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Pareto-Optimal Clustering with the Primal Deterministic Information Bottleneck.

Andrew K Tan1,2, Max Tegmark1,2, Isaac L Chuang1,2,3

  • 1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.

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Summary

This study introduces a new method to map the Pareto frontier for Deterministic Information Bottleneck (DIB) optimization in clustering. The approach reveals hidden trade-offs between data representation fidelity and size, aiding model selection.

Keywords:
Paretobottleneckclusteringfrontierinformationmulti-objectiveoptimization

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Area of Science:

  • Information Theory
  • Machine Learning
  • Data Compression
  • Clustering Algorithms

Background:

  • Lossy compression and clustering rely on balancing data representation fidelity with size.
  • Existing methods often optimize relaxed versions of objectives, potentially missing optimal solutions in discrete spaces.
  • The Deterministic Information Bottleneck (DIB) objective quantifies this trade-off but is challenging to optimize over hard clusterings.

Purpose of the Study:

  • To map and analyze the Pareto frontier quantifying the fidelity-size trade-off in learned representations.
  • To introduce and optimize the primal DIB problem over discrete search spaces for a richer frontier.
  • To develop an algorithm for mapping Pareto frontiers applicable to various two-objective clustering problems.

Main Methods:

  • Formulation of the primal DIB problem for discrete optimization.
  • Development of a novel algorithm to map the Pareto frontier of the primal DIB objective.
  • Analysis of the Pareto frontier's properties, including logarithmic sparsity, and algorithm scaling.
  • Application of the algorithm to diverse datasets: English alphabet compression, image color clustering, and group theory data.

Main Results:

  • The primal DIB problem yields a richer Pareto frontier compared to its Lagrangian relaxation in discrete spaces.
  • The developed algorithm demonstrates polynomial scaling, efficiently navigating super-exponential search spaces.
  • Logarithmic sparsity of the Pareto frontier is evidenced both analytically and numerically.
  • Mapping the DIB frontier for various tasks revealed unique characteristics and facilitated model selection by highlighting previously obscured points.

Conclusions:

  • The primal DIB optimization and the proposed frontier-mapping algorithm offer a more comprehensive approach to understanding data representation trade-offs.
  • The algorithm's efficiency and applicability to diverse problems make it a valuable tool for model selection and analysis.
  • This work provides new insights into the structure of Pareto frontiers in clustering and compression tasks.