Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry

Stefan Klus, Patrick Gelß*, Feliks Nüske, Frank Noé

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)
8 Downloads (Pure)


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.

Original languageEnglish
Article number045016
JournalMachine Learning: Science and Technology
Issue number4
Early online date6 Aug 2021
Publication statusPublished - Dec 2021


  • Quantum chemistry
  • Quantum physics
  • Reproducing kernel Hilbert spaces
  • Symmetry and antisymmetry

ASJC Scopus subject areas

  • Artificial Intelligence
  • Human-Computer Interaction
  • Software


Dive into the research topics of 'Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry'. Together they form a unique fingerprint.

Cite this