# Regularized unbalanced optimal transport as entropy minimization with respect to branching Brownian motion

arXiv: https://arxiv.org/abs/2111.01666 · HTML: https://mlsanity.com/paper/2111.01666/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Nov 2021.

## Built on by (most cited first, 10 of 10)

- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Learning stochastic dynamics from snapshots through regularized unbalanced optimal transport](https://mlsanity.com/paper/2410.00844/) (2024)
- [Trajectory Inference via Mean-field Langevin in Path Space](https://mlsanity.com/paper/2205.07146/) (2022)
- [Modeling Cell Dynamics and Interactions with Unbalanced Mean Field Schr"odinger Bridge](https://mlsanity.com/paper/2505.11197/) (2025)
- [Variational Regularized Unbalanced Optimal Transport: Single Network, Least Action](https://mlsanity.com/paper/2505.11823/) (2025)
- [Integrating Dynamical Systems Modeling with Spatiotemporal scRNA-seq Data Analysis](https://mlsanity.com/paper/2503.11347/) (2025)
- [Branched Schr"odinger Bridge Matching](https://mlsanity.com/paper/2506.09007/) (2025)
- [Inferring stochastic dynamics with growth from cross-sectional data](https://mlsanity.com/paper/2505.13197/) (2025)
- [Beyond Continuity: Simulation-free Reconstruction of Discrete Branching Dynamics from Single-cell Snapshots](https://mlsanity.com/paper/2605.00545/) (2026)
- [Simulation-free Unbalanced Dynamic Optimal Transport with General Growth Penalty](https://mlsanity.com/paper/2609.04710/) (2026)

## Abstract (arXiv)

We consider the problem of minimizing the entropy of a law with respect to the law of a reference branching Brownian motion under density constraints at an initial and final time. We call this problem the branching Schr"odinger problem by analogy with the Schr"odinger problem, where the reference process is a Brownian motion. Whereas the Schr"odinger problem is related to regularized (a.k.a. entropic) optimal transport, we investigate here the link of the branching Schr"odinger problem with regularized unbalanced optimal transport. This link is shown at two levels. First, relying on duality arguments, the values of these two problems of calculus of variations are linked, in the sense that the value of the regularized unbalanced optimal transport (seen as a function of the initial and final measure) is the lower semi-continuous relaxation of the value of the branching Schr"odinger problem. Second, we also explicit a correspondence between the competitors of these two problems, and to that end we provide a fine description of laws having a finite entropy with respect to a reference branching Brownian motion. We investigate the small noise limit, when the noise intensity of the branching Brownian motion goes to 0: in this case we show, at the level of the optimal transport model, that there is convergence to partial optimal transport. We also provide formal arguments about why looking at the branching Brownian motion, and not at other measure-valued branching Markov processes, like superprocesses, yields the problem closest to optimal transport. Finally, we explain how this problem can be solved numerically: the dynamical formulation of regularized unbalanced optimal transport can be discretized and solved via convex optimization.