# A Quasi-Wasserstein Loss for Learning Graph Neural Networks

arXiv: https://arxiv.org/abs/2310.11762 · HTML: https://mlsanity.com/paper/2310.11762/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 44 of its 60 references that are in the map.
- First posted on arXiv 18 Oct 2023.

## Builds on (most cited first, 30 of 44)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Heterogeneous Graph Attention Network](https://mlsanity.com/paper/1903.07293/) (2019)
- [Graph Neural Networks for Social Recommendation](https://mlsanity.com/paper/1902.07243/) (2019)
- [Image-based Recommendations on Styles and Substitutes](https://mlsanity.com/paper/1506.04757/) (2015)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Multi-scale Attributed Node Embedding](https://mlsanity.com/paper/1909.13021/) (2019)
- [Diffusion Improves Graph Learning](https://mlsanity.com/paper/1911.05485/) (2019)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Learning Convolutional Neural Networks for Graphs](https://mlsanity.com/paper/1605.05273/) (2016)
- [Large Scale Learning on Non-Homophilous Graphs: New Benchmarks and Strong Simple Methods](https://mlsanity.com/paper/2110.14446/) (2021)
- [MAGNN: Metapath Aggregated Graph Neural Network for Heterogeneous Graph Embedding](https://mlsanity.com/paper/2002.01680/) (2020)
- [Spatial-Temporal Fusion Graph Neural Networks for Traffic Flow Forecasting](https://mlsanity.com/paper/2012.09641/) (2020)
- [Combining Label Propagation and Simple Models Out-performs Graph Neural Networks](https://mlsanity.com/paper/2010.13993/) (2020)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [BernNet: Learning Arbitrary Graph Spectral Filters via Bernstein Approximation](https://mlsanity.com/paper/2106.10994/) (2021)
- [Graph Neural Networks with convolutional ARMA filters](https://mlsanity.com/paper/1901.01343/) (2019)
- [Optimal Transport for structured data with application on graphs](https://mlsanity.com/paper/1805.09114/) (2018)
- [Interpreting and Unifying Graph Neural Networks with An Optimization Framework](https://mlsanity.com/paper/2101.11859/) (2021)
- [DeepInf: Social Influence Prediction with Deep Learning](https://mlsanity.com/paper/1807.05560/) (2018)

## Built on by (most cited first, 1 of 1)

- [Transport-Coupled Bayesian Flows for Molecular Graph Generation](https://mlsanity.com/paper/2510.10211/) (2025)

## Abstract (arXiv)

When learning graph neural networks (GNNs) in node-level prediction tasks, most existing loss functions are applied for each node independently, even if node embeddings and their labels are non-i.i.d. because of their graph structures. To eliminate such inconsistency, in this study we propose a novel Quasi-Wasserstein (QW) loss with the help of the optimal transport defined on graphs, leading to new learning and prediction paradigms of GNNs. In particular, we design a ``Quasi-Wasserstein'' distance between the observed multi-dimensional node labels and their estimations, optimizing the label transport defined on graph edges. The estimations are parameterized by a GNN in which the optimal label transport may determine the graph edge weights optionally. By reformulating the strict constraint of the label transport to a Bregman divergence-based regularizer, we obtain the proposed Quasi-Wasserstein loss associated with two efficient solvers learning the GNN together with optimal label transport. When predicting node labels, our model combines the output of the GNN with the residual component provided by the optimal label transport, leading to a new transductive prediction paradigm. Experiments show that the proposed QW loss applies to various GNNs and helps to improve their performance in node-level classification and regression tasks. The code of this work can be found at urlhttps://github.com/SDS-Lab/QWLoss.