# 神经网络可证明地学习群组合的谱表示

- 来源：HuggingFace Daily Papers（社区热门论文）
- 发布时间：2026-06-02 08:00
- AIHOT 分数：56
- AIHOT 链接：https://aihot.virxact.com/items/cmpzwl5fs00pksltry1j3nik2
- 原文链接：https://arxiv.org/abs/2606.02993

## AI 摘要

通过群组合任务（预测有限群G中两元素乘积），研究两层神经网络训练中的内部结构涌现。将投影梯度流提升到傅里叶域后，训练动力学由表示论能量泛函上的黎曼梯度上升主导。随机初始化下，每个神经元几乎必然收敛到单个不可约表示，跨层傅里叶系数达到旋转秩一对齐。该框架刻画了矩阵值群表示中的低秩压缩现象。对于阿贝尔群，随机初始化促使非平凡表示均匀多样化并诱导Haar均匀相位，通过多数投票机制逼近指示函数。相位对齐与表示竞争以指数速率出现。

## 正文

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.