This work bridges two important concepts: the Neural Tangent Kernel (NTK),
which captures the evolution of deep neural networks (DNNs) during training,
and the Neural Collapse (NC) phenomenon, which refers to the emergence of
symmetry and structure in the last-layer features of well-trained
classification DNNs. We adopt the natural assumption that the empirical NTK
develops a block structure aligned with the class labels, i.e., samples within
the same class have stronger correlations than samples from different classes.
Under this assumption, we derive the dynamics of DNNs trained with mean squared
(MSE) loss and break them into interpretable phases. Moreover, we identify an
invariant that captures the essence of the dynamics, and use it to prove the
emergence of NC in DNNs with block-structured NTK. We provide large-scale
numerical experiments on three common DNN architectures and three benchmark
datasets to support our theory.

本研究介绍了神经切比雪夫核（NTK）和神经塌缩（NC）现象对深度神经网络（DNN）训练的影响，提出了经验 NTK 应发展为与类标签对齐的块状结构，分析了 DNN 的动态并证明了块状 NTK 中存在 NC 现象。通过数值实验验证了理论的正确性。

神经（正切核）崩塌

Neural (Tangent Kernel) Collapse

We consider the problem associated to recovering the block structure of an
Ising model given independent observations on the binary hypercube. This new
model, called the Ising blockmodel, is a perturbation of the mean field
approximation of the Ising model known as the Curie-Weiss model: the sites are
partitioned into two blocks of equal size and the interaction between those of
the same block is stronger than across blocks, to account for more order within
each block. We study probabilistic, statistical and computational aspects of
this model in the high-dimensional case when the number of sites may be much
larger than the sample size.

研究 Ising 模型的块结构恢复问题，其中 Ising 块模型的区别在于将站点分成两个块，同一块之间的交互比跨块之间的交互更强，以解释每个块内的更多顺序。研究了该模型在高维情况下的概率、统计和计算方面。