Unifying Reconstruction and Density Estimation via Invertible Contraction Mapping in One-Class Classification

Q1. Empirical validation of fixed-point behavior on hard, real-world anomalies. (Weakness 1 and Question 1)

Thank you for your professional comments and insightful questions. The normal distribution of real world is irregular and complex, and we have also found these phenomena in experiments on tabular and image data. Although our Proposition 1 theoretically shows that fixed points will approach the normal manifold with optimization of reconstruction loss, real-world situations are even more complex. Following your comments, we discuss our reconstruction-only part on several specific situations and show corresponding empirical validation. In some simple cases, the reconstruction part is very effective, but in some complex or even extreme cases, the reconstruction part may fail.

(i) Simple and general situations. In actual experiments, we found that common normal manifolds are often connected and do not appear as ring-shaped or multi-ringed structures (see Fig. 3(a)). Under optimizatin of reconstruction loss, the fixed point is not completely located at the geometric center of the distribution, and sometimes , fixed points may even fall on the edges of the distribution. It is worth noting that in this simple case, the fixed points will all fall inside the normal manifold. Anomalies will are contracted towards the normal manifold, leading to more reconstruction error. In addition, some data manifolds are U-shaped (see Fig. 3(d)), the fixed point is located at the bottom of U. We found that after adjusting the random seed or training hyperparameters, the position of the fixed point will change and may not necessarily be located at the bottom position. But the same thing is that as long as enough training epochs are set, the fixed point is located inside or at the edge of U.

(ii) Difficult and failure situations. The normal data manifold in the real world is stochastic and complex, which may be non-connected (see Fig. 4 (left)), ring-shaped, or even multi-ring-shaped. In these complex cases, there is a high probability that the fixed point will be optimized to the non-data region and outside the manifold, and the fixed point may be very close to the abnormal data, resulting in false negative. In addition, if some abnormal data is very close to the normal manifold, almost all reconstruction-based methods will fail. To address these issues on reconstruction part, we introduce backward density estimation part to learn probability distribution of original normal data. Even if the anomaly is close to normal in space, its likelihood may be much lower than that of normal data.

Q2. Whether the mapping suppresses or enhances outlier signals. (Weakness 2)

This is a very insightful observation. You are correct that our mapping 'normalizes' the abnormal features in the output space. However, this feature suppression is precisely the mechanism we leverage to amplify the anomaly signal in the form of reconstruction error. Next, we will show detailed reasons.

The key is that our reconstruction anomaly score is not the output $\mathbf{z}=\varphi(\mathbf{x})$ itself, but the discrepancy between the input and output, i.e., $||\mathbf{z}-\mathbf{x}||_{2}$. For a normal input $\mathbf{x}_n$, the contraction mapping acts nearly as an identity function, so $\mathbf{z}_n ≈ \mathbf{x}_n$ and the reconstruction error is minimal (see lines 190-193). For an abnormal input $\mathbf{x}_a$, the mapping pulls it towards the normal manifold, so $\mathbf{z}_a$ becomes significantly different from $\mathbf{x}_a$. This results in a large reconstruction error. Therefore, by suppressing the anomalous characteristics in the output $\mathbf{z}_a$, we create a large, easily detectable discrepancy signal. This enhances, rather than suppresses, the final anomaly score.

Q3. Limitations of the experimental datasets. (Weakness 3)

Thanks for your professional comments and valuable suggestions.

First of all, it should be noted that that we follow the initial works, DeepSVDD [1], and choose the one-vs-all setting for CIFAR-10 that is the classic setting in the OCC task. Using this established benchmark allows for direct and fair comparison against a wide range of state-of-the-art methods, which is essential for positioning our work. Next, we recognize the limitations of MVTec AD. Following your suggestions, we test our method on an extra non-toy industrial dataset, MPDD [2], with significant intra-class differences and a real-world medical datasets, ISIC [3].

(i) MPDD. MPDD contains 1,346 images from 6 types of industrial metal products with varying lighting conditions, non-uniform backgrounds, and multiple products in each image. We continue to adopt the model and training settings (visual anomaly detection) in the manuscript for this experiment, where only resolution is adjusted to $256\times 256$. As shown in table below, compared to some classic methods, the performance of our method is competitive.

(ii) ISIC. ISIC is a real-word skin lesion dataset, originating from the ISIC 2018 challenge, contains dermoscopic images across seven categories. We employs 6,705 normal images from the official training set for model training and the remaining images, including normal and abnormal ones, for testing, where only classification label can be given and image-level AUROC,F1,ACC are envaluated. We continue to adopt the same model and training settings as (i). As a real-world dataset, defects are more diverse, and all models that perform well on the toy dataset perform poorly on this dataset. However, our method is superior compared to some classic methods.

Finally, it should be noted that due to the limitation of rebuttal time, we did not fine tune more parameters to achieve optimal performance on two new datasets. In addition, the results on MPDD are sourced from [4], and the results on ISIC are sourced from [5].

[1] Lukas Ruff, et al. Deep one-class classification. In ICML, 2018.

[2] Stepan Jezek, et al. Deep learning-based defect detection of metal parts: evaluating current methods in complex conditions. In ICUMT workshop, 2021.

[3] Gutman David, et al. "Skin Lesion Analysis toward Melanoma Detection: A Challenge at ISBI 2016. In ISIC, 2016.

[4] Ximiao Zhang, et al. Realnet: A feature selection network with realistic synthetic anomaly for anomaly detection. In CVPR, 2024.

[5] Chunlei Li, et al. Scale-Aware Contrastive Reverse Distillation for Unsupervised Medical Anomaly Detection. In arXiv, 2025.

Q4. Limitations and societal impact discussion. (Limitations)

Thank you for this valuable suggestion. We also agree that a more concrete discussion of societal impact and practical risks is warranted. In fact, in our used tabular data, 'Arrhythmia' and 'Fraud' datasets are from real medical diagnostic data, where false positives may lead to some negative societal impact. Following your suggestions, we will expand the Limitations section (Appendix E) in the final version. We will add a discussion on the real-world consequences of detection errors (false positives/negatives) in safety-critical domains reflected in our datasets, such as medical misdiagnosis ('Arrhythmia') or failed fraud detection ('Fraud'). We will also connect this back to our method's design, arguing that our unified framework, which combines reconstruction with density estimation, provides a more robust signal than methods relying on a single paradigm, thereby potentially mitigating some of these risks in real-world deployments.

Q1. The trade-off between convergence to a fixed point (iterations of $f$) and bijectivity (each $f$ unit). (Question 1)

This is a very sharp and important observation. You are correct that infinite iterations of a contraction mapping unit $f$ would map all inputs to a single point, breaking bijectivity. In the following response, we first need to clarify a mathematical conclusion that the composition of a finite number of invertible mappings is still an invertible mapping, but the composition of an infinite number of mappings is not necessarily an invertible mapping. Next, our method focuses on leverage a finite number of mapping composition to simulate iteration of a single contraction mapping, rather than directly using iteration of a single contraction mapping.

(i) A general mathematical conclusion. First of all, we need to clarify that that the composition of a finite number of invertible mappings is still an invertible mapping, but the composition of an infinite number of mappings is not necessarily an invertible mapping. A simple counterexample: we consider that $f(x)=x/2$ is invertible with $x\in \mathbb{R}$ and has a unique inverse $f^{-1}(x)=2x$. the composition of an infinite number of $\lim_{n\to\infty} f^n(x) = \lim_{n\to\infty}x/2^n=0$. The infinite composite of this mapping is a constant function that is irreversible. So, in a finite number of composites, there is no contradiction between contraction mapping and reversibility.

(ii) Consideration on mapping structure. According to lines 166-168, we apply $\varphi_{K}(\cdot)=f_{K}\circ f_{K-1}\circ \cdots f_{1}(\cdot)$ to simulate a finite number of iterations of $f$, rather than directly iterating $f$, where each $f_{k}$ has its own parameters $\theta_{k}$ and Lipschitz constant $\eta_{k}$. Lipschitz constant $\eta_{\varphi}\le \eta_{1}\cdot \eta_{2}\cdots \eta_{K}<1$ ensures contracting anomalies towards the normal distribution. We leverage $\mathcal{L}_{\text{rec}}=||\varphi _ {K}(\mathbf{x})-\mathbf{x}|| _ {2}$ to train the entire network $\varphi _ {K}$, rather than firstly training a single $f$ and then iterating it a few times. From an optimization perspective, $\varphi _ {K}$ optimized by $\mathcal{L} _ {\text{rec}}$ tends to restore normal input to normal output instead of contracting all inputs (normal and abnormal) to one fixed point. From an mathematical perspective, a finite and small $K$ value gives $\varphi _ {K}$ the ability to contract the input to the normal manifold, but does not destroy the reversibility of $\varphi _ {K}$ . In summary, selecting $K$ as a finite value, the contraction properties of $\varphi _ {K}$ and the training objective jointly ensure that it can map anomalies to normal manifolds, but it will not completely shrink to a very small region or even a point and destroy its reversibility.

(iii) Consideration on the number of mapping layer. In actual experiments, we intentionally keep $K$ and $R$ small to prevent over-contraction. This allows us to benefit from the "pull" of the contraction towards the normal manifold (which increases reconstruction error for anomalies) while does not destory the reversibility of our INN. The trade-off is managed by the choice of $K$ and $R$, which we study empirically in Table 5 (main manuscript). The results show that a moderate number of iterations yields the best performance (corresponding reasons also are provided in lines 299-309), validating our handling of this trade-off.

Q2. The risk of collapsing to the identity mapping. (Question 2)

The degradation towards identity mapping is a critical concern for contraction mapping trained with reconstruction loss. We show two explicit mechanisms to prevent this as follow, where one is the structure of contraction mapping unit and the other is regularity of our backward density estimation.

(i) According to Eq.(2) and lines 197-198, $f(x)=\left(\frac{\text{Id} + \text{G}}{2}\right)^{-1}(x) - x$, where $\text{G}$ is invertible and $\text{Id}$ is identity mapping. If $f$ degrades into an identity mapping under reconstruction loss, we have $\left(\frac{\text{Id} + \text{G}}{2}\right)^{-1}(x) - x = x$, which means $\left(\frac{\text{Id} + \text{G}}{2}\right)^{-1}(x) = 2x$. Based on definition of inverse mapping, we must have $\frac{\text{Id} + \text{G}}{2}(2x) = x$. This means $\text{Id}(2x) + \text{G}(2x) = 2x$. Then, we can get $\text{G}(2x) = 0$ to make $f(x)$ an identity mapping. Paradoxically, $\text{G}$ is invertible and one-to-one mapping. Therefore, in terms of the structure of $f$, it is difficult to degrade into an identity mapping under the optimization of reconstruction loss.

(ii) The backward density estimation loss $\mathcal{L}_{\text{MLE}}$ provides a powerful regularizer against identity collapse. In the backward process, $\mathcal{L} _ {\text{MLE}}$ pushes $\varphi^{-1}$ to be a mapping that transforms a basic Gaussian distribution $\pi(\mathrm{z})$ to the (typically complex) input distribution $p(\mathrm{x})$. An identity mapping could only achieve this if $p(\mathrm{x})$ is also a Gaussian distribution, which is never the case. This forces $\varphi^{-1}$ to learn a non-trivial, meaningful transformation.

Q3. The conflicting objectives of reconstruction and density estimation. (Question 3)

Your understanding is perfectly correct. $\mathcal{L}_ {\text{rec}}$ pushes $\mathrm{z}$ to be close to $\mathrm{x}$, while $\mathcal{L}_ {\text{MLE}}$ pushes the distribution of $\mathrm{z}$ to be a Gaussian distribution. What we need to clarify is that both forward reconstruction (RE) and backward density estimation (DE) are indispensable, although they may seem contradictory. We have also designed corresponding trade-off schemes in the manuscript.

(i) The motivation of integration of DE and RE. When only using RE, if some anomalies are very close to the normal manifold or obtained fixed point in terms of spatial position, which causes small reconstruction errors and false negatives. In addition, the distribution of normal data may be very complex, chaotic, and disconnected (see Fig 3&4). In this case, fixed points may fall on the edges of the distribution, or even in areas without data. In this case, reconstruction error cannot be used as a good criterion. To address these issues, we introduce a density estimation process to probabilistically identify anomalies under these special circumstances. DE would assign a high likelihood to points on areas with dense data points and a low likelihood to others with low density. An anomaly is thus an input that is both far from its reconstruction (high reconstruction score) and statistically unlikely to have high likelihood (low density estimation score). This synergy makes our method robust.

(ii) Our trade-off strategies. As shown in Fig.2 and lines 224-226, after forward reconstruction process (RE), we apply a stop-gradient operation $\text{sg}(\cdot)$ in learning of Gaussian distribution $\pi(\mathrm{z})$, which allows $\pi(\mathrm{z})$ to be learned independently without affecting the reconstruction training. In addition, the hyperparameter $\lambda$ in our total loss $\mathcal{L}_ \text{total} = \mathcal{L}_ \text{rec} + \lambda\mathcal{L}_ \text{MLE}$ (Eq.(7)) explicitly controls this trade-off. Our sensitivity analysis on $\lambda$ in Table 3 (Appendix,Section D) shows that a balanced value ($\lambda=0.5$) performs best, validating that the two objectives are complementary rather than purely conflicting. The reconstruction provides an error-based score, while the flow provides a likelihood-based score, and their combination is more powerful than either alone.

Q4. Regarding the shallow depth of the flow network. (Question 4)

Your understanding is correct. In fact, our goal is not to build a superior flow model compared to previous density-estimation-only methods and flow-based generative model, which would indeed require a much deeper architecture. According to our answer in Q3.(i), our reconstruction-only method may may encounter some challenges caused by the spatial distribution of data. Therefore, the purpose of the flow component in our framework is precisely to act as an effective complementary scoring mechanism.

(i) The core of our method actually lies in the reconstruction part. A too large $R$ and $K$ will increase the network capacity, making reconstruction training difficult and requiring more training inference costs.

(ii) For fair comparision with previous tabular AD method, MCM [46], the value of $K \cdot R$ are roughly equivalent to the number of network layers of MCM (around six layers). In addition, visual AD method, CFlow [16], adopts the flow structure of 8 layers, while we set $K \cdot R = 9$ on MVTec AD dataset, which can actually meet the accuracy of visual anomaly detection.

Q5. A minor issue of L2 norm notation. (Weakness 2)

We thank the reviewer for their careful reading. We agree and will correct the notation $||\cdot||$ to $||\cdot||_{2}$ in line 183 and ensure consistency throughout the revised manuscript.

Q1. More direct and detailed evaluation on contraction mapping (CM). (Weakness 1 and Question 1)

Thanks for your insightful comments and valuable suggestions. In fact, we compared the reconstruction-only CM (DE process is removed) with MLP-based and CNN-based Autoencoders in variants (A) and (B) in Tables 3&4, but did not provide more detailed explanations for these experiments. Following your professional suggestions, we supplement these sections as follow. Moreover,we also add ablation studies of the components of reconstruction-only CM for your reference.

(i) In Table 3 and lines 288-289, to directly and fairly show the superiority of CM reconstruction, model (A) adopt the MLP structure with the same dimension of input and output, and the same number of layers for reconstruction as our CM. The significant performance improvement from Model (A) (AP/AUROC: 67.44/86.52) to Model (B) (75.21/90.32) represents a gain of +7.77 AP / +3.80 AUROC on all tabular datasets. This improvement can be directly attributed to the effectiveness of the contraction mapping in creating more discriminative reconstruction errors compared to a standard reconstruction approach.

(ii) As shown in Table 4 (A&B), compared with the original CNN-based Autoencoder network, reconstruction-only CM variant (B) obtained gain of image-level AUROC by +4.29 on natural image data and image-level/pixel-level AUROC by +1.4/0.7. All of these directly demonstrate that the reconstruction of CM is superior on image datasets. In our manuscript, due to the space limitations, we did not provide a detailed comparison between variants (A) and (B) in the ablation experiment.

(iii) We directly provide a comparison between a non-CM invertible network and a CM invertible network. According to Eq.(2) in main manuscript lines 197-198, the core of each contraction mapping unit $f$ is the INN $\mathbf{G}$ that is a guarantee of contract mapping. Next, we test general INN network, Glow [1], widely used i-ResNet [2], and our used i-DenseNet [3] as $\mathbf{G}$ respectively in the following table. In these experiments, we only test reconstruction process. Although the structure of Glow is also an INN, each layer does not meet the requirements of CM ($\eta<1$), which makes $f$ no longer a CM. This also means that under the training of reconstruction loss, $f$ may degenerate into an identity map, without solving the 'well reconstructed anomalies' issue. On the contrary, i-ResNet and i-DensNet satisfy the conditions of our CM and achieve higher performance on used datasets. The comparison of the three groups also directly indicates that using CM for reconstruction is effective.

[1] Durk P Kingma, et al. Glow: Generative flow with invertible 1x1 convolutions. In NeurIPS, 2018.

[2] Jens Behrmann, et al. Invertible residual networks. In ICML, 2019.

[3] Yura Perugachi-Diaz, et al. Invertible densenets with concatenated lipswish. In NeurIPS, 2021.

Q2. CM results on different kinds of normal data distributions. (Weakness 2 and Question 2)

These questions you raised about the CM results on different normal data distributions are very crucial and very interesting. Sometimes, extreme and complex distributions may make reconstruction-only part ineffective, which is also the reason why we introduce density estimation process to collaborate with reconstruction process. Following your series of questions, we discussed several situations and provided some special cases of difficult reconstruction.

(i) Ordinary circumstances. In fact, the manuscript visualizes the position of the fixed point in different normal distributions. Generally, the fixed point is not completely located at the geometric center of the distribution, and sometimes due to different data or training settings, fixed points may even fall on the edges of the distribution (see Fig.3 (a)).

Disconnected distribution. As shown in Fig.4 (left), normal distribution is relatively scattered and chaotic, and the fixed point is located at the edge of one of the sub distributions. However, due to the fact that the abnormal distribution is far from the normal one, reconstruction error is still an effective criterion, achieving a good performance under reconstruction-only setting.

U-shaped distribution. As shown in Fig.3 (d), normal distribution is U-shaped and the fixed point is located at the bottom of U. We found that after adjusting the random seed or training hyperparameters, the position of the fixed point will change and may not necessarily be located at the bottom position. But the same thing is that as long as enough training epochs are set, the fixed point is located inside or at the edge of U. As long as the abnormal data is in a distant external area, CM will achieve good and stable performance (AP/AUROC: 88.98 $\pm$ 0.73 /97.18 $\pm$ 0.41 under reconstruction-only setting).

Ring-shaped or multi-ring-shaped distribution. There is currently no typical ring-shaped normal distribution in the used datasets. Fig.3 (b) and (c) show multi-ring and irregular-ring structures, respectively. In fact, through multiple adjustments to the training hyperparameters and random seed, we found that fixed points may be located in some data free areas (central regions of the ring) or within the distribution (internal regions of the ring). If the anomaly is located in the central area of the ring, this method may theoretically fail, but this situation has not been found in experiments yet.

(ii) Special difficult circumstances. In the reconstruction-only experiments, we found that if the abnormal data is very close to the normal distribution in terms of spatial position, the reconstruction errors generated by CM for normal and abnormal data are very close, resulting in false negative and degrading performance. Of course, this is also a challenge for all reconstruction-based methods. In addition, In addition, due to the fact that fixed points may appear at any position on a normal manifold, they may be closer to some anomalies than other normal points, leading to false negatives.

Finally, in the most extreme case, when a few anomalies are within the normal distribution, our framework will fail. However, it should be noted that almost all existing OCC methods cannot address this type of problem. We believe this combined approach makes our method robust even for complex data distributions. We will add a discussion of this aspect to the limitations section in the appendix.

(iii) Strategies for address difficult circumstances. According to difficult circumstances in (ii), CM reconstruction may fail due to the spatial position of fixed points and abnormal data. To address these issues, backward density estimation (DE) is introduced to learn probability distribution of original normal data. Even if the anomaly is close to normal in space, its likelihood may be much lower than that of normal data.

Regarding the typical ring-shaped situation where reconstruction with CM may fail, DE would assign a high likelihood to points on the ring but a low likelihood to points in the empty center. This synergy makes our method robust, as an anomaly would be flagged by either a large reconstruction error, a low likelihood, or both.

Please discuss rebuttal with authors

Q1. Vague term makes it difficult to understand the core motivation of the paper. (Weakness 1)

We thank you very much for pointing out the ambiguity of this term.

(i) Generally, reconstruction-based methods rely on an assumption: reconstruction network trained on normal samples cannot reconstruct abnormal samples. Ideally, the trained network cannot reconstruct an abnormal sample well during inference, so there is a significant difference between the abnormal input and its reconstruction result. However, general reconstruction network (Autoencoder) often fails to achieve the ideal situation mentioned above and still reconstructs some abnormal inputs well. Thus, "well-reconstructed anomalies" refers to a common failure mode in reconstruction-based anomaly detection methods. To make this clearer for all readers, we will explicitly define the term "well-reconstructed anomalies" in the introduction of the revised manuscript and connect it directly to the problem of low reconstruction error for abnormal samples.

(ii) Our core motivation of reconstruction via contraction mapping is to combat the above failure mode. Instead of allowing the network to freely reconstruct inputs, we constrain the output to lie on the normal manifold. As shown empirically in Fig. 1(d) and Fig. 4, our contraction mapping iteratively pulls abnormal inputs towards the normal manifold. This process intentionally increases the reconstruction error for anomalies, making them more detectable and directly addressing the "well-reconstructed anomalies" problem. We will explicitly state this in the introduction of the revised manuscript.

Q2. Explanation of Proposition 1. (Weakness 2)

Thanks for your insightful comments on the depth of our theoretical analysis.

(i) Proposition 1 mainly focuses on establishing a direct causal link between our optimization and the fixed point's location. The proposition (line 151-159) states that $||\mathrm{x}^* -\mathrm{x}|| _ {2}\le \mathcal{L}_{\text{rec}}/(1-\eta )$, which directly and rigorously demonstrates that by minimizing the reconstruction loss $\mathcal{L} _ {\text{rec}}$ on normal samples $\mathrm{x}$ (our training objective), we are guaranteed to bound the distance to the fixed point $\mathrm{x}^*$. The empirical visualizations in Fig. 3 on real datasets strongly support this theoretical claim, showing the fixed point landing within the normal data manifold.

(ii) We are unable to prove that for any mapping and optimization algorithm, $\mathcal{L}_{\text{rec}}$ can be made arbitrarily small, because this depends on model capacity and data distribution complexity. This is a fundamental challenge in deep learning optimization. The empirical visualizations in Fig. 3 on real datasets, which show the fixed point landing within the normal data manifold, provide strong support for this theoretical claim. In addition, as shown in line 153, we only use the term 'a small neighborhood' rather than 'arbitrarily small' or 'complete convergence'.

Q3. Why apply LipSwish activation function. (Weakness 3)

Thanks for your professional comments. We note that the choice of LipSwish is deliberate and critical. LipSwish guarantees the reversibility of the entire network, while ReLU cannot guarantee it. We provide a specific analysis below for your reference.

(i) In this work, we apply an invertible network $\varphi$ as the foundation, where LipSwish activation function is invertible and ensures the invertibility of $\varphi$. In contrast, $\mathbf{ReLU}(x)=\mathbf{Max}(0,x)$ maps all negative inputs to zero, which creates a many-to-one mapping that is mathematically non-invertible. When $x < 0$, the input information is permanently lost, making it impossible to recover the input from the output.

(ii) As mentioned in the appendix (line 45,appendix), we adopt i-DenseNet [1] to parameterize our mapping $\mathbf{G}$ in Eq.(2). According to [1], the use of LipSwish is a standard and effective choice in this context, providing more expressive power than simpler activations. LipSwish is also smooth, continuously differentiable, and invertible, which are essential properties for stable training of INNs.

[1] Yura Perugachi-Diaz, et al. Invertible densenets with concatenated lipswish. In NeurIPS, 2021.

Q4. Why the integration of reconstruction and density estimation should be more effective. (Weakness 4)

The two components are not just used together and they are synergistic.

(i) When only using reconstruction mapping, we found that some anomalies were very close to the normal manifold in terms of spatial position, which caused small reconstruction errors and false negatives. In addition, the distribution of normal data may be very complex, chaotic, and disconnected. In this case, fixed points may fall on the edges of the distribution, or even in areas without data. In this case, reconstruction error cannot be used as a good criterion. To cope with the above situations, we introduce a density estimation process to probabilistically identify anomalies near the normal manifold.

(ii) Generally, an anomaly is thus an input that is both far from its reconstruction (high reconstruction score) and statistically unlikely to have high likelihood (low density estimation score). Therefore, we apply a integration of reconstruction and density estimation to identify test input (see Eq.(8)).

(iii) Our ablation studies (Table 3 & 4) provides strong support for this synergy. The full model (D) consistently and significantly outperforms the reconstruction-only (B) and density-only (C) variants, demonstrating that their combination is more powerful than either part alone.

Please discuss rebuttal with authors