Understanding how structured internal structure emerges during neural network training is central to the study of deep learning. We investigate this phenomenon through the group composition task, where a two-layer neural network is trained to predict $g_1 \star g_2$ for elements of a finite group $G$. By lifting the projected gradient flow to the Fourier domain, we demonstrate that the training dynamics are governed by a Riemannian gradient ascent on a representation-theoretic energy functional. We prove that, under random initialization, this flow drives each neuron to converge almost surely toward a single irreducible representation, while the cross-layer Fourier coefficients achieve a rotational rank-one alignment. This framework provides a representation-theoretic account of feature learning and characterizes a novel low-rank compression phenomenon for matrix-valued group representations. Moreover, for Abelian groups, we provide a complete population-level description: random initialization promotes uniform diversification across nontrivial representations and induces Haar-uniform phases, jointly approximating the indicator via a majority-vote mechanism. We further prove that both phase alignment and representation competition emerge with exponential convergence rates.

翻译：理解神经网络训练过程中结构化内部表征如何涌现，是深度学习研究的核心问题。我们通过群组任务研究这一现象：训练一个双层神经网络来预测有限群$G$中元素的$g_1 \star g_2$。通过将投影梯度流提升至傅里叶域，我们证明训练动力学受表征论能量泛函上的黎曼梯度上升所支配。我们证明，在随机初始化条件下，该梯度流驱动每个神经元几乎必然收敛至单个不可约表示，而跨层的傅里叶系数实现旋转秩一对齐。该框架提供了特征学习的表征论解释，并刻画了矩阵值群表示的新的低秩压缩现象。此外，对于阿贝尔群，我们提供完整的总体描述：随机初始化促进非平凡表示上的均匀多样化，并诱导哈尔均匀相位，通过多数投票机制联合近似指示器。我们进一步证明相位对齐和表示竞争均以指数收敛速率出现。