Generalizing Bottleneck Problems

TL;DR

This paper introduces a generalized framework for bottleneck problems using f-information, providing algorithms and characterizations for specific cases, connecting to classical lemmas and information-theoretic concepts.

Contribution

It generalizes bottleneck functionals with convex functions, offers algorithms for their computation, and characterizes specific cases relating to classical information-theoretic lemmas.

Findings

Characterization of the set for binary symmetric case with specific f-functions.

Algorithms for computing boundaries of the f-information set.

Connections to Mrs. Gerber's Lemma, estimation theory, and Arimoto information measures.

Abstract

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 $\ell^\beta$ norm function for $\beta \geq 2$. 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.