We develop a novel family of metrics over measures, using $p$-Wasserstein style optimal transport (OT) formulation with dual-norm based regularized marginal constraints. Our study is motivated by the observation that existing works have only explored $\phi$-divergence regularized Wasserstein metrics like the Generalized Wasserstein metrics or the Gaussian-Hellinger-Kantorovich metrics. It is an open question if Wasserstein style metrics can be defined using regularizers that are not $\phi$-divergence based. Our work provides an affirmative answer by proving that the proposed formulation, under mild conditions, indeed induces valid metrics for any dual norm. The proposed regularized metrics seem to achieve the best of both worlds by inheriting useful properties from the parent metrics, viz., the $p$-Wasserstein and the dual-norm involved. For example, when the dual norm is Maximum Mean Discrepancy (MMD), we prove that the proposed regularized metrics inherit the dimension-free sample complexity from the MMD regularizer; while preserving/enhancing other useful properties of the $p$-Wasserstein metric. Further, when $p=1$, we derive a Fenchel dual, which enables proving that the proposed metrics actually induce novel norms over measures. Also, in this case, we show that the mixture geodesic, which is a common geodesic for the parent metrics, remains a geodesic. We empirically study various properties of the proposed metrics and show their utility in diverse applications.

翻译：我们研究的动机是,现有工程只探索了以美元为单位的标准化瓦森斯坦标准或Gaussian-Hellinger-Kantorovich标准等标准化瓦森斯坦标准。这是一个开放的问题。例如,如果用非以美元为单位的规范标准来定义瓦森斯坦标准标准,那么我们的工作提供了一个肯定的答案。我们的工作通过证明拟议的配方在温和条件下确实为任何双重规范带来了有效的衡量标准,从而提供了有效的衡量标准。拟议的正规化标准似乎通过从母体的衡量标准中继承有用的属性,即瓦森-Hellinger-Kantorovich标准或Gaussian-Hellinger-Kantorovich标准来实现两个世界的最好。例如,如果双轨标准不是以美元为单位的调和调和度为基础。我们的工作证明,拟议的正规化指标可以继承MMDR正值常规值的无尺寸的采样复杂性;拟议的正规化指标似乎通过从母体的通用标准中继承了两个世界的最佳度指标,与此同时,我们提出的标准能进一步显示我们提出的标准。