Neural collapse vs. low-rank bias: Is deep neural collapse really optimal?

We thank the reviewer for a detailed review. The typos pointed out by the reviewer will be corrected in the revision, and we thank the reviewer for spotting those. We address all the other concerns below (including the technical issue about the Sherman-Morrison formula), and we would like to kindly ask the reviewer to raise the score (together with the soundness score) in case no additional issue remains.

An issue with Sherman-Morrison formula.

We carefully checked our computation and believe there is no error. The matrix we have to invert is indeed $I+\frac{1}{r-2}\mathbb{1}\mathbb{1}^T$ (not $I+\frac{1}{r-1}\mathbb{1}\mathbb{1}^T$, as in the review), as this is the main factor of the previous computation of $T_r T_r^T$, where we exchange the multiplier of $\mathbb{1}\mathbb{1}^T$ from $\frac{1}{r-1}$ to $\frac{1}{r-2}$ by pulling $\frac{r-2}{r-1}$ in front of the parenthesis. Then, we use the Sherman-Morrison formula $(A+uv^T)^{-1}=A^{-1}-\frac{A^{-1}uv^TA^{-1}}{1+u^TA^{-1}v}$, where $A$ is the identity matrix and both $u$ and $v$ are $\frac{1}{\sqrt{r-2}}\mathbb{1}$. Thus, we get $I$ as the left summand, $\frac{1}{r-2}\mathbb{1}\mathbb{1}^T$ as the numerator in the right summand, and $1+\frac{r}{r-2}=\frac{2(r-1)}{r-2}$ as the denominator. The term $r-2$ cancels in numerator and denominator, and we are left with the right hand side we write in our computation.

The definition 4 should be split and clarified.

Thank you for pointing this out, we agree that this definition is rather complicated. In the revision, we will split the definition into different bullet points where we distinguish the construction of the intermediate layers and the final layer, highlight the non-negativity of intermediate activations and more clearly define the trainable parameters which are the scales of the involved matrices.

In Figure 2 the rank is 7-8, which is not K=10. This should be discussed more.

The rank 5-8 we found in our experiments is indeed smaller than the rank of DNC solution, which in this case is 10. This is exactly in agreement with Theorem 5, which shows that NC2 solutions are not globally optimal. We will make this explicit clarification in the revision, adding to the discussion in lines 259-263.

The sentence “solution found by gradient descent is the SRG solution” from line 313 should be more justified.

Thank you for pointing this out, we will clarify the sentence in the revision. What we meant is that the particular solution (i.e. a single run) whose gram matrix is displayed in lower row right of Figure 4 is an SRG solution, not that all solutions that SGD finds are SRG. Regarding the justification of this claim, as the reviewer dZSf correctly pointed out, there is a typo in the label, the plotted matrix is in fact $\tilde{M}_5^T\tilde{M}_5.$ However, we note that the gram matrix looks exactly the same (up to scalar scaling) for all other layers (please see our global response where we upload a PDF showing all 5 layers). This can be also seen theoretically since all the involved matrices have by the construction of the SRG solution the same gram matrix. The reason why the gram matrices look exactly like this can be seen from the fact that the $H_l$ matrices are defined to be incidence matrices of the complete graph and their gram matrix is therefore naturally the adjacency matrix of the triangular graph. For a more detailed explanation, we refer you to Lemma 15 and Lemma 16, which are basically entirely devoted to compute the gram matrices and their entries and singular values explicitly. These calculations exactly agree with the plot in Figure 4. We will explain this in more detail in the revision.

Do numerical experiments include the ReLU activation?

Yes, we include the ReLU activation in all our experiments in all layers.

What do we mean by “the rank deficiency is strongest in the middle layers of the MLP head” and how can one see this?

Thank you for this question, we will clarify the sentence. We do not mean that the rank in these middle layers is smaller than in the other layers, but rather that the tail singular values are closer to zero than the singular values of the other layers (i.e., 0.001 instead of 0.01). In fact, by looking closely at the green and red curves corresponding to layers 3 and 4, one can notice that their right tails are slightly below those of the other curves, meaning that the corresponding singular values are closer to 0 than those in the other layers.

Is the assumption on feature class-means being non-negative valid in experiments?

This is a great question. Indeed, in our experiments we observe that the intermediate features of the MLP head are non-negative before the application of the ReLU, regardless of whether we obtain the SRG solution or not. We will clarify this point in the revision.

Thank you for addressing the concerns. I have increased my score based on the responses.

One more suggestion to consider [optional]

• I think it would be better to also quantitatively measure the rank. For example, a simple measure such as effective-rank can be used. If the notion of a low-rank used here is purely qualitative, and quantitative measures have limitations [if any], then such a discussion can further improve the paper quality.

Thank you for the comments and the positive evaluation of our paper. We address all questions below.

line 209 (clarification) about K=2, L=2

Our statement was only meant to say that the cases $K=2$ and $L=2$ were already treated in prior work [51, 48]. We will clarify this in the manuscript.

Why the MSE loss?

We focus on the MSE loss primarily for theoretical reasons, because the MSE loss allows for a much more explicit analysis of the loss function. However, we expect that this is without much loss of generality, as both the MSE loss and the CE loss have similar behavior for well-fitting solutions. In addition, the low-rank bias does not come from the fit part of the loss, but rather from the regularization.

Typo:

Thank you for pointing this out, we will correct it in the revision.

Thank you for the comments and the positive evaluation of our paper. We address your concerns and questions below:

While the SRG solution has low-rank, it is not clear whether a globally optimal solution also has one.

This is a great point. We agree that we do not rigorously show this in our paper. However, we provide an intuitive explanation in lines 140-150. Our view is also aligned with a line of work on low rank bias [1, 2, 3]. In fact, [2] even proves the global optimality of low-rank solutions for sufficiently deep networks. Moreover, we highlight that the reason why SRG achieves a better loss than DNC is primarily its low-rank structure, as can be seen from our Lemma 13 which shows that the cost of intermediate layers linearly depends on their rank. Nevertheless, proving that global optima have low rank in our setting remains an interesting open problem.

Do the last layer’s post activation features follow the NC geometry? Or a different structure?

In our experiments, we do not recover the exact NC2 geometry, see Figure 2 (right) or Figure 4 (middle). In fact, the singular values are not all the same, and the tail is around 2-3x smaller than the largest singular value, which implies that the matrix cannot be orthogonal. This effect could be due to the large weight decay considered in our experimental setup. We also do not observe any specific type of different structure in the last layer.

Do the solutions that don’t achieve SRG structure outperform the SRG solution?

We did not find a solution that outperforms the SRG construction when the number of classes $K$ and number of layers $L$ are moderate. However, when $K$ and $L$ are very large, there are indeed solutions that outperform the SRG construction. This indeed shows that the SRG solution is not optimal in general, and we suspect that even lower rank structures are optimal in those settings.

Do the weight matrices also exhibit low-rank structure? What about the DNC3?

Yes, the weight matrices exhibit the same low-rank structure as features. This can be seen from the formula $W_l=H_{l+1}H_l^\dagger$ for the min-norm interpolator. This shows that the rank of $W_l$ is at most the minimum between the rank of its input and the rank of the output layer features. If those two are the same, $W_l$ must match the rank of the features exactly. Regarding DNC3, in our numerical experiments we do not observe exact alignment of rows of weight matrices with columns of feature matrices, and we do not expect this to happen theoretically. In fact, the rows of the weight matrices are not aligned with the columns of the feature matrices in our SRG solution.

Typo in Figure 4?

You are right, thank you for spotting this. The layer in question is in fact $\tilde{M}_5$ instead of $M_5.$

[1] G. Ongie and R. Willett. "The role of linear layers in nonlinear interpolating networks." arXiv preprint arXiv:2202.00856 (2022).

[2] A. Jacot. "Implicit bias of large depth networks: a notion of rank for nonlinear functions." ICLR, 2023.

[3] A. Jacot. "Bottleneck structure in learned features: Low-dimension vs regularity tradeoff." NeurIPS, 2023.

Thank you for your positive review. If any questions come up during the discussion phase, we’ll be happy to address them.