The Multi-layer Information Bottleneck Problem

TL;DR

This paper investigates the multi-layer information bottleneck problem, deriving optimal trade-offs between relevance and compression, and establishing conditions for successive refinability across various models, including binary and Gaussian cases.

Contribution

It provides a single-letter characterization of the rate-relevance region and conditions for successive refinability in multi-layer IB problems, extending to Gaussian models.

Findings

Binary source with BSC hidden variables is successively refinable.

Gaussian models are extended and analyzed for successive refinability.

Counterexamples show limitations of successive refinability.

Abstract

The muti-layer information bottleneck (IB) problem, where information is propagated (or successively refined) from layer to layer, is considered. Based on information forwarded by the preceding layer, each stage of the network is required to preserve a certain level of relevance with regards to a specific hidden variable, quantified by the mutual information. The hidden variables and the source can be arbitrarily correlated. The optimal trade-off between rates of relevance and compression (or complexity) is obtained through a single-letter characterization, referred to as the rate-relevance region. Conditions of successive refinabilty are given. Binary source with BSC hidden variables and binary source with BSC/BEC mixed hidden variables are both proved to be successively refinable. We further extend our result to Guassian models. A counterexample of successive refinability is also provided.