  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.
