有限线性逼近类的广义频率原理证明
General Frequency Principle for Linear Approximation Classes
把频率原理做成了可证明的版本：什么条件下低频先被学到、什么时候不成立，用 Legendre 和三角基做了实验验证。
arXiv 论文 2610.41179 研究梯度下降训练下有限线性逼近类的广义频率原理。论文证明在适当假设下，学习振幅相对于目标振幅在高频处消失，且在整个训练过程中一致成立。作者给出了低频优先的充分条件，覆盖单频目标单独训练和两个频率分量在同一轮训练两种情形。论文指出该原理并不意味着低频总是收敛更快，而是满足特定比较条件的两个频率中，较慢学习的分量在常见情形下频率更高。数值实验用 Legendre 基和三角基验证了理论结果。
General Frequency Principle for Linear Approximation Classes
In this paper, we study a general frequency principle under gradient descent training for the finite linear approximation classes. Under suitable assumptions, we prove that the learner amplitude relative to the target amplitude vanishes at high frequencies, uniformly throughout training. We also establish sufficient conditions for low-frequency priority, covering single-frequency targets trained separately and two frequency components trained in the same run. This general frequency principle does not imply that lower frequencies always converge faster. Instead, for two frequencies that satisfy certain comparison conditions, one component will take longer to learn than the other; and in common cases, the longer component has higher frequency. Numerical experiments with Legendre and trigonometric bases validate the theoretical results.