We derive a new Rademacher complexity bound for deep neural networks using Koopman operators, group representations, and reproducing kernel Hilbert spaces (RKHSs). The proposed bound describes why the models with high-rank weight matrices generalize well. Although there are existing bounds that attempt to describe this phenomenon, these existing bounds can be applied to limited types of models. We introduce an algebraic representation of neural networks and a kernel function to construct an RKHS to derive a bound for a wider range of realistic models. This work paves the way for the Koopman-based theory for Rademacher complexity bounds to be valid for more practical situations.

翻译：我们利用Koopman算子、群表示和再生核希尔伯特空间（RKHSs）推导出一种新的深度神经网络Rademacher复杂度界。所提出的界解释了为何具有高秩权重矩阵的模型能实现良好泛化。尽管已有一些界尝试描述这一现象，但这些现有界仅适用于有限类型的模型。我们引入了神经网络的代数表示及核函数来构建RKHS，从而为更广泛的实际模型推导出复杂度界。这项工作为基于Koopman理论的Rademacher复杂度界在更实际场景中的有效性铺平了道路。