Smooth and Sparse Optimal Transport

Entropic regularization is quickly emerging as a new standard in optimal transport (OT). It enables to cast the OT computation as a differentiable and unconstrained convex optimization problem, which can be efficiently solved using the Sinkhorn algorithm. However, the entropy term keeps the transportation plan strictly positive and therefore completely dense, unlike unregularized OT. This lack of sparsity can be problematic for applications where the transportation plan itself is of interest. In this paper, we explore regularizing both the primal and dual original formulations with an arbitrary strongly convex term. We show that this corresponds to relaxing dual and primal constraints with approximate smooth constraints. We show how to incorporate squared 2-norm and group lasso regularizations within that framework, leading to sparse and group-sparse transportation plans. On the theoretical side, we are able to bound the approximation error introduced by smoothing the original primal and dual formulations. Our results suggest that, for the smoothed dual, the approximation error can often be smaller with squared 2-norm regularization than with entropic regularization. We showcase our proposed framework on the task of color transfer.