Bregman Voronoi Diagrams: Properties, Algorithms and Applications (0709.2196v1)

Abstract: The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define many variants of Voronoi diagrams depending on the class of objects, the distance functions and the embedding space. In this paper, we investigate a framework for defining and building Voronoi diagrams for a broad class of distance functions called Bregman divergences. Bregman divergences include not only the traditional (squared) Euclidean distance but also various divergence measures based on entropic functions. Accordingly, Bregman Voronoi diagrams allow to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We define several types of Bregman diagrams, establish correspondences between those diagrams (using the Legendre transformation), and show how to compute them efficiently. We also introduce extensions of these diagrams, e.g. k-order and k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set of points and their connexion with Bregman Voronoi diagrams. We show that these triangulations capture many of the properties of the celebrated Delaunay triangulation. Finally, we give some applications of Bregman Voronoi diagrams which are of interest in the context of computational geometry and machine learning.

• The paper introduces Bregman Voronoi diagrams that generalize traditional Euclidean partitions using divergence measures.
• It details efficient projection-based algorithms with polynomial complexity for constructing first-type, second-type, and symmetrized diagrams.
• Extensions to weighted and k-order diagrams enable enhanced clustering and classification in diverse computational and data-driven fields.

Bregman Voronoi Diagrams: Properties, Algorithms, and Applications

The paper "Bregman Voronoi Diagrams: Properties, Algorithms, and Applications" extends the traditional concept of Voronoi diagrams to a wider set of divergence measures, specifically Bregman divergences. This advancement facilitates the application of Voronoi diagrams in diverse fields such as computational geometry and machine learning.

Overview

Voronoi diagrams traditionally work with Euclidean distances, partitioning a given space into regions based on proximity to a set of points. This concept is generalized in this paper by leveraging Bregman divergences instead of conventional metric distances. Bregman divergences are a broader class that includes the squared Euclidean distance alongside others such as Kullback-Leibler divergence.

Key Contributions

1. Types of Bregman Voronoi Diagrams:
   • The paper defines three types of Bregman Voronoi diagrams: first-type, second-type, and symmetrized diagrams. The first-type has convex polyhedral cells, the second-type's cells may be curved due to divergence asymmetry, and symmetrized diagrams involve a two-fold Euclidean space embedding.
2. Algorithms and Efficiency:
   • The authors present efficient algorithms for constructing these Voronoi diagrams. Specifically, they show that the computation involves the projection of high-dimensional polyhedra onto lower-dimensional spaces, leading to efficient construction times with polynomial complexity bounds.
3. Weighted and k-order Extensions:
   • This work extends the original concept to weighted Bregman Voronoi diagrams and introduces k-order constructs. These extensions enable handling of more complex geometric and optimization problems, such as those involving mixtures of divergences.
4. Correspondence with Power Diagrams:
   • The paper decisively connects Bregman Voronoi diagrams with power diagrams, showing the former as a kind of disguised power diagram. This relationship provides a foundation for exact and efficient computations using existing geometric algorithm frameworks.
5. Bregman Triangulations:
   • The paper discusses two forms of triangulations derived from Bregman Voronoi diagrams. The first form relates to traditional Delaunay triangulations, while the second involves geodesic, or curved-edge, constructions that offer a geometric dual to Bregman Voronoi diagrams.

Implications and Applications

Bregman Voronoi diagrams expand the toolbox available in computational geometry and information geometry. They provide means to handle non-Euclidean distance functions efficiently. For machine learning, these structures facilitate improvements in clustering and classification, where geometric consistency and flexibility are desired. For example, they offer advantages in domains dominated by statistical distances such as the Kullback-Leibler divergence.

Furthermore, the findings could be foundational for developing robust algorithms in data-heavy applications like image processing, data mining, and knowledge discovery. While the computational complexity still presents challenges, particularly in high-dimensional settings, the insights into Bregman divergences promote better handling of metric anomalies and the incorporation of noise and uncertainty in geometric computations.

Future Directions

Future research could focus on resolving high-dimensional challenges, constructing data-optimized structures, and expanding on the theoretical foundation for integrating heterogenous data types within Voronoi frameworks. Advanced algorithms that minimize the practical clustering challenge—high complexity due to dimensional scaling—could also be explored. Experimentation with Bregman Voronoi diagrams in real-world applications may yield new insights and push the boundaries of both theoretical and applied computational geometry.

Conclusion

The paper establishes a comprehensive framework for Bregman Voronoi diagrams, characterized by versatile distance functions. Through a blend of theoretical insight and practical algorithmic adaptations, it opens new avenues for advancing computational geometry's role in data analytics and machine learning, setting the stage for further exploration and innovation in these fields.