The Multi-layer Information Bottleneck Problem
This work addresses theoretical limits in multi-layer information processing for machine learning and communication systems, offering incremental advances in understanding successive refinability.
The paper tackles the multi-layer information bottleneck problem, where information is refined across layers while preserving relevance to hidden variables, and obtains the optimal trade-off between relevance and compression rates, proving successive refinability for specific binary and Gaussian models and providing a counterexample.
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.