The Schrödinger bridge finds the most likely diffusion process interpolating between two distributions — the entropy-regularized optimal transport. Visualize the iterative Sinkhorn algorithm and particle geodesics.
Schrödinger bridge (1931): most probable evolution of a diffusing particle cloud from μ to ν in time T. Solution: P*=argmin_{P:P_0=μ,P_T=ν} D_KL(P||W) where W is Wiener measure. Entropy-regularized OT: min ⟨C,π⟩ + ε·H(π). Solved via Sinkhorn iterations (alternating KL projections). Limit ε→0: Monge-Kantorovich (deterministic). Applications: generative models (diffusion), single-cell trajectory inference, image transport.