Pareto-optimal clustering with the primal deterministic information bottleneck

TL;DR

This paper explores the optimization of the Deterministic Information Bottleneck (DIB) for clustering, introducing a primal formulation and an algorithm to map the Pareto frontier, revealing insights into trade-offs and aiding model selection.

Contribution

It introduces the primal DIB problem, develops an algorithm for Pareto frontier mapping, and analyzes the properties of the DIB trade-off in clustering tasks.

Findings

Primal DIB yields a richer Pareto frontier than Lagrangian relaxation.

The Pareto frontier exhibits logarithmic sparsity.

The proposed algorithm scales polynomially despite the super-exponential search space.

Abstract

At the heart of both lossy compression and clustering is a trade-off between the fidelity and size of the learned representation. Our goal is to map out and study the Pareto frontier that quantifies this trade-off. We focus on the optimization of the Deterministic Information Bottleneck (DIB) objective over the space of hard clusterings. To this end, we introduce the primal DIB problem, which we show results in a much richer frontier than its previously studied Lagrangian relaxation when optimized over discrete search spaces. We present an algorithm for mapping out the Pareto frontier of the primal DIB trade-off that is also applicable to other two-objective clustering problems. We study general properties of the Pareto frontier, and we give both analytic and numerical evidence for logarithmic sparsity of the frontier in general. We provide evidence that our algorithm has polynomial scaling despite the super-exponential search space, and additionally, we propose a modification to the algorithm that can be used where sampling noise is expected to be significant. Finally, we use our algorithm to map the DIB frontier of three different tasks: compressing the English alphabet, extracting informative color classes from natural images, and compressing a group theory-inspired dataset, revealing interesting features of frontier, and demonstrating how the structure of the frontier can be used for model selection with a focus on points previously hidden by the cloak of the convex hull.