Constrained Approximate Optimal Transport Maps

We investigate finding a map g within a function class G that minimises an Optimal Transport (OT) cost between a target measure ν and the image by g of a source measure μ. This is relevant when an OT map from μ to ν does not exist or does not satisfy the desired constraints of G. We address existence and uniqueness for generic subclasses of L-Lipschitz functions, including gradients of (strongly) convex functions and typical Neural Networks. We explore a variant that approaches a transport plan, showing equivalence to a map problem in some cases. For the squared Euclidean cost, we propose alternating minimisation over a transport plan π and map g, with the optimisation over g being the L² projection on G of the barycentric mapping π̄. In dimension one, this global problem equates the L² projection of π^*̄ onto G for an OT plan π^* between μ and ν, but this does not extend to higher dimensions. We introduce a simple kernel method to find g within a Reproducing Kernel Hilbert Space in the discrete case. Finally, we present numerical methods for L-Lipschitz gradients of -strongly convex potentials.