Demeaned Sparse: Efficient Anomaly Detection by Residual Estimate

Dear Reviewer 2ntV:

Firstly, we express our gratitude for your thorough review and insightful comments on our paper. Your recognition of the novelty and contribution of our work is greatly appreciated. We also greatly appreciate your rigorous and professional feedback. Your suggestions are highly meaningful, especially for a theory-driven paper like ours. We have taken your feedback as an opportunity to further refine our manuscript. We address your concerns below:

Q1:” On page 3, line 154, the question arises as to why the first statistic is weakly convergent and, after integration, becomes convergent in distribution”

Thank you for raising this question. Due to space limitations, we did not provide a detailed explanation. In our initial assumption, the Fourier transform is not performed over the $\mathbb{R}$, but rather over a finite interval. However, our integration is carried out over the entire real domain, which corresponds to the weak and strong convergence relationships in functional analysis. This setting is also valid for the weight function $W$.

Q2:” The regularization coefficient alpha has not been discussed”

We have added experiments regarding the regularization coefficient $\alpha$ and provide some of the results here:

W1:” on page 3, in the section on constructing the complex-valued empirical process, it is not explained why the common factors and residuals are combined together”and” this is actually used in the later asymptotic theory section. Another issue is that the title of the paper mentions residual estimation, yet the theoretical section also incorporates common factors”

The purpose of constructing the complex-valued empirical process by combining the common factors and the residuals is to unify the estimated quantities within the test statistics. Additionally, combining the common factors and the residuals helps prevent the residuals from summing to zero. In the process of estimating the residuals, the common factor $F$ is also involved, but $F$ differs across theoretical framework assumptions (as we mentioned in Section 3.1, maybe is not based on the factor structure). However, $F$ can be expressed in the form of residuals in all cases. Therefore, the estimation of the residuals is a central component of the complete theory, with $F$ serving as an intermediate variable.

W2:” the manuscript does not include experiments examining the effects of different sparsity levels on model performance”

Thank you for raising this question. In our theoretical section, our asymptotic theory suggests that as long as the estimated values are smaller than the true values, the anomaly detection task will be effective. In the experimental section, we constrain the sparsity using the regularization coefficient. Since we are applying this in an unsupervised setting, the level of sparsity is learned by the model itself. We believe that the sparsification operation enhances the effectiveness of the anomaly detection task, but the degree of sparsity cannot provide a precise bound. This is because the sparsity degree may vary for different types of unknown anomalies. Nevertheless, this raises an insightful question, and we will continue to explore this in our future research.

W3:” Writing suggestion”

Thank you for carefully pointing out our small mistake. We will make the correction in the subsequent version. Additionally, we will provide the pseudocode of our method to make it more intuitive and clearer for the readers.

Q3:” Whether the module constructed by this method potentially be applicable to a broader range of anomaly detection problems”

We have validated our method on multi-task tasks, and the results are as follows. Theoretically, our method remains effective in more anomaly detection methods. Here, we attempt to apply our method to the multi-class task on MvTec-AD dataset：

Compared to the baseline method, our approach still shows a significant improvement in the multi-class task. We will follow up on this and conduct corresponding tests in the subsequent versions.

We sincerely hope that our clarifications above have increased your confidence in our work. We will be happy to clarify further if needed. We thank you again for sharing your valuable feedback on our work.

Dear Reviewer a5G4:

We appreciate your review of our paper and the insightful comments you provided. We are glad to hear that you find the method we proposed to be both simple and effective. We have addressed the issues you raised as follows:

W1:” Lacking important reviews in the literature”

As mentioned in your comments, we will include a related literature review in the subsequent version. We greatly appreciate your thorough feedback. Unlike the papers mentioned, which operate at the feature level, our method performs a demeaned Fourier transform in the frequency domain and provides a complete theoretical framework and proof, which is not mentioned in previous literature. (We were motivated by the application cases and abstract it into a theoretical model, based on this, we further give the Asymptotic theory of our model. ) Regarding the comment "However, this paper still focuses on one-model-one-task," since our contribution is mainly focus on the theory filed, our initial approach was to validate it on benchmark problems. However, your comment has been very insightful. Based on your suggestion, we have attempted to apply a multi-task, one-model approach on MvTec-AD dataset：

Compared to the baseline method, our approach still shows a significant improvement in the multi-class task. We will refine the complete theoretical framework and methodology in future work. Thank you for your constructive suggestions.

W2:” The theoretical part and the method design part are very disconnected”

We thank the reviewer for raising this question, as it will help others better understand our work. (Maybe some of our expressions in writing let you not clearly see our writing logic, but in fact, the theoretical derivation of the previous sections and the application of the later sections are closely related, you could see when apply the DFT method, our theory clearly explains why the applied method works). As we mentioned in Section 4, the decentralization operation corresponds to the construction of the complex-valued empirical process in our theoretical section. It simultaneously constructs the test statistic, which helps us derive the asymptotic lower bound. Based on our asymptotic theory, in order to establish the quantitative relationship ($K<R$) between the true common factors and the estimated values, we designed the sparsification operation. This operation weakens the main information (non-anomalous parts) in the original image, causing the residuals (anomalous parts) to occupy a larger proportion, making anomalies easier to detect. The experimental results based on the above operations validate our asymptotic theories.

W3:” The paper is very incremental”

Thank you for raising this question, In terms of theory, our contribution is not simply an extension of existing one-dimensional theories to two-dimensional space. We also provide a complete asymptotic theory along with the corresponding proofs. This method and its applications really make sense: first, it requires the development of new hypothesis tests, complex-valued empirical processes, test statistics, and upper and lower bounds for the asymptotic theory. Second, unlike the one-dimensional time series theory, the theory at the two-dimensional image level requires spatial relationships to serve as the central element of the theory. Finally, the proof section involves new derivations and experimental validation of the constructed asymptotic theory, with validation metrics and methods that differ from those in the one-dimensional time series scenario. We are committed to constructing a complete theoretical framework, not merely extending the methods and theories.

W4:”it seems that only DFS Module was added on the U-Net. What if the DFS Module is added on the other architectures, such as UniAD?”

We thank the reviewer for raising this question. Since our main contribution is the theoretical part, it is essential to validate the effectiveness of the theory and its corresponding asymptotic properties using a simple model, this is very common in the Monte Carlo simulation section of such theoretical articles, and our main aim is similar. This allows us to eliminate the additional effects introduced by complex models. Therefore, we opted for the relatively simple U-Net architecture. As pointed out in your comment, adding the DFS module to the UniAD architecture should indeed be effective. However, the UniAD framework operates at the feature token level, which conflicts with our frequency domain operations. This results in our module not being directly applicable to the UniAD framework. We appreciate your valuable suggestion, which will be of great help for our future work.

We sincerely hope that our clarifications above have increased your confidence in our work. We will be happy to clarify further if needed. We thank you again for sharing your valuable feedback on our work.

Dear Reviewer LTm8,

Firstly, we express our gratitude for your thorough review and insightful comments on our paper. Your recognition of the rigor and contribution of our work is greatly appreciated. We also greatly appreciate your rigorous and professional feedback. Your suggestions are highly meaningful, especially for a theory-driven paper like ours. We have taken your feedback as an opportunity to further refine our manuscript. We address your concerns below:

W1:” more test on variable image sizes”

We agree with the viewpoint that the asymptotic global theory should be validated on more image sizes, as you mentioned in your comment: "it could have emphasized how its theoretical edge translates into tangible benefits." To address this, we have applied our method to larger-scale scenarios, with experimental results on MvTec-AD dataset presented for 512×512 and 1024×1024 image sizes. Due to space and time limitations, we only provide the average results of metrics.

results show that our method continues to maintain a high degree of effectiveness and shows greater stability compared to the baseline method. If possible, we will include relevant comparative results in future versions.

W2:” Robustness isn’t fully substantiated.”

We agree that statistical significance tests for the reported metrics are necessary, as the main contribution of this paper is statistical theory. Since we apply our method to practical detection problems, more comprehensive metrics are needed to assess overall performance. Specifically, in the validation of the asymptotic global theory (for the entire image), the p-value contribution of a single pixel might be overlooked (p-values are less adaptive to imbalanced data, as they often fail to account for the sparsity of anomalies). Therefore, we prefer to use metrics commonly employed in the anomaly detection field as our evaluation criteria. We provide partial p-values for our method on MvTec-AD dataset.

As can be seen, after incorporating our method, The p-value is below the significance level of 0.05, indicating that we can reject the null hypothesis (nonanomalous) with a high probability. If possible, we will include corresponding statistical significance tests for both our method and the comparative methods in future versions.

W3:” As mentioned in the Claims and Evidence section”

Please see our response to W1.

W4:”There are some minor issues, which I mentioned in the previous sections”

Please see our response to W1 and W2.

We sincerely hope that our clarifications above have increased your confidence in our work. We will be happy to clarify further if needed. We thank you again for sharing your valuable feedback on our work.