# Tensor-reduced atomic density representations

arXiv: https://arxiv.org/abs/2210.01705 · HTML: https://mlsanity.com/paper/2210.01705/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 77 references that are in the map.
- First posted on arXiv 2 Oct 2022.

## Builds on (most cited first, 12 of 12)

- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Sketching as a Tool for Numerical Linear Algebra](https://mlsanity.com/paper/1411.4357/) (2014)
- [The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentials](https://mlsanity.com/paper/2205.06643/) (2022)
- [Experiments with Random Projection](https://mlsanity.com/paper/1301.3849/) (2013)
- [Ranking the information content of distance measures](https://mlsanity.com/paper/2104.15079/) (2021)
- [TeaNet: universal neural network interatomic potential inspired by iterative electronic relaxations](https://mlsanity.com/paper/1912.01398/) (2019)
- [Gaussian Moments as Physically Inspired Molecular Descriptors for Accurate and Scalable Machine Learning Potentials](https://mlsanity.com/paper/2109.07421/) (2021)
- [WACSF - Weighted Atom-Centered Symmetry Functions as Descriptors in Machine Learning Potentials](https://mlsanity.com/paper/1712.05861/) (2017)
- [Optimal radial basis for density-based atomic representations](https://mlsanity.com/paper/2105.08717/) (2021)

## Built on by (most cited first, 10 of 10)

- [Smooth, exact rotational symmetrization for deep learning on point clouds](https://mlsanity.com/paper/2305.19302/) (2023)
- [Cartesian atomic cluster expansion for machine learning interatomic potentials](https://mlsanity.com/paper/2402.07472/) (2024)
- [A practical guide to machine learning interatomic potentials -- Status and future](https://mlsanity.com/paper/2503.09814/) (2025)
- [Evaluation of the MACE Force Field Architecture: from Medicinal Chemistry to Materials Science](https://mlsanity.com/paper/2305.14247/) (2023)
- [A General Framework for Equivariant Neural Networks on Reductive Lie Groups](https://mlsanity.com/paper/2306.00091/) (2023)
- [Wigner kernels: body-ordered equivariant machine learning without a basis](https://mlsanity.com/paper/2303.04124/) (2023)
- [Equivariant Graph Network Approximations of High-Degree Polynomials for Force Field Prediction](https://mlsanity.com/paper/2411.04219/) (2024)
- [Automatic feature selection and weighting in molecular systems using Differentiable Information Imbalance](https://mlsanity.com/paper/2411.00851/) (2024)
- [Equivariant Matrix Function Neural Networks](https://mlsanity.com/paper/2310.10434/) (2023)
- [Optimal Invariant Bases for Atomistic Machine Learning](https://mlsanity.com/paper/2503.23515/) (2025)

## Abstract (arXiv)

Density based representations of atomic environments that are invariant under Euclidean symmetries have become a widely used tool in the machine learning of interatomic potentials, broader data-driven atomistic modelling and the visualisation and analysis of materials datasets.The standard mechanism used to incorporate chemical element information is to create separate densities for each element and form tensor products between them. This leads to a steep scaling in the size of the representation as the number of elements increases. Graph neural networks, which do not explicitly use density representations, escape this scaling by mapping the chemical element information into a fixed dimensional space in a learnable way. We recast this approach as tensor factorisation by exploiting the tensor structure of standard neighbour density based descriptors. In doing so, we form compact tensor-reduced representations whose size does not depend on the number of chemical elements, but remain systematically convergeable and are therefore applicable to a wide range of data analysis and regression tasks.