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.

翻译：我们通过对称和反对称常规内核并分析其特性来得出对称和反对称内核,特别是,我们计算由此产生的多面内核的空间特性,证明由对称和反对称高斯内核引出的重复的Hilbert内核内核空间在对称和反对称功能空间中密度很大,并提议对反对称内核进行一个Slater决定因素表示,这样就可以有效地评价,即使状态空间是高维的。此外,我们还表明,通过利用对称或反对称,培训数据集的大小可以大大缩小。结果以指导性例子和简单的量子物理和化学应用来说明。