The Information Bottleneck's Ordinary Differential Equation: First-Order Root-Tracking for the IB

TL;DR

This paper introduces a novel first-order ODE for the Information Bottleneck (IB), enabling precise tracking of optimal solutions and bifurcations, thus improving numerical solutions for IB problems.

Contribution

It derives a first-order ODE for IB dynamics, allowing accurate bifurcation detection and solution tracking, advancing numerical methods for IB optimization.

Findings

The IB's solution trajectories follow piecewise smooth paths with bifurcations.

The derived ODE accurately detects and classifies IB bifurcations.

The method enables following the optimal IB curve more reliably.

Abstract

The Information Bottleneck (IB) is a method of lossy compression of relevant information. Its rate-distortion (RD) curve describes the fundamental tradeoff between input compression and the preservation of relevant information embedded in the input. However, it conceals the underlying dynamics of optimal input encodings. We argue that these typically follow a piecewise smooth trajectory when input information is being compressed, as recently shown in RD. These smooth dynamics are interrupted when an optimal encoding changes qualitatively, at a bifurcation. By leveraging the IB's intimate relations with RD, we provide substantial insights into its solution structure, highlighting caveats in its finite-dimensional treatments. Sub-optimal solutions are seen to collide or exchange optimality at its bifurcations. Despite the acceptance of the IB and its applications, there are surprisingly few techniques to solve it numerically, even for finite problems whose distribution is known. We derive anew the IB's first-order Ordinary Differential Equation, which describes the dynamics underlying its optimal tradeoff curve. To exploit these dynamics, we not only detect IB bifurcations but also identify their type in order to handle them accordingly. Rather than approaching the IB's optimal curve from sub-optimal directions, the latter allows us to follow a solution's trajectory along the optimal curve under mild assumptions. We thereby translate an understanding of IB bifurcations into a surprisingly accurate numerical algorithm.