Integral Probability Metric based Regularization for Optimal Transport

Recently it has been shown that Maximum Mean Discrepancy (MMD) based regularization for optimal transport (OT), unlike the popular Kullback Leibler (KL) based regularization, leads to a dimension-free bound on the sample complexity of estimation. On the other hand, interesting classes of metrics like the Generalized Wasserstein (GW) metrics and the Gaussian-Hellinger-Kantorovich (GHK) metrics are defined using Total Variation and KL based regularizations, respectively. It is, however, an open question if appropriate metrics could be defined using the sample-efficient MMD regularization. In this work, we not only bridge this gap, but further consider a generic family of regularizers based on Integral Probability Metrics (IPMs), which include MMD as a special case. We present novel IPM regularized $p$-Wasserstein style OT formulations and prove that they indeed induce metrics over measures. While some of these novel metrics can be interpreted as infimal convolutions of IPMs, interestingly, others turn out to be the IPM-analogues of GW and GHK metrics. Finally, we present finite sample-based formulations for estimating the squared-MMD regularized metric and the corresponding barycenter. We empirically study other desirable properties of the proposed metrics and show their applicability in various machine learning applications.