Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry
This work addresses machine learning challenges in quantum physics and chemistry by leveraging symmetries, though it appears incremental as it builds on existing kernel methods.
The authors derived symmetric and antisymmetric kernels by modifying conventional kernels, enabling efficient evaluation in high-dimensional spaces and reducing training data size through symmetry exploitation, as demonstrated with quantum physics and chemistry examples.
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.