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.

提出了一类基于谱截断的正定核函数，其主要特点是输入和输出都是函数，引入了一个描述核中乘积的非交换性的截断参数n。当n趋于无穷大时，该核函数接近现有的可交换核函数；当n是有限的时候，核函数表现出不同的行为，非交换性在数据函数域上引起了相互作用。通过设置适当的n值，可以在表示能力和表示空间的复杂度之间取得平衡，从而超越以前的可交换核函数。

谱截断核：$C^*$代数核机中的非交换性