单神经元模加法中傅里叶对齐的一个反例
A Counterexample to Fourier Alignment in Single-Neuron Modular Addition
浏览论文内容
中文总结 AI 辅助
该研究针对MAIS-O60问题构造反例，证明单神经元模加法训练中傅里叶对齐不成立，且失效情况在多种条件下普遍存在。
中文摘要 AI 辅助
我们对MAIS-O60问题给出否定解答。我们首先构造一个例子：初始激活的ReLU神经元在有限时间内完全失活，之后保持在一个极限状态，其傅里叶能量在所有非零实频率类中均匀分布。该反例适用于初始条件的开集，因此在高斯初始化下以正概率出现。由GPT-5.6 Sol编写的附录进一步强化了该反例，表明在ReLU'(0)=0的约定下，对于ReLU的光滑死区近似，以及固定步长全批量梯度下降，从初始条件的开集出发的每一条Clarke轨迹都会出现相同的失效情况。因此，在单神经元进行模加法训练时，单频对齐并非普遍结论。
英文摘要
We give a negative solution to MAIS-O60. We first construct an example in which an initially active ReLU neuron becomes completely inactive in finite time and thereafter remains frozen at a limit whose Fourier energy is equally distributed among all nonzero real frequency classes. The counterexample holds on an open set of initial conditions and therefore occurs with positive probability under Gaussian initialization. An appendix prepared by GPT-5.6 Sol strengthens the counterexample by showing that the same failure can occur for every Clarke trajectory from an open set of initial conditions, under the convention $\mathrm{ReLU}'(0)=0$, for smooth dead-zone approximations of ReLU, and for fixed-step full-batch gradient descent. Thus, single-frequency alignment is not a general consequence of training a single neuron on modular addition.
