From Optimal Transport to Discrepancy

A common way to quantify the “distance” between measures is via their discrepancy, also known as maximum mean discrepancy (MMD). Discrepancies are related to Sinkhorn divergences Sε with appropriate cost functions as ε →∞. In the opposite direction, if ε → 0, Sinkhorn divergences approach another important distance between measures, namely, the Wasserstein distance or more generally optimal transport “distance.” In this chapter, we investigate the limiting process for arbitrary measures on compact sets and Lipschitz continuous cost functions. In particular, we are interested in the behavior of the corresponding optimal potentials φ ̂ ε $$\hat \varphi _\varepsilon $$ , ψ ̂ ε $$\hat \psi _\varepsilon $$ , and φ ̂ K $$\hat \varphi _K$$ appearing in the dual formulation of the Sinkhorn divergences and discrepancies, respectively. While part of the results is known, we provide rigorous proofs for some relations which we have not found in this generality in the literature. Finally, we demonstrate the limiting process by numerical examples and show the behavior of the distances when used for the approximation of measures by point measures in a process called dithering.