A central question in vector- and function-valued learning is how to design kernels that capture both local and non-local interactions while remaining computationally tractable. Existing operator-valued kernels offer only partial answers: separable kernels are efficient but fail to model interactions across the function domain, while commutative kernels capture only pointwise structure. To address this, we propose spectral truncation kernels, a new class of positive definite kernels for vector- and function-valued learning based on spectral truncation and $C^*$-algebra. By allowing noncommutative products in the kernel construction, the proposed kernels induce interactions across the data function domain and fill the gap between existing separable and commutative kernels. In addition, by using the $C^*$-algebraic framework, we reduce the computational cost compared to the existing vector-valued RKHS framework with operator-valued kernels.

翻译：向量值与函数值学习中的一个核心问题是如何设计既能捕捉局部与非局部相互作用、又保持计算可行性的核函数。现有算子值核仅能提供部分答案：可分核效率高但无法建模函数域上的交互，而交换核仅能捕捉逐点结构。为解决这一问题，我们提出谱截断核——一种基于谱截断与$C^*$代数的新型向量值与函数值学习的正定核函数。通过允许核构造中的非交换乘积，所提核能诱导函数数据域上的交互作用，填补现有可分核与交换核之间的空白。此外，利用$C^*$代数框架，我们相比现有基于算子值核的向量值再生核希尔伯特空间框架降低了计算成本。