神经网络是如何学会群乘法的 ：一篇论文里的傅里叶、不可约表示与深度的代数学

TL;DR - An exposition of a 2026 arXiv paper showing how neural networks learn sequential finite-group multiplication through irreducible-representation channels. It provides a mathematical account of learning order and why depth can replace exponentially growing width.

• Group Fourier analysis predicts that representation channels are learned sequentially, favoring strong, low-dimensional components.
• A shallow polynomial MLP requires width exponential in sequence length due to the Waring rank of multiplicative interactions.
• RNNs and logarithmic-depth MLPs exploit associativity to reuse binary composition modules at width independent of sequence length.
• Hidden representations become block-diagonal by irreducible representation, separating algebraic channels across layers.