Neural collapse describes the geometry of activation in the final layer of a deep neural network when it is trained beyond performance plateaus. Open questions include whether neural collapse leads to better generalization and, if so, why and how training beyond the plateau helps. We model neural collapse as an information bottleneck (IB) problem in order to investigate whether such a compact representation exists and discover its connection to generalization. We demonstrate that neural collapse leads to good generalization specifically when it approaches an optimal IB solution of the classification problem. Recent research has shown that two deep neural networks independently trained with the same contrastive loss objective are linearly identifiable, meaning that the resulting representations are equivalent up to a matrix transformation. We leverage linear identifiability to approximate an analytical solution of the IB problem. This approximation demonstrates that when class means exhibit $K$-simplex Equiangular Tight Frame (ETF) behavior (e.g., $K$=10 for CIFAR10 and $K$=100 for CIFAR100), they coincide with the critical phase transitions of the corresponding IB problem. The performance plateau occurs once the optimal solution for the IB problem includes all of these phase transitions. We also show that the resulting $K$-simplex ETF can be packed into a $K$-dimensional Gaussian distribution using supervised contrastive learning with a ResNet50 backbone. This geometry suggests that the $K$-simplex ETF learned by supervised contrastive learning approximates the optimal features for source coding. Hence, there is a direct correspondence between optimal IB solutions and generalization in contrastive learning.

翻译：神经坍缩描述了深度神经网络在性能平稳期后继续训练时，其最终层激活的几何结构。待解决的问题包括：神经坍缩是否促进更好的泛化，若如此，为何以及如何通过超越平稳期的训练实现这一效果。我们将神经坍缩建模为信息瓶颈问题，旨在探究是否存在这种紧凑表示，并揭示其与泛化的关联。我们证明，当神经坍缩接近分类问题的最优信息瓶颈解时，它特别能带来良好的泛化。近期研究表明，使用相同对比损失函数独立训练的两个深度神经网络具有线性可识别性，即所得表示在矩阵变换下等价。我们利用线性可识别性近似求解信息瓶颈问题的解析解。这一近似表明，当类别均值呈现 $K$-单纯形等角紧框架行为时（例如，CIFAR10的$K$=10，CIFAR100的$K$=100），这些均值恰好对应信息瓶颈问题的关键相变点。一旦信息瓶颈问题的最优解包含所有这些相变，性能平稳期便会出现。我们还证明，使用ResNet50骨干网络的监督对比学习可将生成的 $K$-单纯形等角紧框架嵌入 $K$维高斯分布。这一几何结构表明，监督对比学习习得的 $K$-单纯形等角紧框架近似于源编码的最优特征。因此，对比学习中的最优信息瓶颈解与泛化之间存在直接对应关系。