to PRNO) due to its use of FFT.
- The proposed technique outperforms baselines such as no regularization, KL divergence, kurtosis, ReNO, and PRNO in experiments on reward alignment for one-step image generation and text-aligned image generation.
- The method is well-motivated and theoretically grounded, with a formal foundation for using spectral loss to enforce spatial neutrality.

(c) Weaknesses
- While theoretical foundations were provided, a deeper analysis of why dual-domain regularization is necessary was requested by reviewers
- The paper appeared to lack clear ablation studies demonstrating the sensitivity of the K (number of moments) and $\lambda$ power parameters.
• The authors acknowledged that the loss function value alone cannot reliably indicate how closely a latent vector matches a true standard Gaussian distribution, which was noted as a limitation.

(d) Reasons to accept: The paper proposes a novel and technically solid regularization method that effectively addresses the important problem of reward hacking in text-to-image models. Its unified framework and efficient dual-domain approach are innovative contributions. The authors provided compelling experimental results demonstrating superior performance and effectively addressed the reviewers' concerns during the rebuttal period.

(e) The authors effectively addressed reviewer concerns, leading to increased scores across the board.
- Reviewer Thdj asked about parameter sensitivity and the necessity of dual-domain regularization. Authors showed stability for $\lambda$ power and explained how moment-based and power-spectrum losses are complementary (spatial for marginal distribution, spectral for spatial neutrality/correlations), and clarified their method is not just KL+PRNO due to FFT efficiency (50x speed-up) and different mechanisms.
- Reviewer LvLi requested clarification on the paper's main application and whether more moments should be enforced. Authors agreed to rewrite for clearer focus on latent regularization for reward alignment and confirmed higher-order moments provided no further performance gain. They also clarified their "fewer updates" claim meant higher rewards for the same iterations.
- Reviewer p3aM sought deeper theoretical analysis for dual-domain and generalization. Authors provided further theoretical justification for dual-domain necessity (complementary limitations of permutation-invariant moment losses and permutation-variant spectral losses). They also presented new results on SDXL-Turbo, demonstrating generalization beyond the initially tested model.
- Reviewer mYPt also requested broader evaluation and more on higher-order moments. Authors provided SDXL-Turbo results, confirming generalization, and reiterated that matching moments beyond the first two did not yield improvements. They also showed that jointly optimizing text embeddings offered no performance benefit.
Decision: Accept (poster)