# Unsupervised Learning of Graph Hierarchical Abstractions with Differentiable Coarsening and Optimal Transport

arXiv: https://arxiv.org/abs/1912.11176 · HTML: https://mlsanity.com/paper/1912.11176/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Dec 2019.

## Built on by (most cited first, 12 of 12)

- [Graph Coarsening with Neural Networks](https://mlsanity.com/paper/2102.01350/) (2021)
- [Accelerating Training and Inference of Graph Neural Networks with Fast Sampling and Pipelining](https://mlsanity.com/paper/2110.08450/) (2021)
- [SizeShiftReg: a Regularization Method for Improving Size-Generalization in Graph Neural Networks](https://mlsanity.com/paper/2207.07888/) (2022)
- [Graph coarsening: From scientific computing to machine learning](https://mlsanity.com/paper/2106.11863/) (2021)
- [Learning Topology-Specific Experts for Molecular Property Prediction](https://mlsanity.com/paper/2302.13693/) (2023)
- [Mirage: Model-Agnostic Graph Distillation for Graph Classification](https://mlsanity.com/paper/2310.09486/) (2023)
- [A Unified Framework for Optimization-Based Graph Coarsening](https://mlsanity.com/paper/2210.00437/) (2022)
- [A Survey on Graph Condensation](https://mlsanity.com/paper/2402.02000/) (2024)
- [Graph Coarsening with Message-Passing Guarantees](https://mlsanity.com/paper/2405.18127/) (2024)
- [Representing Molecules as Random Walks Over Interpretable Grammars](https://mlsanity.com/paper/2403.08147/) (2024)
- [Blockchain Phishing Scam Detection via Multi-channel Graph Classification](https://mlsanity.com/paper/2108.08456/) (2021)
- [Taxonomy of reduction matrices for Graph Coarsening](https://mlsanity.com/paper/2506.11743/) (2025)

## Abstract (arXiv)

Hierarchical abstractions are a methodology for solving large-scale graph problems in various disciplines. Coarsening is one such approach: it generates a pyramid of graphs whereby the one in the next level is a structural summary of the prior one. With a long history in scientific computing, many coarsening strategies were developed based on mathematically driven heuristics. Recently, resurgent interests exist in deep learning to design hierarchical methods learnable through differentiable parameterization. These approaches are paired with downstream tasks for supervised learning. In practice, however, supervised signals (e.g., labels) are scarce and are often laborious to obtain. In this work, we propose an unsupervised approach, coined OTCoarsening, with the use of optimal transport. Both the coarsening matrix and the transport cost matrix are parameterized, so that an optimal coarsening strategy can be learned and tailored for a given set of graphs. We demonstrate that the proposed approach produces meaningful coarse graphs and yields competitive performance compared with supervised methods for graph classification and regression.