Divergence Inequalities from Multivariate Taylor's Theorem

Divergences are a fundamental framework for measuring dissimilarity in information theory and statistics. Here, we use the multivariate Taylor's theorem to obtain new integral representations of twice-differentiable Bregman and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{f}$</tex>-divergences in finite dimensions. This results in input-dependent bounds on many <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$f$</tex>-divergences in terms of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\chi^{2}$</tex>-divergence as well as reverse Pinsker inequalities. As an application, we show how this provides upper bounds on the mixing time of irreducible, scrambling Markov chains under a large class of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$f$</tex> divergences in terms of the input-dependent <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\chi^{2}$</tex> contraction coefficient.