Differentiable Cost-Parameterized Monge Map Estimators
This work addresses the challenge of tailoring optimal transport to specific applications by incorporating prior knowledge, representing an incremental improvement in the field of optimal transport.
The authors tackled the problem of making optimal transport maps more useful for real-world applications by developing a method to simultaneously learn both a transport map estimator and an adapted cost function that incorporates prior information about the map. They constructed a differentiable Monge map estimator using neural ground costs with known Monge map forms, enabling optimization to match known information about optimal transport maps.
Within the field of optimal transport (OT), the choice of ground cost is crucial to ensuring that the optimality of a transport map corresponds to usefulness in real-world applications. It is therefore desirable to use known information to tailor cost functions and hence learn OT maps which are adapted to the problem at hand. By considering a class of neural ground costs whose Monge maps have a known form, we construct a differentiable Monge map estimator which can be optimized to be consistent with known information about an OT map. In doing so, we simultaneously learn both an OT map estimator and a corresponding adapted cost function. Through suitable choices of loss function, our method provides a general approach for incorporating prior information about the Monge map itself when learning adapted OT maps and cost functions.