Spectral Truncation Kernels: Noncommutativity in $C^*$-algebraic Kernel Machines

ICLR 2025 Conference Submission1463 Authors

Keywords: kernel methods, positive definite kernel, spectral truncation

TL;DR: We propose a new class of positive definite kernels based on the spectral truncation, which address two issues regarding vector-valued RKHSs, the choice of the kernel and the computational cost.

Abstract: $C^*$-algebra-valued kernels could pave the way for the next generation of kernel machines. To further our fundamental understanding of learning with $C^*$-algebraic kernels, we propose a new class of positive definite kernels based on the spectral truncation. We focus on kernels whose inputs and outputs are vectors or functions and generalize typical kernels by introducing the noncommutativity of the products appearing in the kernels. The noncommutativity induces interactions along the data function domain. We show that it is a governing factor leading to performance enhancement: we can balance the representation power and the model complexity. We also propose a deep learning perspective to increase the representation capacity of spectral truncation kernels. The flexibility of the proposed class of kernels allows us to go beyond previous separable and commutative kernels, addressing two of the foremost issues regarding learning in vector-valued RKHSs, namely the choice of the kernel and the computational cost.

Primary Area: learning theory

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Submission Number: 1463