Sinkhorn with Memory Solves Control-Affine Schrödinger Bridge Problem

A novel algorithm known as Sinkhorn with memory tackles the control-affine Schrödinger bridge issue in scenarios where control and noise channels are not aligned. The conventional dynamic Sinkhorn recursion, which relies on the Hopf-Cole transform, is only applicable when these channels are in proportion, leading to linear partial differential equations (PDEs). In cases of mismatched channels, the resulting transformed PDEs remain nonlinear, and no existing algorithm addressed this. The researchers developed a Sinkhorn recursion with memory that utilizes the characteristics of these nonlinear PDEs, proving its efficacy. This represents a significant advancement in optimal density steering for controlled diffusions, with potential uses in robotics, control theory, and stochastic systems.