# Learning with a Wasserstein Loss

arXiv: https://arxiv.org/abs/1506.05439 · HTML: https://mlsanity.com/paper/1506.05439/

## Key facts

- Cited by 196 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Jun 2015.

## Built on by (most cited first, 30 of 196)

- [Multimodal Optimal Transport-based Co-Attention Transformer with Global Structure Consistency for Survival Prediction](https://mlsanity.com/paper/2306.08330/) (2023)
- [Recent Advances in Optimal Transport for Machine Learning](https://mlsanity.com/paper/2306.16156/) (2023)
- [Minibatch optimal transport distances; analysis and applications](https://mlsanity.com/paper/2101.01792/) (2021)
- [Unified Optimal Transport Framework for Universal Domain Adaptation](https://mlsanity.com/paper/2210.17067/) (2022)
- [Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections](https://mlsanity.com/paper/2107.01718/) (2021)
- [Unbalanced Optimal Transport, from Theory to Numerics](https://mlsanity.com/paper/2211.08775/) (2022)
- [Global and Local Prompts Cooperation via Optimal Transport for Federated Learning](https://mlsanity.com/paper/2403.00041/) (2024)
- [Improved Fine-Tuning by Better Leveraging Pre-Training Data](https://mlsanity.com/paper/2111.12292/) (2021)
- [Representing Mixtures of Word Embeddings with Mixtures of Topic Embeddings](https://mlsanity.com/paper/2203.01570/) (2022)
- [Convolutional Cross-View Pose Estimation](https://mlsanity.com/paper/2303.05915/) (2023)
- [DL4SciVis: A State-of-the-Art Survey on Deep Learning for Scientific Visualization](https://mlsanity.com/paper/2204.06504/) (2022)
- [Improving Diffusion-Based Image Synthesis with Context Prediction](https://mlsanity.com/paper/2401.02015/) (2024)
- [Towards a mathematical theory of trajectory inference](https://mlsanity.com/paper/2102.09204/) (2021)
- [Unbalanced Optimal Transport through Non-negative Penalized Linear Regression](https://mlsanity.com/paper/2106.04145/) (2021)
- [A framework for data-driven solution and parameter estimation of PDEs using conditional generative adversarial networks](https://mlsanity.com/paper/2105.13136/) (2021)
- [Trajectory Inference via Mean-field Langevin in Path Space](https://mlsanity.com/paper/2205.07146/) (2022)
- [Semi-Supervised Multi-Modal Multi-Instance Multi-Label Deep Network with Optimal Transport](https://mlsanity.com/paper/2104.08489/) (2021)
- [Graph Auto-Encoder Via Neighborhood Wasserstein Reconstruction](https://mlsanity.com/paper/2202.09025/) (2022)
- [A Unified Framework for Implicit Sinkhorn Differentiation](https://mlsanity.com/paper/2205.06688/) (2022)
- [Meta Optimal Transport](https://mlsanity.com/paper/2206.05262/) (2022)
- [GENOT: Entropic (Gromov) Wasserstein Flow Matching with Applications to Single-Cell Genomics](https://mlsanity.com/paper/2310.09254/) (2023)
- [Addressing Misspecification in Simulation-based Inference through Data-driven Calibration](https://mlsanity.com/paper/2405.08719/) (2024)
- [Pre-training for Speech Translation: CTC Meets Optimal Transport](https://mlsanity.com/paper/2301.11716/) (2023)
- [Faster Unbalanced Optimal Transport: Translation invariant Sinkhorn and 1-D Frank-Wolfe](https://mlsanity.com/paper/2201.00730/) (2022)
- [Learning to Predict Graphs with Fused Gromov-Wasserstein Barycenters](https://mlsanity.com/paper/2202.03813/) (2022)
- [PatchCT: Aligning Patch Set and Label Set with Conditional Transport for Multi-Label Image Classification](https://mlsanity.com/paper/2307.09066/) (2023)
- [Sliced Optimal Partial Transport](https://mlsanity.com/paper/2212.08049/) (2022)
- [Implicit Graphon Neural Representation](https://mlsanity.com/paper/2211.03329/) (2022)
- [SOOD: Towards Semi-Supervised Oriented Object Detection](https://mlsanity.com/paper/2304.04515/) (2023)
- [Expected Sliced Transport Plans](https://mlsanity.com/paper/2410.12176/) (2024)

## Abstract (arXiv)

Learning to predict multi-label outputs is challenging, but in many problems there is a natural metric on the outputs that can be used to improve predictions. In this paper we develop a loss function for multi-label learning, based on the Wasserstein distance. The Wasserstein distance provides a natural notion of dissimilarity for probability measures. Although optimizing with respect to the exact Wasserstein distance is costly, recent work has described a regularized approximation that is efficiently computed. We describe an efficient learning algorithm based on this regularization, as well as a novel extension of the Wasserstein distance from probability measures to unnormalized measures. We also describe a statistical learning bound for the loss. The Wasserstein loss can encourage smoothness of the predictions with respect to a chosen metric on the output space. We demonstrate this property on a real-data tag prediction problem, using the Yahoo Flickr Creative Commons dataset, outperforming a baseline that doesn't use the metric.