CAS: A Probability-Based Approach for Universal Condition Alignment Score

We thank the reviewer for the interest in our work and the valuable feedback that strengthens our paper. The segments of the revised paper that reflect the received feedback are marked in blue color.

W1. More investigation into the hypothesis which provided in Sec. 4.1. with toy experiments. (Appendix A.1)

As requested, we have newly conducted an additional toy experiment to support our claim in Sec. 4.1. Our objective is to demonstrate that 1) since the user-specified condition is usually not observed in the training procedure of diffusion, a generated image $\mathbf{x}$ complying with the given condition is actually often out-of-distribution in the training distribution, resulting in a lower measured $p_\mathbf{\theta}(\mathbf{x})$ , 2) this effect is so strong that it biases $p_\mathbf{\theta}(\mathbf{x}\vert\mathbf{c})$ , and 3) by debiasing the conditional probability with the probability, i.e., $CAS(\mathbf{x},\mathbf{c},\theta)=\log p_\mathbf{\theta}(\mathbf{x}\vert\mathbf{c}) - \log p_\mathbf{\theta}(\mathbf{x})$ , it facilitates to measure alignment between image and conditions.

We have conducted the following experiment: We generated 100 images each from two prompts, $\mathbf{c}_1=$ "a woman with black hair" and $\mathbf{c}_2=$"a woman with rainbow hair," denoted as $\mathbf{x}_1$ and $\mathbf{x}_2$, respectively. $\mathbf{c}_2$ represents an unlikely scenario in the real world and $\mathbf{c}_1$ represents a more plausible scenario.

We then measured the average $\log p_\mathbf{\theta}(\mathbf{x}|\mathbf{c})$, $\log p_\mathbf{\theta}(\mathbf{x})$, and $CAS(\mathbf{x},\mathbf{c},\theta)$ for these images.

The results, as shown in the table above, satisfy all aspects of our hypothesis: 1) $\mathbb{E}[\log p_\theta(x_1)] > \mathbb{E}[\log p_\theta(x_2)]$, indicating that less observed conditions result in a lower probability, 2) the similarity between $\mathbb{E}[\log p_\mathbf{\theta}(\mathbf{x})]$ and $\mathbb{E}[\log p_\mathbf{\theta}(\mathbf{x}|\mathbf{c})]$ suggests that $\log p_\mathbf{\theta}(\mathbf{x}|\mathbf{c})$ is biased by $\log p_\mathbf{\theta}(\mathbf{x})$, and 3) $\mathbb{E}[CAS(\mathbf{x_1},\mathbf{c_1},\theta)]>\mathbb{E}[CAS(\mathbf{x_2},\mathbf{c_1},\theta)]$ and $\mathbb{E}[CAS(\mathbf{x_2},\mathbf{c_2},\theta)]>\mathbb{E}[CAS(\mathbf{x_1},\mathbf{c_2},\theta)]$ demonstrate that CAS effectively measures image condition alignment. We have newly added this experiment to Appendix A.1.

W2. Why NRMSE at the edge is big? How practitioners will select $\sigma$? (Appendix A.3)

The $\sigma$ is a hyper-parameter that controls the approximation quality of gradient by numerical differentiation. According to the definition of the numerical differentiation, it is obvious for Normalized Root Mean Square Error (NRMSE) to be large when $\sigma=10^{-1}$.

However, the reason for the high value of NRMSE when $\sigma=10^{-7}$ is different, because the NRMSE should decrease, as $\sigma$ approaches zero. It has been well-known that with the limited precision number like float32 we used, when $\sigma$ is set to be too low, the numerical differentiation causes significant rounding errors, leading to the gradient being calculated as zero [C1]. Thus, the widely accepted and common practice is $\sigma=10^{-3}$ (refer to [C1]), which is consistent with our numerical analysis. This may hint that the sigma is a hyper-parameter that does not require intensive tuning at all.

Furthermore, the table below shows the NRMSE between log probability computed by backpropagation and our approximation with various $\sigma$ for the Dreamlike Photoreal 2.0 model (Stable Diffusion v1.5 was employed in the NRMSE table in the main paper.) It can be observed that the approximation reaches the best performance when $\sigma=10^{-3}$. The results also support that practitioners can set $\sigma=10^{-3}$ regardless of the model. We have newly included these experiments in Appendix A.3 with additional details. Please note that this segment in the revised paper is colored green since there was the same question asked by Reveiwer uTB1.

• [C1] https://en.wikipedia.org/wiki/Numerical_differentiation

W3. Provide other metrics that can show the correlation between human preference and CAS in modalities experiments. (Appendix B.2)

Note: CAS(0 ~ 25%) represents the lower 25% of CAS scores when histogram is divided; CAS(25 ~ 50%), CAS(50 ~ 75%), and CAS(75 ~ 100%) represent the 25 ~ 50%, 50 ~ 75%, and top 25% percentiles, respectively.

To recap, in Sec. 5.2, the performance of CAS across various modalities was measured under the preference of five participants for each test case. Therefore, there are average human scores for each image. Based on this, we have newly analyzed the correlation between CAS and average human scores. We divided the CAS scores into histogram intervals based on percentiles and measured the corresponding average human score. As shown in the table, there is a tendency for the average human score to increase with higher CAS scores, indicating a correlation between CAS and human preference.

We have newly added this result in Appendix B.2.

Q1. Will the dataset of human evaluations (used in Sec. 5.2) be released along with the code?

We will! Thanks for your interest.

W5. How was ground truth estimated from Van Gogh dataset?

As stated in “Experiment setting in Van Gogh dataset” of Sec. 5.1, there are two types of generated images: 1) the images generated directly from the “Van Gogh style, {additional prompts}” themselves, and 2) the images generated from same prompts but with de-weighted Van Gogh style using diffusers module [C1]. We designated the ground truth for the first group as Van Gogh and for the second group as non-Van Gogh. Examples of this can be found in Fig. 5, where all samples are clearly distinguished. Additionally, we conducted a human verification step.

We have detailed this implementation in Sec. 5.1 of the revised paper. We will also provide the dataset if accepted.

• [C1] https://huggingface.co/docs/diffusers/using-diffusers/weighted_prompts

Q1. In Table 1, why NRMSE is the biggest at the edge ($\sigma=10^{-1}$, $\sigma=10^{-7}$)?

The sigma is a hyper-parameter that controls the approximation quality of gradient by numerical differentiation. According to the definition of the numerical differentiation, it is obvious for Normalized Root Mean Square Error (NRMSE) to be large when $\sigma=10^{-1}$.

However, the reason for the high value of NRMSE when $\sigma=10^{-7}$ is different, because the NRMSE should decrease, as $\sigma$ approaches zero. It has been well-known that with the limited precision number like float32 we used, when $\sigma$ is set to be too low, the numerical differentiation causes significant rounding errors, leading to the gradient being calculated as zero [C2]. Thus, the widely accepted and common practice is $\sigma=10^{-3}$ (refer to [C2]), which is consistent with our numerical analysis. We have newly added this phenomenon with a more detailed explanation in Appendix A.3 with a newly attached histogram in Fig. 7.

Furthermore, the table below shows the NRMSE between log probability computed by backpropagation and our approximation with various $\sigma$ for the Dreamlike Photoreal 2.0 model (Stable Diffusion v1.5 was employed in the NRMSE table in the main paper.) It can be observed that the approximation reaches the best performance when $\sigma=10^{-3}$. The results also support that practitioners can set $\sigma=10^{-3}$ regardless of the model. We have newly included these experiments in Appendix A.3 with additional details.

• [C2] https://en.wikipedia.org/wiki/Numerical_differentiation

We thank the reviewer for the interest in our work and the valuable feedback that strengthens our paper. The segments of the revised paper that reflect the received feedback are marked in green color.

W1. Report an Equal Error Rate (EER) for the Pick Score dataset experiment.

Note that measuring EER for the experiment in the Pick Score dataset is non-trivial. To recap, each test case of the Pick Score dataset involves a pair of images generated from the same prompt and the label, which image is preferred by the human. Then, each T2I alignment score was measured for these images, and we measured whether the score of the human-preferred image is higher than the other one or not. We reported this as accuracy in the experiment. This is markedly different from binary classification, which involves determining whether a single image is positive or negative. This discrepancy makes it hard to measure EER directly in the provided experiment.

Nevertheless, we have measured EER as requested, albeit with a slight modification. First, suppose there is a pair of images (image 1, image 2) generated for a single prompt, and there is a human preference for one of these images. We set the answer as 1 for the preferred image and 0 for the less preferred one. Next, we set the prediction of each score for image 1 as (score(image 1) - score(image 2)) and the prediction for image 2 as (score(image 2) - score(image 1)). After setting up answers and predictions for each image in this manner, we measured the EER.

As shown in the table, EER measured by this scheme follows the behavior of accuracy. We have added this new result in Appendix B.1. Thanks for suggesting this experiment, which improves the evidence of our claim in this work.

W2. Provide the complete derivations of main results (CAS) and more explanation of Equation 9 in the Appendix.

We have newly added full derivations of CAS and more explanation of Equation 9 in Appendix D.1/D.2.

W3. Notational issues. Vectors should be marked with bold symbols.

We have reflected all the comments in the revised paper.

W4. What is the intuition that in Fig. 2, $-\log p_\mathbf{\theta}(\mathbf{x})$ shows better performance than $\log p_\mathbf{\theta}(\mathbf{x})$?

As stated in the main paper, we believe that the conditions, which users typically desire to generate images for, are likely to be unobserved in the training distribution. Therefore, our hypothesis is that if a generated image complies with the input condition well, it may turn out a sample from out-of-distribution in terms of the diffusion model's training data, so that the low value of $\log p_\mathbf{\theta}(\mathbf{x})$ is measured. The results in Fig.2, which shows $-\log p_\mathbf{\theta}(\mathbf{x})$ tends to perform better than $\log p_\mathbf{\theta}(\mathbf{x})$, may hint to support this hypothesis.

This observation is important as it underpins our rationale for debiasing $\log p_\mathbf{\theta}(\mathbf{x})$ from $\log p_\mathbf{\theta}(\mathbf{x}|\mathbf{c})$. The second observation in Fig. 2 was that $\log p_\mathbf{\theta}(\mathbf{x}|\mathbf{c})$ is highly biased to $\log p_\mathbf{\theta}(\mathbf{x})$. Therefore, $\log p_\mathbf{\theta}(\mathbf{x})$, which shows a lower value for images well aligned with the condition, degrades the effectiveness of $\log p_\mathbf{\theta}(\mathbf{x}|\mathbf{c})$ as a measurement for alignment score.

This phenomenon is detailed in the response to the W1 asked by Reviewer cbcH with additional toy experiments, which we recommend the reviewer to check it. With newly conducted toy experiments, we have shown that 1) since the user-specified condition is usually not observed in the training procedure of diffusion, a generated image $\mathbf{x}$ complying with the given condition is actually often out-of-distribution in the training distribution, resulting in a lower measured $p_\mathbf{\theta}(\mathbf{x})$ , 2) this effect is so strong that it biases $p_\mathbf{\theta}(\mathbf{x}|\mathbf{c})$ , and 3) by debiasing the conditional probability with the probability, i.e., $CAS(\mathbf{x},\mathbf{c},\theta)=\log p_\mathbf{\theta}(\mathbf{x}|\mathbf{c}) - \log p_\mathbf{\theta}(\mathbf{x})$ , it facilitates to measure alignment between image and conditions.

If there are any additional discussion points or questions, we are happy to discuss. Thank you.

W3. Provide the limitations of CAS.

After receiving a request about limitations, we have conducted additional analysis and have newly discovered a new interesting finding that the performance of CAS decreases when comparing images sourced from different generative models.

Specifically, the HPS dataset, a T2I alignment score benchmark, includes images synthesized by 20 different models, and our technique showed lower accuracy compared to the baseline on this dataset. We have revealed that this limitation stems from a new interesting bias of CAS. CAS is measured higher for images that are generated from the same diffusion model on which CAS is computed on, while leading to lower accuracy for those generated from other generative models.

We see this intriguing property as a favorable property, rather than just a weakness. Thereby, we advantageously leverage and extend this characteristic to the other challenging applications of fake detection and image generation source detection, of which details and further discussion can be found in the response to Q3.

W4. Typos.

We have polished and fixed some typos. We will further polish in the camera ready. Thanks.

Q1. Could you clarify How the probabilities in Fig. 2 were calculated? Are they derived using Equation 4 in conjunction with the Skilling-Hutchinson trace estimator?

Yes. As mentioned, they are derived using Equation 4 in conjunction with the Skilling-Hutchinson trace estimator.

Q2. Has the DDIM Recursive Inversion been introduced in previous literature?

To the best of our knowledge, our paper is the first to introduce DDIM recursive inversion. Please inform us if there is any other related work that the reviewer thinks we need to add.

Q3. The unexpected poor performance of baselines on the Van Gogh dataset – Why CAS perform best in the Van Gogh dataset? (Appendix A.2)

To address the question, we have investigated the training set of ImageReward. We found that out of the total 9K training prompts, only 21 are related to Van Gogh, which we suspect contributes to the lower performance. This is a common issue shared by all learning-based metrics; they tend to underperform on data they have not frequently encountered. Despite this, the superior performance of CAS is intriguing, and we delve into analyzing it.

This hints to us that our CAS is sensitive to out-of-distribution cases. By leveraging this observation, we have newly revealed the strong capability of CAS to detect out-of-distribution samples. For the analysis, we have generated 200 images from each of the three different diffusion models. We have also prepared 200 real images. Then, we measured the average CAS of each group of images according to the diffusion models for image generation, and CAS is computed with respective diffusion models that is used for generating each group of images. The following table presents the results of the experiment.

As shown in the above table, CAS is notably higher for images generated by the same diffusion model that CAS employs. The characteristic of CAS producing high values for in-distribution samples and low values can be straightforwardly applied to fake image detection and image generation source model detection tasks.

We simply implement the detection methods as follows: 1) we compute CAS values from various diffusion models for an input sample, 2) we use the computed CAS values as an input feature vector, and 3) we train a MLP layer with a supervised dataset of respective tasks.

For the fake image detection application, the accuracy of 0.92 was measured, outperforming the current SOTA fake detection model [C2] whose accuracy is 0.87, trained on the same data. The remarkable point here is that diffusion models leveraged to generate samples in the test data differ from those in train data.

For the source model detection, we reached an accuracy of 0.90.

More detailed results have newly included in Appendix A.2 of the revised paper. Our future work is to further refine this technique (applying MLP to CAS from various models) to better utilize it in fake detection and image generation source detection.

• [C2] Ojha et al., “Towards universal fake image detectors that generalize across generative models.” CVPR, 2023.

We thank the reviewer for the interest in our work and the valuable feedback that strengthens our paper. The segments of the revised paper that reflect the received feedback are marked in red color.

W1-1. When referring to a cited paper, provide a small summary or refer by their method names.

As requested, we have partially reflected this in Sec. 2 (Related Work). We already referred the prior works in the experiment section by their names in the initial submission.

However, if the request was to explain or name the ScoreSDE [C1] paper specifically, we intentionally avoided doing so to prevent misunderstanding. Although we cited the ScoreSDE paper multiple times for the idea of probability computation, the core of the ScoreSDE paper and probability computation are almost unrelated. The referred probability computation can be found at the end of the appendix of the ScoreSDE paper. Therefore, instead of explicitly referring to the ScoreSDE paper by its method name, we described it as "the method proposed by Song et al." If there are any references we might have missed that could aid understanding, we would appreciate the advice.

• [C1] Song et al., “Score-based generative modeling through stochastic differential equations.” ICLR, 2021.

W1-2. The captions can be improved, e.g., Fig. 3 and Table 1.

We have revised captions (Fig. 3, Table 1, and Table 2) in this revision. Thanks.

W1-3. Provide more ranking examples in Appendix.

As requested, we have added more ranking examples in Appendix E of this revision.

W2-1. Report the standard deviation for experiments in Pick Score dataset and Van Gogh dataset. (Table 2)

As requested, we have repeated the experiment in Sec. 5.1 five times to calculate the standard deviation in Table 2 of the revised paper. On the Pick Score dataset, CAS with DDIM inversion had an average accuracy of $0.593$ (std: $0.004$), while ours with the 2nd order recursive inversion yields $0.622$ (std: $0.005$). On the Van Gogh dataset, the average accuracy for CAS using DDIM inversion was $0.401$ (std: $0.033$), and with the 2nd order recursive inversion, it was $0.412$ (std: $0.013$). These results indicate our technique's higher and stabler performance and show the effectiveness of our 2nd order recursive inversion.

Regarding the results of the 2nd order recursive inversion, to apparently present the statistical significance of our results, we have added them as confidence interval in Table 2 of the revised paper. With 95% confidence interval, the accuracy is $0.622 \pm 0.004$ on the Pick Score dataset, and $0.412 \pm 0.011$ on the Van Gogh dataset.

W2-2. The number of test samples is small in Table 3 (other modalities experiment).

As requested, we have re-experimented Table 3 by adding additional 80 test samples to the test set of each modality and by measuring accuracy on the new test set. The results are reported in Table 3 of the revised paper, which shows the accuracy of CAS is around or higher than 0.6 in most of the modalities. Note that the accuracy for CAS on each modality is based on 5 participants for 180 samples now, which deems sufficient. We have newly reported the p-values as well, which imply the probability of obtaining results of other methods better than our accuracy when participants make random 50:50 choices. The measured p-value is $1.74\cdot10^{-8}$ for InstructPix2Pix, $3.18\cdot10^{-18}$ for ControlNet (Canny Edge), $2.16\cdot10^{-11}$ for ControlNet (Scribble), and $3.22\cdot10^{-7}$ for AudioLDM, demonstrating the statistical significance of our results. We have newly included this result in Appendix B.2.

Also, note that our evaluation is not entirely based on human studies. In fact, most of our evaluations are quantitative and extensive, which are sufficient to support our contribution and findings across our paper. Through this rebuttal, we have added more results requested by the reviewers, which improve our work meaningfully. Thanks for the suggestion.

W2-3. The ablation on $\lambda$ (Appendix C.1)

As requested, we have newly conducted the ablation study of the accuracy of CAS according to $\lambda$ on the Pick Score dataset, and have added this result with discussions in Appendix C.1 of this revision.

We found that $\lambda$ is optimal at 1, which directly measures the ratio of two probability $\log (p_\mathbf{\theta}(\mathbf{x}|\mathbf{c})/p_\mathbf{\theta}(\mathbf{x}))$. We postulate that this balance is favored, because of the quantity balance between $\log p_\mathbf{\theta}(\mathbf{x}|\mathbf{c})$ and $\log p_\mathbf{\theta}(\mathbf{x})$, which is discussed with toy experiments in the response to the question W1 by Reviewer cbcH. We invite the reviewer to also check the response to W1 of Reviewer cbcH.