# Entropy Partial Transport with Tree Metrics: Theory and Practice arXiv: https://arxiv.org/abs/2101.09756 ยท HTML: https://mlsanity.com/paper/2101.09756/ ## Key facts - Cited by 13 papers in the map (data through 14 Oct 2026). - Builds on 27 of its 74 references that are in the map. - First posted on arXiv 24 Jan 2021. ## Builds on (most cited first, 27 of 27) - [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013) - [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) - [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017) - [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015) - [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019) - [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015) - [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015) - [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019) - [Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions](https://mlsanity.com/paper/1806.08141/) (2018) - [Sliced Gromov-Wasserstein](https://mlsanity.com/paper/1905.10124/) (2019) - [On Unbalanced Optimal Transport: An Analysis of Sinkhorn Algorithm](https://mlsanity.com/paper/2002.03293/) (2020) - [Tree-Sliced Variants of Wasserstein Distances](https://mlsanity.com/paper/1902.00342/) (2019) - [Learning Generative Models across Incomparable Spaces](https://mlsanity.com/paper/1905.05461/) (2019) - [Asymptotic Guarantees for Learning Generative Models with the Sliced-Wasserstein Distance](https://mlsanity.com/paper/1906.04516/) (2019) - [Scalable Unbalanced Optimal Transport using Generative Adversarial Networks](https://mlsanity.com/paper/1810.11447/) (2018) - [Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams](https://mlsanity.com/paper/1802.03569/) (2018) - [Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport](https://mlsanity.com/paper/1805.08331/) (2018) - [Inference for Empirical Wasserstein Distances on Finite Spaces](https://mlsanity.com/paper/1610.03287/) (2016) - [Kernel method for persistence diagrams via kernel embedding and weight factor](https://mlsanity.com/paper/1706.03472/) (2017) - [Wasserstein regularization for sparse multi-task regression](https://mlsanity.com/paper/1805.07833/) (2018) - [Dynamical Optimal Transport on Discrete Surfaces](https://mlsanity.com/paper/1809.07083/) (2018) - [Fast Unbalanced Optimal Transport on a Tree](https://mlsanity.com/paper/2006.02703/) (2020) - [Unnormalized Optimal Transport](https://mlsanity.com/paper/1902.03367/) (2019) - [On properties of the Generalized Wasserstein distance](https://mlsanity.com/paper/1304.7014/) (2013) - [Imaging with Kantorovich-Rubinstein discrepancy](https://mlsanity.com/paper/1407.0221/) (2014) - [Parallel Unbalanced Optimal Transport Regularization for Large Scale Imaging Problems](https://mlsanity.com/paper/1909.00149/) (2019) ## Built on by (most cited first, 13 of 13) - [Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics](https://mlsanity.com/paper/2202.10723/) (2022) - [Fixed Support Tree-Sliced Wasserstein Barycenter](https://mlsanity.com/paper/2109.03431/) (2021) - [Scalable Unbalanced Sobolev Transport for Measures on a Graph](https://mlsanity.com/paper/2302.12498/) (2023) - [Approximating 1-Wasserstein Distance with Trees](https://mlsanity.com/paper/2206.12116/) (2022) - [Generalized Sobolev Transport for Probability Measures on a Graph](https://mlsanity.com/paper/2402.04516/) (2024) - [On Multimarginal Partial Optimal Transport: Equivalent Forms and Computational Complexity](https://mlsanity.com/paper/2108.07992/) (2021) - [Optimal Transport for Measures with Noisy Tree Metric](https://mlsanity.com/paper/2310.13653/) (2023) - [Scalable Counterfactual Distribution Estimation in Multivariate Causal Models](https://mlsanity.com/paper/2311.00927/) (2023) - [Distance-Based Tree-Sliced Wasserstein Distance](https://mlsanity.com/paper/2503.11050/) (2025) - [Spherical Tree-Sliced Wasserstein Distance](https://mlsanity.com/paper/2503.11249/) (2025) - [Scalable Sobolev IPM for Probability Measures on a Graph](https://mlsanity.com/paper/2502.00737/) (2025) - [An Empirical Study of Self-supervised Learning with Wasserstein Distance](https://mlsanity.com/paper/2310.10143/) (2023) - [An Efficient Orlicz-Sobolev Approach for Transporting Unbalanced Measures on a Graph](https://mlsanity.com/paper/2502.00739/) (2025) ## Abstract (arXiv) Optimal transport (OT) theory provides powerful tools to compare probability measures. However, OT is limited to nonnegative measures having the same mass, and suffers serious drawbacks about its computation and statistics. This leads to several proposals of regularized variants of OT in the recent literature. In this work, we consider an textitentropy partial transport (EPT) problem for nonnegative measures on a tree having different masses. The EPT is shown to be equivalent to a standard complete OT problem on a one-node extended tree. We derive its dual formulation, then leverage this to propose a novel regularization for EPT which admits fast computation and negative definiteness. To our knowledge, the proposed regularized EPT is the first approach that yields a textitclosed-form solution among available variants of unbalanced OT. For practical applications without priori knowledge about the tree structure for measures, we propose tree-sliced variants of the regularized EPT, computed by averaging the regularized EPT between these measures using random tree metrics, built adaptively from support data points. Exploiting the negative definiteness of our regularized EPT, we introduce a positive definite kernel, and evaluate it against other baselines on benchmark tasks such as document classification with word embedding and topological data analysis. In addition, we empirically demonstrate that our regularization also provides effective approximations.