Generalizing Bottleneck Problems
This work addresses theoretical bottlenecks in information theory, offering incremental extensions to existing concepts with specific characterizations.
The paper tackles the problem of generalizing bottleneck problems by introducing bottleneck functionals for pairs of random variables and convex functions, providing an algorithm to compute boundaries and fully characterizing the set in the binary symmetric case for specific functions. The results connect these boundaries to known lemmas and variants like Mrs. Gerber's Lemma and Information Bottleneck.
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 $\left(I_{f_1}(W; X), I_{f_2}(W; Y)\right)$, where $I_f$ denotes $f$-information and $W$ varies over the set of all discrete random variables satisfying the Markov condition $W \to X \to Y$. 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 $f_2$ are both $\ell^β$ norm function for $β\geq 2$. 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.