Deep Learning with Kernels through RKHM and the Perron-Frobenius Operator

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.