Generalizing Bottleneck Problems

Given a pair of random variables $(X,\ Y)\sim P_{XY}$ and two convex functions $f_{1}$ and $f_{2}$, we introduce two bottleneck functionals as the lower and upper boundaries of the two-dimensional convex set that consists of the pairs $(I_{f_{1}}(W;X),\ I_{f_{2}}(W;Y))$, where $I_{f}$ denotes $f$-information and $W$ varies over the set of all discrete random variables satisfying the Markov condition $W\rightarrow X\rightarrow Y$. Applying …