Ordinary differential equations for regularized variational problems involving semi-discrete optimal transport
Provides a new theoretical and numerical framework for solving regularized variational problems involving optimal transport, benefiting researchers in computational optimal transport and related fields.
The authors prove that solutions to entropically regularized semi-discrete optimal transport problems satisfy ordinary differential equations in the regularization parameter, enabling efficient numerical solution via ODE methods that are more robust than Newton's method.
We consider entropically regularized, semi-discrete versions of variational problems on the set of probability measures involving optimal transport as well as other terms. We prove that the solutions can be characterized by well-posed ordinary differential equations in the regularization parameter. The initial conditions for these equations, corresponding to solutions to completely regularized problems, are typically known explicitly. The ODE can then be solved to recover the solution for an arbitrary degree of regularization; we verify that the solution is continuous in the regularization parameter, implying that taking the limit of the trajectory yields the solution to the fully unregularized problem. We establish analogous results for a version of the problem when the non-optimal transport term is not scaled with the regularization parameter. We exploit our characterization to numerically solve several example problems using standard ODE methods; this strategy exhibits superior robustness to alternatives such as Newton's method, as arbitrary initializations are not required.