Variational Inference for Uncertain Optimal Transport via Sinkhorn Parametrization

Optimal Transport (OT) traditionally relies on a fixed ground cost to produce a single deterministic transport plan, a practice that overlooks the inherent variability and noise in real-world data. While recent sampling-based approaches to OT offer a principled way to quantify this uncertainty, they are computationally prohibitive and struggle to scale. In this paper, we introduce Sinkhornparameterized Variational Inference (SPVI), the first scalable variational framework for performing posterior inference over transport plans. Our key insight is that the Sinkhorn map can be treated as a differentiable reparameterization of the set of entropic plans. This enables the use of flexible generative models like normalizing flows to approximate distributions over transport plans while enforcing marginal constraints. We experimentally demonstrate that our method matches the quality of intensive sampling techniques at a fraction of the computational cost, scaling effectively to large-scale problems.