Learning in RKHM: a $C^*$-Algebraic Twist for Kernel Machines

TL;DR

This paper extends supervised learning frameworks from RKHS and vvRKHS to RKHM, leveraging $C^*$-algebras to create more powerful kernels capable of capturing complex data interactions, such as in image analysis.

Contribution

It introduces a novel $C^*$-algebraic framework for kernel machines, expanding the representation capacity beyond traditional RKHS-based methods.

Findings

Enables construction of kernels with greater expressive power.

Allows analysis of data interactions, e.g., Fourier components in images.

Demonstrates potential for improved learning performance.

Abstract

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.