Reproducing kernel Hilbert $C^*$-module (RKHM) is a generalization of reproducing kernel Hilbert space (RKHS) by means of $C^*$-algebra, and the Perron-Frobenius operator is a linear operator related to the composition of functions. Combining these two concepts, we present deep RKHM, a deep learning framework for kernel methods. We derive a new Rademacher generalization bound in this setting and provide a theoretical interpretation of benign overfitting by means of Perron-Frobenius operators. By virtue of $C^*$-algebra, the dependency of the bound on output dimension is milder than existing bounds. We show that $C^*$-algebra is a suitable tool for deep learning with kernels, enabling us to take advantage of the product structure of operators and to provide a clear connection with convolutional neural networks. Our theoretical analysis provides a new lens through which one can design and analyze deep kernel methods.

翻译：再生核希尔伯特$C^*$-模（RKHM）是通过$C^*$-代数对再生核希尔伯特空间（RKHS）的推广，而Perron-Frobenius算子是与函数复合相关的线性算子。结合这两个概念，我们提出了deep RKHM——一种用于核方法的深度学习框架。我们在此框架下推导了新的Rademacher泛化界，并通过Perron-Frobenius算子对良性过拟合现象给出了理论解释。借助$C^*$-代数的性质，该界对输出维度的依赖性较现有结果更为宽松。我们证明$C^*$-代数是适用于核深度学习的工具，它使我们能够利用算子的乘积结构，并与卷积神经网络建立清晰的联系。我们的理论分析为设计和分析深度核方法提供了新的视角。