Deep learning with kernels through RKHM and the Perron-Frobenius operator

Yuka Hashimoto, Masahiro Ikeda, Hachem Kadri

Keywords: kernel method. generalization bound. C*-algebra. Perron-Frobenius operator and Koopman operator.

TL;DR: We propose deep RKHM and analyzed it through $C^*$-algebra and the Perron-Frobenius operators.

Abstract: 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.

Submission Number: 3970