Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry. Issue 4 (6th August 2021)

• Record Type: Journal Article
• Authors: Klus, Stefan; Gelß, Patrick; Nüske, Feliks; Noé, Frank
• Is Part Of: Machine learning: science and technology. Volume 2:Issue 4(2021)
• Publication Date: 2021-08-06
• Subjects: symmetry and antisymmetry -- reproducing kernel Hilbert spaces -- quantum physics -- quantum chemistry; 006.31
• DOI: 10.1088/2632-2153/ac14ad
• Languages: English
• ISSNs: 2632-2153

Abstract

Abstract: 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.