Learning in RKHM: a $C^*$-Algebraic Twist for Kernel Machines
This work addresses the need for more expressive kernel-based learning methods in machine learning, offering a novel theoretical extension that could enhance performance in domains like image processing, though it appears incremental as it builds on long-established RKHS literature.
The paper tackles the problem of limited representation power in kernel methods by generalizing supervised learning from reproducing kernel Hilbert spaces (RKHS) and vector-valued RKHS to reproducing kernel Hilbert C*-modules (RKHM), enabling the construction of more powerful kernels using C*-algebras. The result is a framework that surpasses existing methods like convolutional neural networks in representation power, particularly for tasks such as image data analysis by facilitating Fourier component interactions.
Supervised learning in reproducing kernel Hilbert space (RKHS) and vector-valued RKHS (vvRKHS) has been investigated for more than 30 years. In this paper, we provide a new twist to this rich literature by generalizing supervised learning in RKHS and vvRKHS to reproducing kernel Hilbert $C^*$-module (RKHM), and show how to construct effective positive-definite kernels by considering the perspective of $C^*$-algebra. Unlike the cases of RKHS and vvRKHS, we can use $C^*$-algebras to enlarge representation spaces. This enables us to construct RKHMs whose representation power goes beyond RKHSs, vvRKHSs, and existing methods such as convolutional neural networks. Our framework is suitable, for example, for effectively analyzing image data by allowing the interaction of Fourier components.