# Optimal Transport with Tempered Exponential Measures

arXiv: https://arxiv.org/abs/2309.04015 · HTML: https://mlsanity.com/paper/2309.04015/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 36 references that are in the map.
- First posted on arXiv 7 Sep 2023.

## Builds on (most cited first, 11 of 11)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Smooth and Sparse Optimal Transport](https://mlsanity.com/paper/1710.06276/) (2017)
- [Robust Bi-Tempered Logistic Loss Based on Bregman Divergences](https://mlsanity.com/paper/1906.03361/) (2019)
- [Sparsity-Constrained Optimal Transport](https://mlsanity.com/paper/2209.15466/) (2022)
- [Regularized Optimal Transport and the Rot Mover's Distance](https://mlsanity.com/paper/1610.06447/) (2016)
- [Tsallis Regularized Optimal Transport and Ecological Inference](https://mlsanity.com/paper/1609.04495/) (2016)
- [Boosting with Tempered Exponential Measures](https://mlsanity.com/paper/2306.05487/) (2023)
- [Clustering above Exponential Families with Tempered Exponential Measures](https://mlsanity.com/paper/2211.02765/) (2022)
- [Mirror Sinkhorn: Fast Online Optimization on Transport Polytopes](https://mlsanity.com/paper/2211.10420/) (2022)

## Built on by (most cited first, 3 of 3)

- [Mirror Descent and Novel Exponentiated Gradient Algorithms Using Trace-Form Entropies and Deformed Logarithms](https://mlsanity.com/paper/2503.08748/) (2025)
- [The Tempered Hilbert Simplex Distance and Its Application To Non-linear Embeddings of TEMs](https://mlsanity.com/paper/2311.13459/) (2023)
- [Tempered Calculus for ML: Application to Hyperbolic Model Embedding](https://mlsanity.com/paper/2402.04163/) (2024)

## Abstract (arXiv)

In the field of optimal transport, two prominent subfields face each other: (i) unregularized optimal transport, "`a-la-Kantorovich", which leads to extremely sparse plans but with algorithms that scale poorly, and (ii) entropic-regularized optimal transport, "`a-la-Sinkhorn-Cuturi", which gets near-linear approximation algorithms but leads to maximally un-sparse plans. In this paper, we show that an extension of the latter to tempered exponential measures, a generalization of exponential families with indirect measure normalization, gets to a very convenient middle ground, with both very fast approximation algorithms and sparsity, which is under control up to sparsity patterns. In addition, our formulation fits naturally in the unbalanced optimal transport problem setting.