On the Hopf-Cole Transform for Control-affine Schrödinger Bridge
This work addresses a theoretical limitation in optimal transport and control theory, but it is incremental as it clarifies existing conditions rather than proposing new methods.
The paper tackles the problem of solving control-affine Schrödinger bridge problems without assuming a specific proportionality condition between control and noise coefficients, showing that this leads to nonlinear PDEs that are not easily solvable, whereas the condition simplifies them to linear PDEs solvable by dynamic Sinkhorn recursions.
The purpose of this note is to clarify the importance of the relation $\boldsymbol{gg}^{\top}\propto \boldsymbol{σσ}^{\top}$ in solving control-affine Schrödinger bridge problems via the Hopf-Cole transform, where $\boldsymbol{g},\boldsymbolσ$ are the control and noise coefficients, respectively. We show that the Hopf-Cole transform applied to the conditions of optimality for generic control-affine Schrödinger bridge problems, i.e., without the assumption $\boldsymbol{gg}^{\top}\propto\boldsymbol{σσ}^{\top}$, gives a pair of forward-backward PDEs that are neither linear nor equation-level decoupled. We explain how the resulting PDEs can be interpreted as nonlinear forward-backward advection-diffusion-reaction equations, where the nonlinearity stem from additional drift and reaction terms involving the gradient of the log-likelihood a.k.a. the score. These additional drift and reaction vanish when $\boldsymbol{gg}^{\top}\propto\boldsymbol{σσ}^{\top}$, and the resulting boundary-coupled system of linear PDEs can then be solved by dynamic Sinkhorn recursions. A key takeaway of our work is that the numerical solution of the generic control-affine Schrödinger bridge requires further algorithmic development, possibly generalizing the dynamic Sinkhorn recursion or otherwise.