# Optimal Transport for Probabilistic Circuits

arXiv: https://arxiv.org/abs/2410.13061 · HTML: https://mlsanity.com/paper/2410.13061/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 34 references that are in the map.
- First posted on arXiv 16 Oct 2024.

## Builds on (most cited first, 16 of 16)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Improving GANs Using Optimal Transport](https://mlsanity.com/paper/1803.05573/) (2018)
- [A Differential Approach to Inference in Bayesian Networks](https://mlsanity.com/paper/1301.3847/) (2013)
- [Generative Modeling with Optimal Transport Maps](https://mlsanity.com/paper/2110.02999/) (2021)
- [Distributionally Robust Optimization: A Review](https://mlsanity.com/paper/1908.05659/) (2019)
- [A Wasserstein-type distance in the space of Gaussian Mixture Models](https://mlsanity.com/paper/1907.05254/) (2019)
- [Tree-Sliced Variants of Wasserstein Distances](https://mlsanity.com/paper/1902.00342/) (2019)
- [FlipTest: Fairness Testing via Optimal Transport](https://mlsanity.com/paper/1906.09218/) (2019)
- [Optimal transport for Gaussian mixture models](https://mlsanity.com/paper/1710.07876/) (2017)
- [Tractable Regularization of Probabilistic Circuits](https://mlsanity.com/paper/2106.02264/) (2021)
- [Statistical Optimal Transport via Factored Couplings](https://mlsanity.com/paper/1806.07348/) (2018)
- [Strudel: Learning Structured-Decomposable Probabilistic Circuits](https://mlsanity.com/paper/2007.09331/) (2020)
- [Scaling Tractable Probabilistic Circuits: A Systems Perspective](https://mlsanity.com/paper/2406.00766/) (2024)
- [On Relaxing Determinism in Arithmetic Circuits](https://mlsanity.com/paper/1708.06846/) (2017)
- [Learning Arbitrary Sum-Product Network Leaves with Expectation-Maximization](https://mlsanity.com/paper/1604.07243/) (2016)

## Built on by (most cited first, 1 of 1)

- [PeTeR: Post-Training Robustification of Probabilistic Circuits](https://mlsanity.com/paper/2607.07671/) (2026)

## Abstract (arXiv)

We introduce a novel optimal transport framework for probabilistic circuits (PCs). While it has been shown recently that divergences between distributions represented as certain classes of PCs can be computed tractably, to the best of our knowledge, there is no existing approach to compute the Wasserstein distance between probability distributions given by PCs. We propose a Wasserstein-type distance that restricts the coupling measure of the associated optimal transport problem to be a probabilistic circuit. We then develop an algorithm for computing this distance by solving a series of small linear programs and derive the circuit conditions under which this is tractable. Furthermore, we show that we can easily retrieve the optimal transport plan between the PCs from the solutions to these linear programs. Lastly, we study the empirical Wasserstein distance between a PC and a dataset, and show that we can estimate the PC parameters to minimize this distance through an efficient iterative algorithm.