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

arXiv: https://arxiv.org/abs/2103.17233 · HTML: https://mlsanity.com/paper/2103.17233/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 46 references that are in the map.
- First posted on arXiv 31 Mar 2021.

## Builds on (most cited first, 11 of 11)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [Kernel Flows: from learning kernels from data into the abyss](https://mlsanity.com/paper/1808.04475/) (2018)
- [Kernel-based approximation of the Koopman generator and Schr"odinger operator](https://mlsanity.com/paper/2005.13231/) (2020)
- [Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1807.09331/) (2018)
- [On Representing (Anti)Symmetric Functions](https://mlsanity.com/paper/2007.15298/) (2020)
- [Convergence to the fixed-node limit in deep variational Monte Carlo](https://mlsanity.com/paper/2010.05316/) (2020)
- [Approximating the Permanent by Sampling from Adaptive Partitions](https://mlsanity.com/paper/1911.11856/) (2019)
- [Spectral Analysis of Symmetric and Anti-Symmetric Pairwise Kernels](https://mlsanity.com/paper/1506.05950/) (2015)
- [Symmetric and antisymmetric properties of solutions to kernel-based machine learning problems](https://mlsanity.com/paper/1606.08501/) (2016)

## Built on by (most cited first, 5 of 5)

- [Smooth, exact rotational symmetrization for deep learning on point clouds](https://mlsanity.com/paper/2305.19302/) (2023)
- [Algorithmic Differentiation for Automated Modeling of Machine Learned Force Fields](https://mlsanity.com/paper/2208.12104/) (2022)
- [Scalable learning of potentials to predict time-dependent Hartree-Fock dynamics](https://mlsanity.com/paper/2408.04765/) (2024)
- [Uniform Ck Approximation of G-Invariant and Antisymmetric Functions, Embedding Dimensions, and Polynomial Representations](https://mlsanity.com/paper/2403.01339/) (2024)
- [Physically-informed change-point kernels for structural dynamics](https://mlsanity.com/paper/2506.11625/) (2025)

## Abstract (arXiv)

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.