Generalizing Bottleneck Problems

Given a pair of random variables (X, Y) similar to 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-f1 (W;X); I-f2 (W; Y)), where I-f denotes f-information and W varies over the set of all discrete random variables satisfying the Markov condition W -> X -> Y. Applying Witsenhausen and Wyner's approach, we provide an algorithm for computing boundaries of this set for f(1), f(2), and discrete P-XY. In the binary symmetric case, we fully characterize the set when (i) f(1)(t) = f(2)(t) = t log t, (ii) f(1)(t) = f(2)(t) = t(2) - 1, and (iii) f(1) and f2 are both l(beta) norm function for beta >= 2. We then argue that upper and lower boundaries in (i) correspond to Mrs. Gerber's Lemma and its inverse (which we call Mr. Gerber's Lemma), in (ii) correspond to estimation-theoretic variants of Information Bottleneck and Privacy Funnel, and in (iii) correspond to Arimoto Information Bottleneck and Privacy Funnel. An extended version of this paper is available in [1].