A Study of Neural Collapse Phenomenon: Grassmannian Frame, Symmetry and Generalization

Papyan et al. (2020) conducted extensive experiments to reveal the NeurCol phenomenon. This phenomenon occurs during the Terminal Phase of Training (TPT), which starts from the epoch that the training accuracy has reached $100\%$. During TPT, training error is zero, but cross-entropy loss keeps decreasing. To describe this phenomenon more clearly, we introduce several necessary notations first. We denote the class number as $C$ and feature dimension as $d$. Here, we consider classifiers with the form $logit=\boldsymbol{M}\boldsymbol{z}=[\langle M_{1},\boldsymbol{z}\rangle,\dots,\langle M_{C},\boldsymbol{z}\rangle]^{T}$, where $\boldsymbol{M}\in\mathbb{R}^{d\times C}$ is the linear classifier, and $\boldsymbol{z}$ is the feature of a smaple obtained from a deep feature extractor. The classification result is given by selecting the maximum score of $logit$. Given a balanced dataset, we denote the feature of $i$-th sample in $y$-th category as $\boldsymbol{z}_{y,i}$. Specifically, when the model is trained on a balanced dataset, its last layer would converge to the following manifestations:

• NC1

  Variability Collapse All samples belonging the same class converge to the class mean: $\|\boldsymbol{z}_{y,i}-\bar{\boldsymbol{z}}_{y}\|\rightarrow 0,\forall y,\forall i$ where $\bar{\boldsymbol{z}}_{y}=\text{Ave}_{i}\left(\boldsymbol{z}_{y,i}\right)$ denote the class-center of $y$-th class;

• NC2

  Convergence to Self Duality The samples and classifier belonging the same class converge to the same: $\|\boldsymbol{z}_{y,i}-M_{y}\|\rightarrow 0,\forall y,\forall i$;

• NC3

  Convergence to Simplex ETF The classifier weight converges to the vertices of Simplex ETF;

• NC4

  Nearest Classification The learned classifier behaves like the nearest classifier, $i.e.\arg\max_{y}\langle M_{y},z\rangle\rightarrow\arg\min_{y}\|z-\bar{\boldsymbol{z}}_{y}\|$.

In NC3, Simplex ETF is an interesting structure. Note that there exists two different objects: Simplex ETF and ETF. ETF is rooted from Frame Theroy (refer to next subsection), while Papyan et al. (2020) introduced the Simplex ETF as a new definition in the context of the NeurCol phenomenon. They made some extensions to the original definition of ETF by introducing an orthogonal projection matrix and a scale factor. Here, we provide the definition of Simplex ETF:

Simplex ETF is a structure with high symmetry, as every pair of vectors has equal angles (Equiangular property). However, it has a limitation: it only exists when feature dimension $d$ is larger than class number $C$. Recently, a work Liu et al. (2023) removed this restriction by proposing Generalized Neural Collapse hypothesis. In their hypothesis, Hyperspherical Uniformity is introduced to generalize Equiangular property. Hyperspherical Uniformity means features of every class is distributed uniformly on a hypersphere with maximal distance.

We have discovered that Grassmannian Frame, a mathematical object from Frame Theory, is a suitable candidate for meeting Hyperspherical Uniformity. Frame Theory is a fundamental research area in mathematics Casazza and Kutyniok (2012) that provides a framework for the analysis of signal transmission and reconstruction. In communication field, certain frame properties have been shown to be optimal configurations in various transmission problems. For instance, Uniform and Tight frame are optimal codes in erasure problem Casazza and Kovačević (2003) and non-orthogonal communication schemes Ambrus et al. (2021). Grassmannian Frame Holmes and Paulsen (2004); Strohmer and Heath Jr (2003), as a more specialized example, not only satisfies the Uniform and Tight properties but also meets the minimized maximal correlation property. This property makes us confident that Grassmannian Frame satisfies Hyperspherical Uniformity.

Here, we provide a series of definitions in Frame Theory¹¹1 Frame Theory has more general definitions in Hilbert space. Definitions we provided here are actually based on Euclidean space. See Strohmer and Heath Jr (2003) for general definitions. :

We can now define the Grassmannian frame:

We also introduce Equiangular Tight Frame (ETF).

The following theorem relates ETF and Grassmannian Frame:

This theorem tells ETF is the special case of Grassmannian Frame and how a Grassmannian frame $\{\zeta_{i}\}_{i=1}^{C}$ can be Equiangular: if and only if it can achieve the Welch Bound. Intuitively, if $d$ is large enough, the correlation between every two vectors in $\{\zeta_{i}\}_{i=1}^{C}$ can be minimized equally to achieve Equiangular property. Otherwise, Equiangular property can not be guaranteed.

Motivation Our motivation for considering Grassmannian Frame as a potential structure for the Generalized Neural Collapse hypothesis is that it satisfies two important properties: it has minimized maximal correlation ($i.e.$, Hyperspherical Uniformity) and exists for any vector number $C$ and dimension $d$. We will provide theoretical supports for this argument in the next section.

In communication field, Grassmannian frame is not only the optimal $2$-erasure code Holmes and Paulsen (2004), but also the optimal code in Gaussian Channel Papyan et al. (2020); Shannon (1959):

This theorem is essentially adopted from the Corollary.4 of Papyan et al. (2020), which is the first study to identify NeurCol phenomenon. However, they only validated this result with Simplex ETF and did not further investigate this structure.

Then we consider the classification from the optimization perspective. We start from the cross-entropy minimization problem.

Notations Denote feature dimension of the last layer as $d$ and class number as $C$. The linear classifier is $\boldsymbol{M}=[M_{1},\cdots,M_{C}]\in\mathbb{R}^{d\times C}$. Given a balanced dataset $\boldsymbol{Z}=\{\{\boldsymbol{z}_{y,i}\in\mathbb{R}^{d}\}_{i=1}^{N/C}\}_{y=1}^{C}$ where every class has $N/C$ samples, we use $\boldsymbol{z}_{y,i}$ to represent the feature of $i$-th sample in $y$-th class.

Optimization Objective Since modern deep models are highly overparameterized, we assume the models have infinite capacity and can fit any dataset. Therefore, we directly optimize sample features to simplify analysis Yang et al. (2022); Zhu et al. (2021):

As the Proposition.2 of Fang et al. (2021) highlighted, NeurCol occurs only if features and classifiers are $\ell_{2}$ norm bounded. Therefore, following previous work Lu and Steinerberger (2022); Fang et al. (2021); Yaras et al. (2022b), we introduce $\ell_{2}$ norm constraint into (1):

where the norms of features and linear classifiers are bounded by $\rho$. Then to perform optimization directly, we turn to the following unconstrained feature models Zhu et al. (2021); Zhou et al. (2022a):

Compared with (1), $\|M_{y}\|^{2}$ and $\|\boldsymbol{z}_{y,i}\|^{2}$ are added in (3). In deep learning, the newly added terms can be seen as the weight decay, while $\lambda$ and $\omega$ are factors of weight decay. This can limit the norm of the sample features $\boldsymbol{Z}$ and linear classifier $\boldsymbol{M}$, that is, given any $\lambda,\omega\in(0,\infty)$, there exists a $\rho(\lambda,\omega)$ such that $\rho(\lambda,\omega)\geq\|M_{y}\|$ and $\rho(\lambda,\omega)\geq\|\boldsymbol{z}_{y,i}\|$ when (3) converges.

Gradient Descent Ji et al. (2022) analyses Gradient Flow of (1), while different from them, we consider the Gradient Descent of (3):

where we use upper script $(t)$ to denote optimized variables in the $t$-th iterations, $\alpha$ and $\beta$ are learning rates. Note that when $\lambda,\omega\rightarrow 0$, the optimization problem (3) is equivalent to (1) since norm constraint condition (2) vanishes, $i.e.\rho\rightarrow\infty$.

We conduct a simulation experiment to visualize the convergence of Generalized Neural Collapse in a $2$-dimensional feature space with $4$ classes. A GIF animation can be found HERE. Please refer to Appendix.G for a detailed description and visual representation of this experiment.

Next, we consider the generalization of classification problems. While correlation measures the similarity between two vectors in Frame Theory, in classification problems, margin is a similar concept but with an opposite degree. Therefore, our analysis of generalization focuses on margin.

Notations Consider a $C$ classes classification problem. Suppose sample space is $\mathcal{X}\times\mathcal{Y}$, where $\mathcal{X}$ is data space and $\mathcal{Y}=\{1,\dots,C\}$ is label space. We assume class distribution is $\mathcal{P}_{\mathcal{Y}}=\left[p(1),\dots,p(C)\right]$, where $p(c)$ denote the proportion of class $c$. Let the training set $S=\{(\boldsymbol{x}_{i},y_{i})\}_{i=1}^{N}$ be drawn i.i.d from probability $\mathcal{P}_{\mathcal{X}\times\mathcal{Y}}$. For $y$-class samples in $S$, we denote $S_{y}=\{\boldsymbol{x}|(\boldsymbol{x},y)\in S\}$ and $|S_{y}|=N_{y}$. The form of classifiers is $logit=\boldsymbol{M}^{T}f(\boldsymbol{x};\boldsymbol{w})=[\langle M_{1},f(\boldsymbol{x};\boldsymbol{w})\rangle,\dots,\langle M_{C},f(\boldsymbol{x};\boldsymbol{w})\rangle]$, where $\boldsymbol{M}\in\mathbb{R}^{d\times C}$ is the last linear layer, and $f(\cdot;\boldsymbol{w})\in\mathbb{R}^{d}$ is the feature extractor parameterized by $\boldsymbol{w}$. We use a tuple $(\boldsymbol{M},f(\cdot;\boldsymbol{w}))$ to denote the classifier.

First, give the definition of margin:

The following lemma establishes the relationship between margin and correlation in NeurCol:

By substituting the conclusions of NC1-3 into the definition of margin, we can prove this lemma straightforwardly. It says given the maximal norm $\rho$, margin $\gamma_{i,j}$ and correlation $\langle M_{i},M_{j}\rangle$ is a pair of opposite quantities. Then we propose the Multiclass Margin Bound.

Recall NeurCol occurs when class distribution is uniform. We consider this case.

Minority Collapse Fang et al. (2021) have identified a related phenomenon called Minority Collapse, which is an imbalanced version of NeurCol. Specifically, they observed that when the training set is extremely imbalanced, the features of the minority class tend to converge to be parallel. Our Multiclass Margin Bound can offer explanations for the generalization of this phenomenon.

First, we introduce two kinds of equivalence of frame Holmes and Paulsen (2004); Bodmann and Paulsen (2005):

The Grassmannian Frame is a geometrically symmetrical structure, with its symmetry stemming from the invariance of two transformations: rotation and permutation. Specifically, after a rotation or permutation (or their combination), the frame still satisfies the minimized maximal correlation property, as only the frame’s direction and order change. We are curious about how these equivalences affect the performance of models. In machine learning, we typically consider two aspects of a model: optimization and generalization. Given that the training set and classification model (backbone and capacity) are the same, we argue that models with equivalent Grassmannian Frames will exhibit the same optimization performance. However, there is no reason to believe that these models will also have the same generalization performance. As such, we propose the following question:

Is generalization of models invariable to symmetric transformations of Grassmannian Frame?

To explore this question, we conduct a series of experiments. Our experimental results lead to an interesting conclusion:

Optimization performance of models is not affected by Rotation and Permutation transformation of Grassmannian Frame, but generalization performance is.

This newly discovered phenomenon, which we call Symmetric Generalization, contradicts a recent argument made in Yang et al. (2022). The authors of that work claimed that since modern neural networks are often overparameterized and can learn features with any direction, fixing linear classifiers as a Simplex ETF is sufficient, and there is no need to learn it. Our findings challenge this viewpoint.

To investigate the impact of Rotation and Permutation transformations of Grassmannian Frame on the generalization performance of deep neural networks, we conducted a series of experiments.

How to Reveal the Anwser We generate $10$ Grassmannian Frame with different Rotation and Permutation. Then, we train the same network architecture 10 times. In each time, the linear classifier are loaded from a pre-generated equivalent Grassmannian Frame and fixed during training. To ensure a completely same optimization process (mini-batch, augmentation, and parameter initialization), we use the same random seed for each training. Once the NeurCol phenomenon occurrs, we know that the $10$ models have learned different Grassmannian Frames. Finally, we compare their generalization performances by evaluating the cross-entropy loss and accuracy on a test set.

Generation of Equivalent Grassmannian Frame Our Theorem.3.2 naturally offers a numerical method for generating Grassmannian Frame. Given class number and feature dimension, we use Gradient Descent (4) on the unconstrained feature models (3) to generate Grassmannian Frame. Then given a Grassmannian Frame $\{M_{y}\}_{y=1}^{C}$ in $\mathbb{R}^{d}$, if it is denoted as $\boldsymbol{M}=[M_{1},\cdots,M_{C}]\in\mathbb{R}^{d\times C}$, we can use $R\boldsymbol{M}P$ to denote its equivalent frame. where $P\in Permutation(C)$ and $R\in SO(d)$:

Note that $Permutation(C)$ and $SO(d)$ act on vector orders and directions of $\{M_{y}\}_{y=1}^{C}$ respectively. Refer to Appendix.F.2 for code implementation.

Models with Different Features Yang et al. (2022) point out in their Theorem.1: if the linear classifier is fixed as Simplex ETF, then the final features learned by the model would converge to be Simplex ETF with the same direction to classifier. Following their work, to let models to learn equivalent Grassmannian Frame, we initialize the linear classifier as equivalent Grassmannian Frame and do not perfrom optimization on it during training. In this way, when NeurCol occurs, models have learned equivalent Grassmannian Frame.

Network Architecture and Dataset Our experiments involve two image classification datasets: CIFAR10/100 Krizhevsky (2009). And for every dataset, we use three different convolutional neural networks to verify our finding, including ResNet He et al. (2016), Vgg Simonyan and Zisserman (2014), DenseNet Huang et al. (2017). Both datasets are balanced with 10 and 100 classes respectively, each having $500$ and $5,000$ training images per class. To meet the larger number of classes than feature dimensions, we use $6$ and $64$ as the feature dimensions, respectively. Then, to obtain different dimensional feature for every backbone, we attach a linear layer after the end of backbone, which can transform feature dimensions.

Training To make NeurCol appear, we follow Papyan et al. (2020)’s practice. More details refer to Appendix.F.1.

Results Table.1 presents results on CIFAR10 and Vgg11 with different Permutation. All metrics are given when the model converges to NeurCol ($100\%$ accuracy and zero loss on training set). We observe that, even though all experiments achieved $0$ cross-entropy loss and $100\%$ accuracy, they still exhibit significant differences in test loss and accuracy. This implies although permutation hardly affects optimization, it has a significant impact on generalization. Table.2 provides results of different dataset, backbone and two transformations to reveal the same phenomenon. These experimental results answer the question we posed in Section.4.1, demonstrating that different feature order and direction can influence generalization performance of models.

We aim to explain why Grassmannian Frame with different Permutation can lead to different generalization performance theoretically. Most of symbol definitions are adopted from Section.3.3.

Permutation We provide an example to give the insight of Symmetric Generalization of Permutation. Consider a Grassmannian Frame $\{M_{i}\}_{i=1}^{4}$ living in $\mathbb{R}^{2}$, it resembles a cross (Theorem III.1 of Benedetto and Kolesar (2006)). As shown in Figure.1(a), there are four classes that correspond to different feature $M_{i}$. Obviously, since Dog and Cat look similar to each other, they deserve to have a smaller margin in the feature space (near to each other). The same goes for the other two categories. However, if we swap the features of Truck and Dog to increase the distance between dogs and cats, as shown in Figure.1(b), the semantic relationship in the feature space would be disrupted. We argue that this can harm the model’s training and lead to worse generalization.

Rotation Symmetric Generalization of Rotation has been completely beyond our initial expectation. We believe that the margin or correlation between features is the most effective tool to understand NeurCol phenomenon. However, it fails to explain why different directions of features have different generalization. In Deep Learning community, Implicit Bias phenomenon is a possible way to approach this finding. Soudry et al. (2018) has proved that the Gradient Descent optimization leads to weight direction that maximizes the margin when using logistic loss and linear models. As a further progress, Lyu and Li (2020) extends this result to homogeneous neural networks. We speculate that the explanations for Symmetric Generalization of Rotation may be hidden within the layers of neural networks. Therefore, studying the Implicit Bias of deep models layer by layer could be a promising direction for future research. This is beyond the scope of our current work, and we leave it as a topic for our future work.