In this paper, we propose a new class of positive definite kernels based on the spectral truncation, which has been discussed in the fields of noncommutative geometry and $C^*$-algebra. We focus on kernels whose inputs and outputs are functions and generalize existing kernels, such as polynomial, product, and separable kernels, by introducing a truncation parameter $n$ that describes the noncommutativity of the products appearing in the kernels. When $n$ goes to infinity, the proposed kernels tend to the existing commutative kernels. If $n$ is finite, they exhibit different behavior, and the noncommutativity induces interactions along the data function domain. We show that the truncation parameter $n$ is a governing factor leading to performance enhancement: by setting an appropriate $n$, we can balance the representation power and the complexity of the representation space. The flexibility of the proposed class of kernels allows us to go beyond previous commutative kernels.

翻译：本文基于非交换几何与$C^*$代数领域讨论过的谱截断概念，提出了一类新的正定核。我们关注输入输出均为函数的核，通过引入描述核中乘积非交换性的截断参数$n$，推广了现有核（如多项式核、乘积核与可分离核）。当$n$趋于无穷时，所提出的核收敛于现有的交换核；当$n$有限时，它们表现出不同行为，且非交换性会沿数据函数域诱导相互作用。我们证明截断参数$n$是导致性能提升的主导因素：通过设置合适的$n$，可以平衡表示能力与表示空间的复杂度。所提出核类的灵活性使我们能够超越以往的交换核。