Paper: Pushing Boundaries: Mixup's Influence on Neural Collapse (ICLR-2024)

Reviewer comment (Evaluation):
Despite stating the study aims to explore how the proven geometric configurations in the paper can shed light on the success of mixup, the current iteration of the paper falls short in this regard. Mixup is a well-established augmentation technique with a lot of empirical success (e.g., generalization, adversarial robustness mentioned in this work), but it's hard for the reviewer to establish the connections between these empirical successes and the theoretical results shown in the paper.

Author response:
We have updated the paper to include more experimentation and analysis relating to calibration. We believe that this addition provides a connection between our initial findings and the empirical success of mixup.

Reviewer comment (Evaluation):
The reviewer believes the paper needs additional work to discuss its practical implications.

Author response:
Discussed the practical significance of our findings in the context of calibration improvements.

Reviewer comment (Presentation):
Sections 1.2-1.5 could be placed before the figures on pages 2 and 3 for better clarity.

Author response:
We have updated the formatting of the paper in the way that the reviewer has suggested to improve the clarity.

Reviewer comment (Experiments):
Why do the authors keep the models in train mode when plotting? Does this make a difference in the results?

Author response:
Train mode was chosen mostly for consistency among plots. Yes, there is a slight difference in the plots for the resnets. We have included a figure for the network in eval mode in the appendix to show this difference. We believe this difference is due to batch normalization but we have not proved such a claim.

Reviewer comment (Presentation):
Poor layout of the text and figures. The figures don't align well with their appearances in the sections.

Author response:
1, 3,  4. Upon your suggestion, we have made significant changes to the formatting of the paper as well as the coloration of the plots presented. We hope that this new format and visualization have resolved any readability issues you encountered.

Reviewer comment (Evaluation):
The connection between neural collapse in mixup and the generalization and calibration improvement of mixup is not further investigated.

Author response:
2. This is a valid point to raise, and in response to it, we have added an additional section investigating and explaining the relationship that this configuration has with calibration.

Reviewer comment (Presentation):
In Figures 1, 2, 3, 5 and 7, although the literal descriptions in the captions are clear, the gradient effects of the colors in the plots are not distinct. It is hard to tell how the $\lambda$'s vary from the plots.

Author response:
1, 3,  4. Upon your suggestion, we have made significant changes to the formatting of the paper as well as the coloration of the plots presented. We hope that this new format and visualization have resolved any readability issues you encountered.

Reviewer comment (Presentation):
In these figures, both the same-class and the different-class activations with $\lambda=0.5$ use the color black. Though the plots are clearly described in words, the colorations may still bring misunderstandings.

Author response:
1, 3,  4. Upon your suggestion, we have made significant changes to the formatting of the paper as well as the coloration of the plots presented. We hope that this new format and visualization have resolved any readability issues you encountered.

Reviewer comment (Presentation):
How the activations are projected into 2D planes for visualization in the figures is not clearly explained.  
a. Is it training data or test data that is used to generate these plots?  
b. In the caption of Figure 1, what does it mean by saying "project activations onto the classifier"? From my understanding of the initial observation of neural collapse, we just need to project the last layer activations and the components of the classifier onto the same low-dimension space. Is that correct? And if so, is the projecting method in this paper different from that?  
c. How is the conclusion "last layer activations align with the classifier" justified from the figures? Or from other observations presented in this paper?

Author response:
5.  We have added additional descriptions of the last-layer activations being plotted as well as a detailed explanation of the projection method in the appendix. You are correct about the Neural Collapse projections. The only difference between ours is that in the Neural Collapse paper, to the best of our knowledge, everything is projected onto the span of the class means, whereas we are projecting onto the span of the classifiers. Indeed, we have neglected to display how the shape aligns with the classifier so we have updated the figures to include the classifiers.

Reviewer comment (Theory):
The theoretical explanation has made an assumption that the classifier is a simplex ETF. This would be a very strong assumption if further proof or demonstration of it is not provided.

Author response:
6. It would be a strong assumption, so we have incorporated experiments illustrating the convergence of the classifier to a simplex ETF (Figure 5). However, we acknowledge the importance of theoretical proof of the characterization of optimal features without making this assumption, and we have designated this as a direction for future work.

Reviewer comment (Theory):
The theoretical explanation only considers the last layer features and the training targets, but not the inputs. In mixup, sometimes the mixed training target of a mixed input may not in fact be the ground-truth target of the input. Consider a dataset with three clusters of points side by side, where the clusters on the two sides are class 1 and the cluster in the middle is class 2. It is possible that a point mixed from two points in class 1 falls into the cluster of class 2; in other words, this point may be labelled either class 1 or class 2. These two scenarios will result in different training targets, which will then result in different optimal last layer features. However, the point can have only one possible last layer feature output in a single model. Does this contradict the theoretical explanations in this paper? Should this point be aligned with the classifier component corresponding to class 1 or that corresponding to class 2?

Author response:
7. If the scenario given by the reviewer were to occur, then let $x_1$ and $x_1’$ be training examples of class 1, $\tilde{\lambda}$ be such that $\tilde{\lambda}x_1 + (1-\lambda)x_1’ = x$, a training example of class 2. 

Our objective function involves an expectation over $\lambda$, and since the set {$\{ \tilde{\lambda} \}$} is of Lebesgue measure 0, the case that $x$ is labeled as class 1 is negligible compared to when $x$ is assigned class 2, so $x$ would be aligned with class 2. Thus, we don’t consider this in our theoretical results.

Since in practice lambda is sampled, we can also consider a more discrete approach. This would result in two terms involving the input $x$ contributing to the overall loss, one with label $y_1$ and the other with label $y_2$. That is, there would be a term $\operatorname{CE}(x, y_1)$ and another term $\operatorname{CE}(x, y_2)$. If the first term appears with probability $P$ and the second with probability $(1-P)$, then overall we get:

$P \operatorname{CE}(x, y_1) + (1-P) \operatorname{CE}(x, y_2) = \operatorname{CE}(x, Py_1 + (1-P)y_2)$

Then, the example overall will have an effective label of $Py_1 + (1-P)y_2$, which corresponds to a mixup label with mixup coefficient P, which is covered by our theoretical result.

Interestingly, [1] considers the general setting proposed by the reviewer called manifold intrusion. There the authors claimed that mixup can be detrimental for cases where mixed up examples align with the original training data, exactly the same way the reviewer is proposing.
The authors showed that this can occur in datasets like MNIST and that this causes a deterioration in test performance. However, it is unclear to what extent this occurs in more complicated datasets which is why in our paper we did not consider this case.

Reviewer comment (Novelty):
Although the reported observations are new, they are nevertheless superficious and brief. It is not much of a surprise that same-class mixed activations perform similarly to conventional neural collapse in ERM and that different-class mixed activations align with the decision boundaries, since different-class mixup induces linear combinations on the training targets while same-class mixup normally performs similarly to conventional data augmentations. The effort the authors have put into the investigations is appreciated, but the presented results don't seem adequate to constitute a rich-content formal paper.

Author response:
9. We appreciate the different perspectives on the results. Our team holds the view that the result is not superficial. We believe the expected result is a triangular configuration. In order to illustrate this, we conducted an additional experiment incorporating mixup with the MSE loss. The resulting last-layer activations are arranged as convex combinations of the classifier, aligning with what we perceive as a superficial outcome. We hope that this additional experiment emphasizes the unexpected nature of the observed configuration.

Reviewer comment (Evaluation):
In section 1.5, how do the authors justify if a trained model shows "good generalization"? Is it based on some well-defined metrics or solely based on intuition?

Author response:
1. We are basing this claim on the test accuracies presented in Table 1. To improve the paper we have clarified this in the paper and mentioned that these accuracies are being compared to those found in other papers.

Reviewer comment (Evaluation):
In Figure 6, second row, the characteristics exhibit an interesting behavior: they are low at the very beginning of the training processes, then increase, then decrease again, and their final level may even be higher than their initial level. Can the authors give some explanations or intuitive insights into this trend?

Author response:
2. Though this is an interesting behaviour, and could potentially be investigated further, the authors would note that monotonicity is not necessarily a characteristic of Neural Collapse. Furthermore our measured values are comparable to that of the values found in the Neural Collapse paper. [2]

Reviewer comment (Evaluation):
In section 4.2, how is the statement "... reaped the minimum benefits" justified?

Author response:
3. We were referring to the fact that, in Table 1, the Fashion MNIST mixup experiments saw the least improvement in test accuracy over the baseline. We have clarified this further in the paper following your question.

Reviewer comment (Writing):
In section 4.2, what is channel collapse?

Author response:
5. In this case, we are referring to the activations collapsing to the decision boundary. In hindsight, this phrasing is unclear and has since been changed.

Reviewer comment (Presentation):
The organization of the paper could benefit from a more conventional structure. The introduction figures and the placement of experimental results in Section 2 present challenges to readability and could be restructured for clarity.

Author response:
Upon your suggestion, we have revised the formatting of this paper and hope that it has improved its overall readability. Specifically, section 2 has been reworked with figures appearing closer to their relevant text sections.

Reviewer comment (Presentation):
A clearer presentation of the problem's structure, particularly how it has been addressed numerically, would enhance the reader's comprehension of the methodology and findings.

Author response:
We have re-written the problem statement of the paper and additional explanations for the numerical experiments have been incorporated, including a detailed explanation of the projection method.

Reviewer comment (Evaluation):
While the analytical proof is a notable element of the paper, the practical significance of its findings could be better articulated to highlight their impact and possibly extend the manuscript's reach to a broader audience.

Author response:
We have included a clearer explanation of the interpretation of the theorem after the theorem statement.

Reviewer comment (Presentation):
The rationale for projecting features into 2D, as depicted in the introductory figure, is not immediately clear and warrants further explanation. Given the dimension-specific uniqueness of the regular d-Simplex ETF, further details on the projection process would be beneficial. It should be clarified whether the projection includes all classes or if any are excluded, and if so, whether this exclusion is systematic or random. Employing a pair plot or scatter plot matrix, as referenced in [1], may provide a more conventional representation of the multidimensional data and ensure clarity regarding the inclusion of all classes. Given that datasets such as CIFAR10 and FashionMNIST comprise 10 classes, they could be visualized directly without the need for projection.

Author response:
We choose to project the features into 2D mainly to be consistent with the original Neural Collapse visualizations as well as other neural collapse-related papers and visualizations such as those cited in [1] - [4]. To address a more substantial set of classes, we have included a larger subset of classes in the appendix of the paper.

Reviewer comment (Experiments):
The part on 'amplification' related to decision boundaries would benefit from a more thorough explanation. This concept is central to the paper, and a detailed discussion is essential, extending beyond the brief description in Figure 4's caption. Providing clear evidence or a solid rationale for this phenomenon is crucial to enhance the paper's validity. To better understand the amplification effect, it may be helpful to initiate training with a regular d-simplex ETF that is fixed, as in [A] and [B]. By 'fixed,' it is meant that the parameters do not change during the training. This could provide clearer insights into the behavior of decision boundaries.

Author response:
We have incorporated a more comprehensive exploration of the differences in the theoretical features in comparison to the empirical features. In response to the reviewer's suggestion, we conducted an experiment with the classifier fixed as a simplex ETF, contributing valuable insights to our analysis (as the resulting features align closer to the theoretical features). Additionally, we have included the details of the amplification used to generate Figure 6 in the Appendix.