神经网络可证明地学习群组构成的谱表示

摘要

理解神经网络训练过程中结构化内部结构的涌现是深度学习研究的核心问题。我们通过群组合成任务探究这一现象——训练一个双层神经网络预测有限群G中元素g₁★g₂的结果。通过将投影梯度流提升至傅里叶域，我们证明训练动力学由表示论能量泛函上的黎曼梯度上升控制。我们证明，在随机初始化条件下，该流驱动每个神经元几乎必然收敛至单一不可约表示，同时跨层傅里叶系数实现旋转秩一的排列对齐。该框架为特征学习提供了表示论解释，并刻画了矩阵值群表示的一种新型低秩压缩现象。此外，对于阿贝尔群，我们给出了完整的总体层面描述：随机初始化促进了非平凡表示上的均匀多样化，并诱导出哈达玛均匀相位，通过多数投票机制共同逼近指示函数。我们进一步证明相位对齐与表示竞争均以指数收敛速度涌现。

English

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.