Lipschitz Variational Approximation of Total Variation Distance

In this work, we build on the family of Integral Probability Metrics and design a new distance metric between probability distributions that belong to this family.The new metric is termed Lipschitz Variational Total Variation Distance and is a relaxation of the integral probability metric representation of the well-known total variation distance.We propose simple procedures to estimate this distance metric and demonstrate its convergence.Based on the Lipschitz smoothness of the proposed metric family, the proposed metrics, hence its empirical estimate, can provide meaningful and tight lower bounds for the total variation distance between two probability distributions.Finally, we extend our results to general measures and provide an application of the proposed estimators to bounding the Neyman-Pearson region.