# The Density Fingerprint of a Periodic Point Set

arXiv: https://arxiv.org/abs/2104.11046 · HTML: https://mlsanity.com/paper/2104.11046/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Apr 2021.

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

- [Complete and Efficient Graph Transformers for Crystal Material Property Prediction](https://mlsanity.com/paper/2403.11857/) (2024)
- [Recognizing Rigid Patterns of Unlabeled Point Clouds by Complete and Continuous Isometry Invariants with no False Negatives and no False Positives](https://mlsanity.com/paper/2303.15385/) (2023)
- [Fast predictions of lattice energies by continuous isometry invariants of crystal structures](https://mlsanity.com/paper/2108.07233/) (2021)
- [Material Property Prediction using Graphs based on Generically Complete Isometry Invariants](https://mlsanity.com/paper/2212.11246/) (2022)
- [Accelerating Material Property Prediction using Generically Complete Isometry Invariants](https://mlsanity.com/paper/2401.15089/) (2024)

## Abstract (arXiv)

Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functions that facilitates the efficient search for new materials and material properties. We prove invariance under isometries, continuity, and completeness in the generic case, which are necessary features for the reliable comparison of crystals. The proof of continuity integrates methods from discrete geometry and lattice theory, while the proof of generic completeness combines techniques from geometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and related inclusion-exclusion formulae. We have implemented the algorithm and describe its application to crystal structure prediction.