# Geometric Losses for Distributional Learning

arXiv: https://arxiv.org/abs/1905.06005 · HTML: https://mlsanity.com/paper/1905.06005/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 May 2019.

## Built on by (most cited first, 7 of 7)

- [Geometric Entropic Exploration](https://mlsanity.com/paper/2101.02055/) (2021)
- [Learning to Predict Graphs with Fused Gromov-Wasserstein Barycenters](https://mlsanity.com/paper/2202.03813/) (2022)
- [Sparse Continuous Distributions and Fenchel-Young Losses](https://mlsanity.com/paper/2108.01988/) (2021)
- [Action Redundancy in Reinforcement Learning](https://mlsanity.com/paper/2102.11329/) (2021)
- [MANO: Exploiting Matrix Norm for Unsupervised Accuracy Estimation Under Distribution Shifts](https://mlsanity.com/paper/2405.18979/) (2024)
- [Joint Learning of Energy-based Models and their Partition Function](https://mlsanity.com/paper/2501.18528/) (2025)
- [Mirror Bridges Between Probability Measures](https://mlsanity.com/paper/2410.07003/) (2024)

## Abstract (arXiv)

Building upon recent advances in entropy-regularized optimal transport, and upon Fenchel duality between measures and continuous functions , we propose a generalization of the logistic loss that incorporates a metric or cost between classes. Unlike previous attempts to use optimal transport distances for learning, our loss results in unconstrained convex objective functions, supports infinite (or very large) class spaces, and naturally defines a geometric generalization of the softmax operator. The geometric properties of this loss make it suitable for predicting sparse and singular distributions, for instance supported on curves or hyper-surfaces. We study the theoretical properties of our loss and show-case its effectiveness on two applications: ordinal regression and drawing generation.