# A cohomology-based Gromov-Hausdorff metric approach for quantifying molecular similarity

arXiv: https://arxiv.org/abs/2411.13887 · HTML: https://mlsanity.com/paper/2411.13887/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 73 references that are in the map.
- First posted on arXiv 21 Nov 2024.

## Builds on (most cited first, 8 of 8)

- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [Topological Data Analysis](https://mlsanity.com/paper/1609.08227/) (2016)
- [Attending to Topological Spaces: The Cellular Transformer](https://mlsanity.com/paper/2405.14094/) (2024)
- [Simplicial complexes: higher-order spectral dimension and dynamics](https://mlsanity.com/paper/2001.05934/) (2020)
- [The Gromov-Hausdorff distance between ultrametric spaces: its structure and computation](https://mlsanity.com/paper/2110.03136/) (2021)

## Abstract (arXiv)

We introduce, for the first time, a cohomology-based Gromov-Hausdorff ultrametric method to analyze 1-dimensional and higher-dimensional (co)homology groups, focusing on loops, voids, and higher-dimensional cavity structures in simplicial complexes, to address typical clustering questions arising in molecular data analysis. The Gromov-Hausdorff distance quantifies the dissimilarity between two metric spaces. In this framework, molecules are represented as simplicial complexes, and their cohomology vector spaces are computed to capture intrinsic topological invariants encoding loop and cavity structures. These vector spaces are equipped with a suitable distance measure, enabling the computation of the Gromov-Hausdorff ultrametric to evaluate structural dissimilarities. We demonstrate the methodology using organic-inorganic halide perovskite (OIHP) structures. The results highlight the effectiveness of this approach in clustering various molecular structures. By incorporating geometric information, our method provides deeper insights compared to traditional persistent homology techniques.