@article{10.1145/3811314,
author = {Mattos Da Silva, Leticia and Nabizadeh, Mohammad Sina and Solomon, Justin},
title = {Schr\"{o}dinger Bridges on Discretized Geometric Domains},
year = {2026},
issue_date = {July 2026},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {45},
number = {4},
issn = {0730-0301},
url = {https://doi.org/10.1145/3811314},
doi = {10.1145/3811314},
abstract = {We introduce a spatially discrete formulation of the Schr\"{o}dinger bridge problem on meshes and grids that enables structure-preserving and scalable interpolation between probability distributions. Our approach builds on the duality between entropy-regularized optimal transport and the log-heat equation, deriving a discrete theory that is compatible with mesh-based finite element discretizations. The resulting Sinkhorn algorithm alternates application of the heat kernel with multiplicative updates to enforce marginal constraints. Compared to interpolation via Wasserstein barycenters, our formulation produces sharper interpolants for a given level of regularization and enforces exact endpoint marginals, in addition to enjoying faster computation. It also scales to high-resolution meshes and finer temporal discretizations, avoiding the prohibitive cost of directly discretizing dynamical transport. We demonstrate our approach across mesh- and grid-based applications, including displacement interpolation, shape interpolation, and color histogram manipulation, highlighting its ability to achieve geometric fidelity with computational efficiency.},
journal = {ACM Trans. Graph.},
month = jul,
articleno = {114},
numpages = {16},
keywords = {optimal transport, sinkhorn algorithm, structure-preserving discretization}
}