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At the heart of both lossy compression and clustering is a trade-off betweenthe fidelity and size of the learned representation. Our goal is to map out andstudy the Pareto frontier that quantifies this trade-off. We focus on theoptimization of the Deterministic Information Bottleneck (DIB) objective overthe space of hard clusterings. To this end, we introduce the primal DIBproblem, which we show results in a much richer frontier than its previouslystudied Lagrangian relaxation when optimized over discrete search spaces. Wepresent an algorithm for mapping out the Pareto frontier of the primal DIBtrade-off that is also applicable to other two-objective clustering problems.We study general properties of the Pareto frontier, and we give both analyticand numerical evidence for logarithmic sparsity of the frontier in general. Weprovide evidence that our algorithm has polynomial scaling despite thesuper-exponential search space, and additionally, we propose a modification tothe algorithm that can be used where sampling noise is expected to besignificant. Finally, we use our algorithm to map the DIB frontier of threedifferent tasks: compressing the English alphabet, extracting informative colorclasses from natural images, and compressing a group theory-inspired dataset,revealing interesting features of frontier, and demonstrating how the structureof the frontier can be used for model selection with a focus on pointspreviously hidden by the cloak of the convex hull.

Pareto-optimal clustering with the primal deterministic information bottleneck