Complex Schrödinger Bridges

Diffusion bridges have emerged as versatile diffusion-based frameworks for facilitating transformations between arbitrary data distributions. However, these approaches are hindered by the immensity of the optimization space without consideration of underlying high-order structures. This paper explores a complex-valued variant of diffusion bridges for solving physical problems using conformal field theory and mean field games. To this end, we propose a new class of dynamical diffusion bridges, called complex Schr\"odinger Bridges (CSB), whose derived algorithms are able to handle diffusion-based tasks using Hamiltonian-based geometric prior. Additionally, we propose image-to-image CSB (I$^2$CSB), tractable diffusion bridge techniques with a spiral analytic posterior. We verify our theoretical claims and practicality of our Hamiltonian based approach with a wide range of experiments.