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# System prepended metadata

title: icml24_ood_theory

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## [General Response]
 
We thank all the reviewers for their time and valuable comments. We are encouraged to see that reviewers find our theoretical analysis **formal** and **rigorous** (iCEA, kB2A, GsDV), and our paper **novel, important** and **significant** for the community (kB2A, GsDV, rY6z). Reviewers also recognize our paper presentation to be **excellent, well-written/organized** (iCEA, GsDV, rY6z). 
 
As recognized by multiple reviewers, the significance of our work can be summarized as follows:

- Our work offers a provable understanding of when and how the in-distribution labels help OOD detection, which is an important research question. 
- Our analytical framework is based on a graph-theoretic approach to analyze the separability of ID data from OOD data, which is novel in the context of the graph-aided representation learning theory.
- We provide supportive experiments with both illustrative examples and real-world data, precisely highlighting how the theory manifests in practice.

We respond to each reviewer's comments in detail below. We are happy to revise the manuscript according to the reviewers' suggestions, and we believe this makes our paper stronger.

## [Individual responses to each reviewer]


#### Response to Reviewer iCEA:
We thank the reviewer for the comments and suggestions. We are encouraged that you recognize our analysis to be clear and well-written. We address your questions below:


**A1. Practical grounding of the in-distribution (ID) connection**

We are glad you bring that up! In practice, a closer ID connection occurs when ID data has a higher intra-class similarity. For example, the data for fine-grained classification, such as the CUB dataset, would be connected more closely than the CIFAR datasets. 

Here we provide a verification experiment by contrasting the performance of the CUB dataset with CIFAR. Specifically, we train a ResNet-18 using the same unlabeled and labeled contrastive losses on 100 CUB classes for 200 epochs and test on the remaining 100 classes as OOD (to simulate the near-OOD scenario). The comparison with the results on CIFAR-100 in the near-OOD scenario is shown below. The result shows that both the difference in linear probing error and FPR&AUROC metrics is smaller for the CUB dataset, where the ID connections are denser, and that aligns with our theory.   

|     ID dataset   | ID labels |OOD dataset|    FPR95     | AUROC  |           Linear probing error     |
| ------ | ----- | ----- |----- | ----- |----- | 
|CIFAR-100|- |CIFAR-10| 62.20|85.93|0.27|
||+ |CIFAR-10|**58.28**|**89.01**| **0.19**|
|CUB-ID|-|  CUB-OOD| 83.49| 75.11 | 0.62|
||+ |CUB-OOD| **82.17**|**77.92** | **0.57**|


In addition to this, we also ablate on different sparsities of ID connection in our original submission, though in a different experiment setting. Specifically, we would like to point the reviewer to **Tables 3 and 6** where we show the result of our bound when the ID data (CIFAR-100) are connected with different sparsities through different training epochs. We show both intuitively and practically, that the CIFAR-100 representations became more dispersed throughout the contrastive training procedure, thus leading to a sparser ID adjacency matrix. We observe that the linear probing error difference $\mathcal{G}$ increases when the ID connections become sparser.

We also provide the visualizations with different sparsities on the ID connections for the illustrative example in **Appendix G, Figures 3 (a)-(d)**, where the ID connections are denser in Figure 3 (a) than Figure 3 \(c\). The denser connection leads to a smaller error difference on linear probing $\mathcal{G}$ of 0.09 (Figure 3 (b)) compared to 0.14 in Figure 3 (d). Both the results on the illustrative example and the real data are consistent with our main Theorem 2.

<!-- Recall that in theory, we measure the sparsity of the ID connections by the Frobenius norm of the unlabeled ID adjacency matrix $\|\tilde{\mathbf{A}}^{(u)}\|_F$, i.e., the dot product of the data representations  -->

<!-- based on if we train the neural feature extractor with the contrastive loss function outlined in Equations 4 and 5,  -->


<!-- and therefore a sparse ID connection is equivalent to a sparse dot product matrix (similarity) on ID representations.  -->




**A2. Generation of the data in the illustrative example**


We provided the details on how we generate the data (particularly the ID adjacency matrices $\mathbf{A}^{(u)}, \mathbf{A}^{(l)}$ and the OOD-ID adjacency matrices $\tilde{\mathbf{A}}_{\mathrm{OI}}^{(u)}, \tilde{\mathbf{A}}_{\mathrm{OI}}^{(l)}$) for the illustrative example in **Appendix Section F** of our submission. As a brief summary,

- We start by formulating the augmentation graph $\mathcal{T}$ for the ID data, with several block-wise square matrices introduced for encoding the augmentation probability $\mathcal{T}(\mathbf{x}|\mathbf{\bar{x}})$ within each ID class (Appendix Equation 83). Specifically, we simulate 3 ID classes and 40 data points per class. The square matrices are generated by Equation 84. Then, according to Definitions 3-6, we can calculate the adjacency matrices $\mathbf{A}^{(u)}$ and $\mathbf{A}^{(l)}$ for the unlabeled and labeled cases based on $\mathcal{T}$. These adjacency matrices are depicted in Figure 2 (a) and (b).
- We simulate the OOD-ID adjacency matrix $\tilde{\mathbf{A}}_{\mathrm{OI}}^{(u)}$ by sampling from a truncated normal distribution. In the far-OOD scenario (Figure 2 (d)), the sparsity of $\tilde{\mathbf{A}}_{\mathrm{OI}}^{(u)}$ is set to be lower than the near-OOD scenario (Figure 2 \(c\)). In the labeled case, the OOD-ID adjacency matrix $\tilde{\mathbf{A}}_{\mathrm{OI}}^{(l)}$ can be calculated by Equation 23 based on the augmentation matrix $\mathcal{T}$ and $\tilde{\mathbf{A}}_{\mathrm{OI}}^{(u)}$. 
- Finally, we can calculate the ID and OOD representations by Equations 6 and 8, respectively, and visualize them in Figure 2 (e) and (f).

For clarity, we have included these descriptions in our manuscript. Thank you again for your question. 


<!-- As explained in **Appendix F** in the original submission, we generate the augmentation graph $\mathcal{T}$ with several block-wise square matrices, where both the ID adjacency matrix and the OOD-ID adjacency matrix can then be defined or calculated to simulate the different scenarios analyzed in our paper.  -->

#### Response to Reviewer kB2A:

We are glad to see that the reviewer finds our work novel and formal from various perspectives. We thank the reviewer for the thorough comments and suggestions. We are happy to clarify as follows:

**A1. More results on near-OOD detection**

Thank you for your suggestion! As suggested, we have included results on another near-OOD dataset, i.e., TinyImageNet following [1]. Under the same training and evaluation setting as our current submission, the results on ID data, CIFAR-10, and CIFAR-100 are shown below. We found that the advantage of training with ID labels particularly in the near-OOD scenario still holds.


| ID dataset                                                                               | ID labels | OOD dataset  | FPR95     | AUROC     | Linear probing error |
| ---------------------------------------------------------------------------------------- | --------- | ------------ | --------- | --------- | -------------------- |
| $\mathbb{P}_{\mathrm{ood}}^{\mathrm{test}} = \mathbb{P}_{\mathrm{ood}}^{\mathrm{LP}}$    |           |              |           |           |                      |
| CIFAR-10                                                                                 | -         | TinyImageNet | 10.28     | 95.27     | 0.18                 |
|                                                                                          | +         | TinyImageNet | **6.71**  | **98.40** | **0.11**             |
| $\mathbb{P}_{\mathrm{ood}}^{\mathrm{test}} \neq \mathbb{P}_{\mathrm{ood}}^{\mathrm{LP}}$ |           |              |           |           |                      |
| CIFAR-10                                                                                 | -         | TinyImageNet | 77.64     | 68.92     | 0.85                 |
|                                                                                          | +         | TinyImageNet | **71.19** | **74.90** | **0.69**             |
| $\mathbb{P}_{\mathrm{ood}}^{\mathrm{test}} = \mathbb{P}_{\mathrm{ood}}^{\mathrm{LP}}$    |           |              |           |           |                      |
| CIFAR-100                                                                                | -         | TinyImageNet | 15.76     | 90.01     | 0.26                 |
|                                                                                          | +         | TinyImageNet | **9.39**  | **94.98** | **0.17**             |
| $\mathbb{P}_{\mathrm{ood}}^{\mathrm{test}} \neq \mathbb{P}_{\mathrm{ood}}^{\mathrm{LP}}$ |           |              |           |           |                      |
| CIFAR-100                                                                                | -         | TinyImageNet | 85.53     | 60.20     | 0.91                 |
|                                                                                          | +         | TinyImageNet | **76.17** | **69.92** | **0.72**             |




**A2. Discussion on the computational cost**

You raise a great point. We concur with your opinion on the computational cost of spectral decomposition for large datasets. In fact, in **lines 165-170** of our submission, we acknowledge the concern about the complexity of a large matrix (dataset) and instead present a surrogate contrastive learning objective to recover the feature representations. We show in Lemma 1 that minimizing this surrogate objective, which can be efficiently trained end-to-end using a neural net, is equivalent to performing spectral decomposition on the graph. In other words, **the realization of our algorithm does not require an explicit spectral decomposition, which bypasses the computational concern**. Importantly, the training cost of our approach should be similar to that of the other contrastive losses, such as SimCLR loss [2]. As a verification, we report the training time on CIFAR-10/100 for our loss and SimCLR loss with a single NVIDIA GeForce RTX 2080 Ti for 200 epochs below.

|     ID dataset   | Training time (Ours) |Training time (SimCLR) |
| ------ | ----- | ----- |
|CIFAR-10| 4.1 hours | 4.0 hours|
|CIFAR-100| 4.2 hours |4.3 hours |

For very large datasets, we believe it will be helpful to refer to fast contrastive learning methods, such as [3] that introduced additional intermediate contrastive losses for speeding up, and are happy to further improve our implementation based on these methods. 




**A3. Assumption sensitivity**

Another great point! We verify our key assumption of the graph structure on real-world datasets in **Appendix Section E**, where the result shows that the largest eigengap of the unlabeled ID adjacency matrix can be much larger than the feature dimension after spectral decomposition. For the similarity matrices used for constructing the graphs, we follow the original spectral contrastive loss paper [4] and define the adjacency matrix based on the augmentation graph. The particular graph construction approach is evaluated on multiple real-world datasets as well in [4], which proves its real-world robustness. 



**A4. Discussion on integration with existing OOD detection techniques**

We are glad you bring that up! We do believe our graph-theoretic approach can be integrated with existing OOD detection techniques. In fact, we have provided the experimental results in **Appendix Section I** where the OOD detection performance is evaluated using a non-parametric OOD detection score [5], i.e., k-NN score. 

In addition to that extension, we believe the spectral decomposition, i.e., the spectral contrastive learning objective in our framework can be easily applied to the model training with the in-distribution data for the current OOD detection methods. During inference time, the model can also be evaluated by other different OOD scores (rather than the k-NN score), such as energy score [6], etc.

For the integration with the existing OOD theories, we do think it will be a very promising future work to introduce a PAC learning-based analysis framework [7] to analyze our linear probing module after spectral decomposition. 

**A5. Detailed explanation of the illustrative example**

Happy to elaborate on this further. Firstly, we provided the details on how we generate the data (particularly the ID adjacency matrices $\mathbf{A}^{(u)}, \mathbf{A}^{(l)}$ and the OOD-ID adjacency matrices $\tilde{\mathbf{A}}_{\mathrm{OI}}^{(u)}, \tilde{\mathbf{A}}_{\mathrm{OI}}^{(l)}$) for the illustrative example in **Appendix Section F**. As a brief summary,

- We start by formulating the augmentation graph $\mathcal{T}$ for the ID data, with several block-wise square matrices introduced for encoding the augmentation probability $\mathcal{T}(\mathbf{x}|\mathbf{\bar{x}})$ within each ID class (Appendix Equation 83). Specifically, we simulate 3 ID classes and 40 data points per class. The square matrices are generated by Equation 84. Then, according to Definitions 3-6, we can calculate the adjacency matrices $\mathbf{A}^{(u)}$ and $\mathbf{A}^{(l)}$ for the unlabeled and labeled cases based on $\mathcal{T}$. These adjacency matrices are depicted in Figure 2 (a) and (b).
- We simulate the OOD-ID adjacency matrix $\tilde{\mathbf{A}}_{\mathrm{OI}}^{(u)}$ by sampling from a truncated normal distribution. In the far-OOD scenario (Figure 2 (d)), the sparsity of $\tilde{\mathbf{A}}_{\mathrm{OI}}^{(u)}$ is set to be lower than the near-OOD scenario (Figure 2 \(c\)). In the labeled case, the OOD-ID adjacency matrix $\tilde{\mathbf{A}}_{\mathrm{OI}}^{(l)}$ can be calculated by Equation 23 based on the augmentation matrix $\mathcal{T}$ and $\tilde{\mathbf{A}}_{\mathrm{OI}}^{(u)}$. 
- Finally, we can calculate the ID and OOD representations by Equations 6 and 8, respectively, and visualize them in Figure 2 (e) and (f).

Our findings in Figure 2 are mainly based on the representation contrast in Figure 2 (e) and (f). 
- In the far-OOD scenario, we observe that the ID representations (colored in purple) can already be well-separated from the OOD representations (colored in orange) without introducing the ID labels (Figure 2 (f) left). Therefore, the benefit of ID labels is marginal for OOD detection in this particular scenario.
- In the near-OOD scenario, the ID and OOD representations are mixed together in the unlabeled case (Figure 2 (e) left), making the OOD detection a hard task. In the labeled case, the decision boundary of a linear probing module can clearly separate the ID and OOD representations (Figure 2 (e) right). This justifies the benefit of the ID labels in the near-OOD scenario for detecting the OOD data.

For clarity, we have already included these descriptions in our manuscript. Thank you again for your question!

[1] Yang et al., OpenOOD: Benchmarking Generalized Out-of-Distribution Detection, NeurIPS 2022.

[2] Chen et al., A Simple Framework for Contrastive Learning of Visual Representations, ICML 2020.

[3] Gong et al., Fast Training of Contrastive Learning with Intermediate Contrastive Loss.

[4] HaoChen et al., Provable guarantees for self-supervised deep learning with spectral contrastive loss, NeurIPS 2021.

[5] Sun et al., Out-of-Distribution Detection with Deep Nearest Neighbors, ICML 2022.

[6] Liu et al., Energy-based Out-of-distribution Detection, NeurIPS 2020.

[7] Fang et al., Is Out-of-Distribution Detection Learnable? NeurIPS 2022.

#### Response to Reviewer GsDV:
We thank you for recognizing our paper to be well-organized and answer an important research question. We thank the reviewer for the comments and questions, which we address below:

**A1. The connection between the surrogate objective and the low-rank approximation form**

We would like to refer the reviewer to **Lemma 1** in our paper (**with full derivation in Appendix Section D.1**), where we rigorously justify the theoretical equivalence between the surrogate objective (Equation 4) and the spectral decomposition form (Equation 3). 

The surrogate objective is relevant and essential to our studied problem for the following reasons:
- **First, the surrogate objective is important to enable efficient training in practice, thus enables bridging the gap between theory and practice**. This motivation has been clearly stated in our manuscript **lines 165-170**, "_In practice, directly solving objective 3 can be computationally expensive for an extremely large matrix. To circumvent this, the feature representations can be equivalently recovered by minimizing the following contrastive learning objective, which can be efficiently trained end-to-end using a neural net parameterized by $\mathbf{w}$_".
- **Secondly, the surrogate objective serves as the critical foundation of our subsequent theoretical analysis in Section 4**. Because training with the surrogate objective is equivalent to solving the spectral decomposition objective in Equation 3, we can rigorously derive the representations of the ID data in closed form. After that, we can then proceed to compute the OOD representations based on the ID representations. With both ID and OOD representations in closed form, we are able to evaluate the OOD detection by linear probing error (Section 4.2), and analyze the effect of ID labels in different scenarios (Section 4.3). 
<!-- - and OOD data, which are used in the downstream linear probing analysis.  -->


**A2. Tightness of our bound**

We provide two evidence to show that our bound is indeed tight. 

- Firstly, we provide numerical results on our illustrative example to show the proximity between the value of the error difference $\mathcal{G}$ and the bound in our Theorem in the following table. Specifically, we set a different value of $\|\tilde{\mathbf{A}}_{\mathrm{OI}}^{(u)} \|_F$ and observed our lower bound is sufficiently close to the error difference $\mathcal{G}$ (The details of the dataset used are the same as those described in Appendix Section F except for the Frobenius norm of the OOD-ID adjacency matrix).

|  $\|\tilde{\mathbf{A}}_{\mathrm{OI}}^{(u)} \|_F$    | $\mathcal{G}$ |  Our bound|
| ------ |  ----- |  ----- |
| 60| 0.09| 0.07|
| 72|0.16 |0.12 |
| 84| 0.21 |0.16|
| 96| 0.39|0.37 |
| 108|0.40 | 0.34|
| 120| 0.61| 0.56|


- Moreover, by definition of the Moore-Penrose inverse and the fact that matrix trace is a linear operator, we can get $\operatorname{Tr}(\mathbf{Z}_{\mathrm{all}}^{\dagger}) = O((M+N)^3)$. Furthermore, according to Equation 10 and the shape of the linear probing label vector $\mathbf{y}$, we can get $\mathcal{G} = O((M+N)^5)$. For the derived bound in Equation 12 and the definitions in Theorem 1, we have the first term in the $\epsilon$ function equivalent to $O(MN^2)$, the second term of the $\epsilon$ function equivalent to a value of $O(MN^5)$ and the third term of the $\epsilon$ function equivalent to $O(MN^4)$. Divide them by $N+M$ and since in practice the number of ID and OOD samples $N, M$ can be of the same order of magnitude, we have the bound equivalent to $O(N^5)$, which grows at the same rate as the error difference $\mathcal{G}$. Therefore, **our lower bound is asymptotically tight**.

**A3. Discussion on the performance improvement**

Firstly, we kindly refer the reviewer to take a close look at Table 1, where **training with the in-distribution labels does not necessarily lead to a performance improvement for OOD detection**, especially in the far-OOD scenario. Therefore, we believe claim such as "of course, the performance will be improved" might be oversimplified. This phenomenon echoes well with our theoretical analysis results, and in fact, illustrates the intricacy and nontriviality of our studied problem on *when and how the in-distribution labels help OOD detection*.

Moreover, we point the reviewer to our theory and interpretations in **Section 4.3**, which rigorously explain the different OOD detection performances with and without in-distribution labels in different scenarios. Therefore, we believe the experimental results are certainly relevant to our paper and are able to highlight precisely how the effects of ID labels manifest in practice. The importance and relevance of our experiments are also recognized by other reviewers, such as Reviewer rY6z.


#### Response to Reviewer rY6z:
We are deeply encouraged that you recognize our method to be novel, significant, and with excellent presentations.

Your summary and comments are insightful and spot-on :)

**A1. Presentation of the results**
Thank you for the suggestion! We agree with your suggestion and are happy to 1) revise the structure of Table 1 by removing the bolding fonts and highlighting the error difference for linear probing, and 2) add more discussion on how the results in Tables 2 and 3 reflect the visualization on the illustrative example. We believe that can make our paper logically clearer in storytelling.


**A2. Verification experiments on increasingly far OOD datasets**
Great suggestion! As suggested, we adopt CIFAR-100 as the in-distribution dataset and use CIFAR-10 corrupted by Gaussian noise with zero mean and different variances ($\sigma^2$) as the OOD data. Since the Gaussian noise is known as far-OOD, changing the variance (increasing $\sigma^2$) of the noise added is equivalent to interpolating between far-OOD data (Gaussian) and near-OOD data (CIFAR-10), and therefore is able to simulate the setting of increasingly far OOD datasets.

Specifically, we follow the same experimental setting as in Table 1 except for the test OOD datasets. The results of the OOD detection are shown below, where the OOD detection performance increases when the variance of the Gaussian noise increases, i.e., the OOD datasets are more separable from the ID datasets.  In addition, we observe that both the linear probing error difference and the difference in FPR&AUROC decrease when the OOD data are further away from the ID datasets, which aligns with our theoretical results.

|     ID dataset   | OOD dataset| ID labels |   FPR95     | AUROC  |           Linear probing error     | ID labels |FPR95     | AUROC  |           Linear probing error    |
| ------ | ----- | ----- |----- | ----- |----- |  ----- |----- | ----- |----- |
|$\mathbb{P}_{\mathrm{ood}}^{\mathrm{test}} = \mathbb{P}_{\mathrm{ood}}^{\mathrm{LP}}$|
|CIFAR-100|CIFAR-10 ($\sigma^2=0$) |-|62.20 |85.93 | 0.27| + |53.28 |89.01 |0.19 |
|CIFAR-100|CIFAR-10 ($\sigma^2=0.2$) |-| 60.04 | 87.72 | 0.27 |+ |51.26 |90.93 |0.18| 
|CIFAR-100|CIFAR-10 ($\sigma^2=0.4$) |-| 49.01| 89.92 | 0.24|+ |44.81| 92.87| 0.15|
|CIFAR-100|CIFAR-10 ($\sigma^2=0.6$) |-| 37.01 |93.92 | 0.18 | + |35.72 |95.04 |0.12 |
|CIFAR-100|CIFAR-10 ($\sigma^2=0.8$) |-| 32.58 | 95.45| 0.13 | + | 26.10| 96.23|0.10 | 
|CIFAR-100|CIFAR-10 ($\sigma^2=1.0$) |-| 19.97 | 96.28 | 0.09 | + | 19.99| 98.78| 0.06| 
|$\mathbb{P}_{\mathrm{ood}}^{\mathrm{test}} \neq \mathbb{P}_{\mathrm{ood}}^{\mathrm{LP}}$|
|CIFAR-100|CIFAR-10 ($\sigma^2=0$) |-|94.54 |55.42 | 0.97| + |91.07 |67.72 |0.85 |
|CIFAR-100|CIFAR-10 ($\sigma^2=0.2$) |-| 86.29| 60.01| 0.90| + | 84.15| 69.25| 0.82 |
|CIFAR-100|CIFAR-10 ($\sigma^2=0.4$) |-| 70.28| 67.94| 0.84| + | 65.79| 73.38|0.76 |
|CIFAR-100|CIFAR-10 ($\sigma^2=0.6$) |-| 68.92|73.75 | 0.80 | +| 59.33 |77.21 |0.70|  
|CIFAR-100|CIFAR-10 ($\sigma^2=0.8$) |-| 58.21 | 74.40| 0.78| +| 54.83| 80.97| 0.69|
|CIFAR-100|CIFAR-10 ($\sigma^2=1.0$) |-| 50.26| 75.37| 0.63 | + | 47.08| 81.47| 0.60|

**A3. Applicability on different data types**
Great point! We are open to discussing the applicability of our approach to different data types. In particular, our paper is based on the core theoretical definition of the augmentation graph, which was originally proposed in [1]. Therefore, one simple extension to data with different modalities, such as texts, videos, or graph-structured data, is to formulate the augmentations between these data points and properly define their augmentation probabilities $\mathcal{T}(\mathbf{x}|\mathbf{\bar{x}})$. After that, we can train a neural net with the same contrastive loss as in our current submission to get their feature representations, which is equivalent to performing spectral decomposition on the adjacency matrix. However, how to properly formulate the augmentations and the augmentation probability is an open research question, which we believe is a promising and interesting problem to work on in the future, and are happy to add discussions on this in the revised version.

**A4. The extension to other contrastive losses**
We are glad you bring that up! We do see the possibility of extending our spectral contrastive loss to common contrastive losses, such as SimCLR [2]. Here is a brief extension idea.

Note that the difference between the unsupervised spectral contrastive loss and the SimCLR loss (See the last paragraph on page 9 of paper [1]) is that the former one removes a term that is related to $\mathbf{h}_{\mathbf{w}}(\mathbf{x})^\top\mathbf{h}_{\mathbf{w}}(\mathbf{x}^+)$, and replace the log sum of exponential terms with the average of the squares of $\mathbf{h}_{\mathbf{w}}(\mathbf{x})^\top\mathbf{h}_{\mathbf{w}}(\mathbf{x})$. Therefore, we can change the derivation accordingly in Lemma 1 for an extension to SimCLR loss, and we expect the change to the equivalent spectral decomposition objective is that the specific form of the representations $\mathbf{F}^{(l)}$ in Equation 3 will involve additional terms related to the adjacency matrices and the representations. After that, we can go from there and calculate the OOD representations and analyze the lower bound of the linear probing error difference $\mathcal{G}$ similar to the current analysis procedure.

**A5. Clarification on the caption of Table 2**
Thank you for your question! Here we use the CIFAR-100 as the in-distribution data for training and CIFAR-10 (abbreviated as C10 in Table 2) as the near-OOD data for evaluation. We will be sure to clarify the difference between ID training and OOD evaluation data for Table 2 in the revised version. 

**A6. Typos**
Great catch! We will fix this in the revised version. 


[1] HaoChen et al., Provable guarantees for self-supervised deep learning with spectral contrastive loss, NeurIPS 2021.

[2] Chen et al., A Simple Framework for Contrastive Learning of Visual Representations, ICML 2020.

## [Note to AC]

Dear Area Chair and Program Chair,

We thank the chairs and reviewers for their hard work and valuable service to the community.

We are writing to express our deep concerns regarding the quality of the review from R3 (Reviewer GsDV). Specifically:

R3 didn't appear to understand our paper and theory, and raised questions that have been extensively discussed and proved in our paper (e.g., Lemma 1). Unlike the other three reviews, R3 wrote fairly short comments that significantly lack depth. 

More conerningly, R3 completely overlooked our discussion on the experimental results and their connection to our theory (Figure 2 and Section 5), and raised false claim such as "with labels, of course, the performance will be improved". Contrary to this, our key contribution is providing new & important theories for analyzing when and how the ID labels help OOD detection in different scenarios, which is well recognized by all other reviewers (iCEA, kB2A, rY6z).

In light of the situation, we kindly request a careful re-evaluation of the comments by R3, considering the positive feedback from other reviewers and the thoroughness of our revisions and responses. We sincerely hope our manuscript can receive a fair assessment, given both its theoretical and empirical significance for the field.

Thank you for your attention to this matter.


Sincerely,

Authors