Refined Pinsker's and reverse Pinsker's inequalities for probability distributions of different dimensions

TL;DR

This paper introduces refined versions of Pinsker's inequalities that provide optimal bounds for the augmented Kullback-Leibler divergence based on the augmented total variation distance between probability measures in different-dimensional Euclidean spaces.

Contribution

It presents the first optimal bounds for the augmented divergence measures, extending Pinsker's inequalities to probability distributions of different dimensions.

Findings

Derived optimal bounds for augmented Kullback-Leibler divergence

Extended Pinsker's inequalities to multi-dimensional probability measures

Provided theoretical framework for divergence measures across different dimensions

Abstract

We provide optimal lower and upper bounds for the augmented Kullback-Leibler divergence in terms of the augmented total variation distance between two probability measures defined on two Euclidean spaces having different dimensions. We call them refined Pinsker's and reverse Pinsker's inequalities, respectively.