Linear Bounds between Contraction Coefficients for $f$-Divergences

Data processing inequalities for $f$-divergences can be sharpened using contraction coefficients to produce strong data processing inequalities. These contraction coefficients turn out to provide useful optimization problems for learning likelihood models. Moreover, the contraction coefficient for $\chi^2$-divergence admits a particularly simple linear algebraic solution due to its relation to maximal correlation. Propelled by this context, we analyze the relationship between various contraction coefficients for $f$-divergences and the contraction coefficient for $\chi^2$-divergence. In particular, we prove that the latter coefficient can be obtained from the former coefficients by driving the input $f$-divergences to zero. Then, we establish linear bounds between these contraction coefficients. These bounds are refined for the KL divergence case using a well-known distribution dependent variant of Pinsker's inequality.