# Pareto-optimal clustering with the primal deterministic information bottleneck

arXiv: https://arxiv.org/abs/2204.02489 · HTML: https://mlsanity.com/paper/2204.02489/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 40 references that are in the map.
- First posted on arXiv 5 Apr 2022.

## Builds on (most cited first, 19 of 19)

- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [Deep Variational Information Bottleneck](https://mlsanity.com/paper/1612.00410/) (2016)
- [Deep Learning and the Information Bottleneck Principle](https://mlsanity.com/paper/1503.02406/) (2015)
- [On Variational Bounds of Mutual Information](https://mlsanity.com/paper/1905.06922/) (2019)
- [Emergence of Invariance and Disentanglement in Deep Representations](https://mlsanity.com/paper/1706.01350/) (2017)
- [Learning the Pareto Front with Hypernetworks](https://mlsanity.com/paper/2010.04104/) (2020)
- [Stochastic gradient descent performs variational inference, converges to limit cycles for deep networks](https://mlsanity.com/paper/1710.11029/) (2017)
- [Taming VAEs](https://mlsanity.com/paper/1810.00597/) (2018)
- [The Conditional Entropy Bottleneck](https://mlsanity.com/paper/2002.05379/) (2020)
- [Controllable Pareto Multi-Task Learning](https://mlsanity.com/paper/2010.06313/) (2020)
- [The deterministic information bottleneck](https://mlsanity.com/paper/1604.00268/) (2016)
- [Caveats for information bottleneck in deterministic scenarios](https://mlsanity.com/paper/1808.07593/) (2018)
- [The Hardness of Approximation of Euclidean k-means](https://mlsanity.com/paper/1502.03316/) (2015)
- [Learnability for the Information Bottleneck](https://mlsanity.com/paper/1907.07331/) (2019)
- [Phase Transitions for the Information Bottleneck in Representation Learning](https://mlsanity.com/paper/2001.01878/) (2020)
- [TherML: Thermodynamics of Machine Learning](https://mlsanity.com/paper/1807.04162/) (2018)
- [The information bottleneck and geometric clustering](https://mlsanity.com/paper/1712.09657/) (2017)
- [Perturbation Theory for the Information Bottleneck](https://mlsanity.com/paper/2105.13977/) (2021)
- [Pareto-optimal data compression for binary classification tasks](https://mlsanity.com/paper/1908.08961/) (2019)

## Abstract (arXiv)

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.