# Tsallis Regularized Optimal Transport and Ecological Inference

arXiv: https://arxiv.org/abs/1609.04495 · HTML: https://mlsanity.com/paper/1609.04495/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Sep 2016.

## Built on by (most cited first, 15 of 15)

- [On Unbalanced Optimal Transport: Gradient Methods, Sparsity and Approximation Error](https://mlsanity.com/paper/2202.03618/) (2022)
- [Semi-Discrete Optimal Transport: Hardness, Regularization and Numerical Solution](https://mlsanity.com/paper/2103.06263/) (2021)
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- [MANO: Exploiting Matrix Norm for Unsupervised Accuracy Estimation Under Distribution Shifts](https://mlsanity.com/paper/2405.18979/) (2024)
- [Optimization for Classical Machine Learning Problems on the GPU](https://mlsanity.com/paper/2203.16340/) (2022)
- [Submodular Framework for Structured-Sparse Optimal Transport](https://mlsanity.com/paper/2406.04914/) (2024)
- [Optimal transport with f-divergence regularization and generalized Sinkhorn algorithm](https://mlsanity.com/paper/2105.14337/) (2021)
- [Discrete Optimal Transport with Independent Marginals is #P-Hard](https://mlsanity.com/paper/2203.01161/) (2022)
- [Optimal Transport with Tempered Exponential Measures](https://mlsanity.com/paper/2309.04015/) (2023)
- [Learning non-equilibrium diffusions with Schr"odinger bridges: from exactly solvable to simulation-free](https://mlsanity.com/paper/2505.16644/) (2025)
- [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025)
- [Sparse Partial Optimal Transport via Quadratic Regularization](https://mlsanity.com/paper/2508.08476/) (2025)
- [When Should Graph Attention Be Sparse? Learning a Per-Edge Tsallis Index](https://mlsanity.com/paper/2608.02938/) (2026)
- [Convergence rate of Tsallis entropic regularized optimal transport](https://mlsanity.com/paper/2304.06616/) (2023)
- [Accelerated Algorithm for Sparse Regularized Partial Optimal Transport](https://mlsanity.com/paper/2609.40075/) (2026)

## Abstract (arXiv)

Optimal transport is a powerful framework for computing distances between probability distributions. We unify the two main approaches to optimal transport, namely Monge-Kantorovitch and Sinkhorn-Cuturi, into what we define as Tsallis regularized optimal transport (trot). trot~interpolates a rich family of distortions from Wasserstein to Kullback-Leibler, encompassing as well Pearson, Neyman and Hellinger divergences, to name a few. We show that metric properties known for Sinkhorn-Cuturi generalize to trot, and provide efficient algorithms for finding the optimal transportation plan with formal convergence proofs. We also present the first application of optimal transport to the problem of ecological inference, that is, the reconstruction of joint distributions from their marginals, a problem of large interest in the social sciences. trot~provides a convenient framework for ecological inference by allowing to compute the joint distribution --- that is, the optimal transportation plan itself --- when side information is available, which is textite.g. typically what census represents in political science. Experiments on data from the 2012 US presidential elections display the potential of trot~in delivering a faithful reconstruction of the joint distribution of ethnic groups and voter preferences.