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Supervised learning in reproducing kernel Hilbert space (RKHS) andvector-valued RKHS (vvRKHS) has been investigated for more than 30 years. Inthis paper, we provide a new twist to this rich literature by generalizingsupervised learning in RKHS and vvRKHS to reproducing kernel Hilbert$C^*$-module (RKHM), and show how to construct effective positive-definitekernels by considering the perspective of $C^*$-algebra. Unlike the cases ofRKHS and vvRKHS, we can use $C^*$-algebras to enlarge representation spaces.This enables us to construct RKHMs whose representation power goes beyondRKHSs, vvRKHSs, and existing methods such as convolutional neural networks. Ourframework is suitable, for example, for effectively analyzing image data byallowing the interaction of Fourier components.

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