We derive symmetric and antisymmetric kernels by symmetrizing and antisymmetrizing conventional kernels and analyze their properties. In particular, we compute the feature space dimensions of the resulting polynomial kernels, prove that the reproducing kernel Hilbert spaces induced by symmetric and antisymmetric Gaussian kernels are dense in the space of symmetric and antisymmetric functions, and propose a Slater determinant representation of the antisymmetric Gaussian kernel, which allows for an efficient evaluation even if the state space is high-dimensional. Furthermore, we show that by exploiting symmetries or antisymmetries the size of the training data set can be significantly reduced. The results are illustrated with guiding examples and simple quantum physics and chemistry applications.

我们通过对称和反对称常规核来推导对称和反对称核，并分析它们的性质。特别地，我们计算所得多项式核的特征空间维数，证明对称和反对称高斯核引起的再生核希尔伯特空间在对称和反对称函数空间中是稠密的，并提出反对称高斯核的 Slater 行列式表示内核，即使状态空间是高维的，它也可以进行有效的评估。此外，我们表明，通过利用对称性或反对称性，可以显着减少训练数据集的大小。结果通过指导性示例和简单的量子物理和化学应用进行说明。