The information bottleneck method

TL;DR

The paper introduces the information bottleneck method, a framework for extracting relevant information from signals by compressing data while preserving information about another related signal, with applications in signal processing and learning.

Contribution

It formalizes the information bottleneck principle as a constrained optimization problem and provides exact self-consistent equations and a convergent algorithm for its solution.

Findings

Provides a set of self-consistent equations for the coding rules.

Introduces a convergent re-estimation algorithm generalizing Blahut-Arimoto.

Framework applicable to various problems in signal processing and learning.

Abstract

We define the relevant information in a signal $x\in X$ as being the information that this signal provides about another signal $y\in \Y$. Examples include the information that face images provide about the names of the people portrayed, or the information that speech sounds provide about the words spoken. Understanding the signal $x$ requires more than just predicting $y$, it also requires specifying which features of $\X$ play a role in the prediction. We formalize this problem as that of finding a short code for $\X$ that preserves the maximum information about $\Y$. That is, we squeeze the information that $\X$ provides about $\Y$ through a `bottleneck' formed by a limited set of codewords $\tX$. This constrained optimization problem can be seen as a generalization of rate distortion theory in which the distortion measure $d(x,\x)$ emerges from the joint statistics of $\X$ and $\Y$. This approach yields an exact set of self consistent equations for the coding rules $X \to \tX$ and $\tX \to \Y$. Solutions to these equations can be found by a convergent re-estimation method that generalizes the Blahut-Arimoto algorithm. Our variational principle provides a surprisingly rich framework for discussing a variety of problems in signal processing and learning, as will be described in detail elsewhere.