# Learning with a Wasserstein Loss

@inproceedings{Frogner2015LearningWA, title={Learning with a Wasserstein Loss}, author={Charlie Frogner and Chiyuan Zhang and Hossein Mobahi and Mauricio Araya-Polo and Tomaso A. Poggio}, booktitle={NIPS}, year={2015} }

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…

## 443 Citations

### The Wasserstein Loss Function

- Computer Science
- 2015

This project would like to explore the properties of this Wasserstein Loss function by comparing its accuracy, convergence rates etc. against other loss functions, and by evaluating how changes in parameters and the distance metric impact its performance.

### The Cramer Distance as a Solution to Biased Wasserstein Gradients

- Computer ScienceArXiv
- 2017

This paper describes three natural properties of probability divergences that it believes reflect requirements from machine learning: sum invariance, scale sensitivity, and unbiased sample gradients and proposes an alternative to the Wasserstein metric, the Cramer distance, which possesses all three desired properties.

### Wasserstein of Wasserstein Loss for Learning Generative Models

- Computer ScienceICML
- 2019

The Wasserstein distance serves as a loss function for unsupervised learning which depends on the choice of a ground metric on sample space and the new formulation is more robust to the natural variability of images and provides for a more continuous discriminator in sample space.

### A Simulated Annealing Based Inexact Oracle for Wasserstein Loss Minimization

- Computer ScienceICML
- 2017

A stochastic approach based on simulated annealing for solving WLMs is introduced and a Gibbs sampler is developed to approximate effectively and efficiently the partial gradients of a sequence of Wasserstein losses.

### Wasserstein Distance Measure Machines

- Computer ScienceArXiv
- 2018

A distance-based discriminative framework for learning with probability distributions is presented and it is proved that, for some learning problems, Wasserstein distance achieves low-error linear decision functions with high probability.

### Heterogeneous Wasserstein Discrepancy for Incomparable Distributions

- Computer Science
- 2021

A novel extension of Wasserstein distance is proposed to compare two incomparable distributions, that hinges on the idea of distributional slicing, embeddings, and on computing the closed-form Wassertein distance between the sliced distributions.

### The Fisher-Rao Loss for Learning under Label Noise

- Computer ScienceInformation Geometry
- 2022

It is argued that the Fisher-Rao loss provides a natural trade-oﬀ between robustness and training dynamics, and Numerical experiments with synthetic and MNIST datasets illustrate this performance.

### Quantifying the Empirical Wasserstein Distance to a Set of Measures: Beating the Curse of Dimensionality

- Computer ScienceNeurIPS
- 2020

The formulation provides insights that help clarify why the Wasserstein distance enjoys favorable empirical performance across a wide range of statistical applications and establishes a strong duality result that generalizes the celebrated Kantorovich-Rubinstein duality.

### Wasserstein Training of Restricted Boltzmann Machines

- Computer ScienceNIPS
- 2016

This work proposes a novel approach for Boltzmann machine training which assumes that a meaningful metric between observations is known, and derives a gradient of that distance with respect to the model parameters from the Kullback-Leibler divergence.

### The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation

- Computer ScienceNeurIPS
- 2021

Two Unbalanced Gromov-Wasserstein formulations are introduced: a distance and a more computationally tractable upper-bounding relaxation that allow the comparison of metric spaces equipped with arbitrary positive measures up to isometries.

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