Bregman Voronoi Diagram

• Bregman Voronoi diagrams are geometric partitions that use Bregman divergences derived from convex generator functions to measure proximity in vector spaces.
• They leverage Legendre duality to relate affine first-type cells with curved dual cells, enabling efficient computation via reduction to power diagrams.
• Applications include prototype-based machine learning, vector quantization, and TV-regularized estimation, making them vital for modern clustering and data compression.

A Bregman Voronoi diagram is a fundamental geometric structure that partitions an ambient vector space according to proximity with respect to a Bregman divergence, a class of distance-like measures derived from convex functions. Formally, given a strictly convex, differentiable “generator” $F: \mathbb{R}^d \to \mathbb{R}$, the associated Bregman divergence between $p, q \in \mathbb{R}^d$ is defined by $$D_F(p\,\|\,q) = F(p) - F(q) - \langle \nabla F(q), p-q\rangle$$. Classical Voronoi diagrams for the squared Euclidean metric are thus a special case. This generalization enables Voronoi-type partitioning in statistical and information geometric spaces, including relative entropy (Kullback-Leibler divergence), Mahalanobis distance, and other entropic measures. Bregman Voronoi diagrams provide the geometric infrastructure for prototype assignment, loss functions, and clustering algorithms in a variety of parametric and geometric regimes (0709.2196).

1. Bregman Divergences and Affine Bisectors

Let $F: \mathbb{R}^d \to \mathbb{R}$ be a strictly convex, continuously differentiable generator. The Bregman divergence $D_F(p\,\|\,q)$ is nonnegative, vanishing only if $p=q$, but is generally not symmetric and does not obey the triangle inequality. Given a finite set of sites or prototypes $Q=\{q_1,\dots, q_n\}$, the primal (first-type) Bregman Voronoi cell for $q_j$ is defined by

$$V_F(q_j) = \left\{x \in \mathbb{R}^d : D_F(x\,\|\,q_j) \leq D_F(x\,\|\,q_i) \quad \forall i \neq j \right\}.$$

The bisector between $q_j$ and $p, q \in \mathbb{R}^d$0 is given by the affine hyperplane

$p, q \in \mathbb{R}^d$1

which can be rewritten in the form

$p, q \in \mathbb{R}^d$2

Therefore, each $p, q \in \mathbb{R}^d$3 is a convex polyhedron in $p, q \in \mathbb{R}^d$4-space. In contrast, the dual (second-type) cell

$p, q \in \mathbb{R}^d$5

generally has curved boundaries in $p, q \in \mathbb{R}^d$6-space, but becomes linear in the dual coordinates $p, q \in \mathbb{R}^d$7.

2. Legendre Duality and Combinatorial Equivalence

The Legendre transform $p, q \in \mathbb{R}^d$8 defines the convex conjugate of $p, q \in \mathbb{R}^d$9, and establishes a mapping between the primal and dual spaces via $$D_F(p\,\|\,q) = F(p) - F(q) - \langle \nabla F(q), p-q\rangle$$0 and $$D_F(p\,\|\,q) = F(p) - F(q) - \langle \nabla F(q), p-q\rangle$$1. The dual Bregman divergence,

$$D_F(p\,\|\,q) = F(p) - F(q) - \langle \nabla F(q), p-q\rangle$$2

obeys the identity $$D_F(p\,\|\,q) = F(p) - F(q) - \langle \nabla F(q), p-q\rangle$$3 for $$D_F(p\,\|\,q) = F(p) - F(q) - \langle \nabla F(q), p-q\rangle$$4. Consequently, the first-type Voronoi diagram of $$D_F(p\,\|\,q) = F(p) - F(q) - \langle \nabla F(q), p-q\rangle$$5 under $$D_F(p\,\|\,q) = F(p) - F(q) - \langle \nabla F(q), p-q\rangle$$6 is combinatorially equivalent to the second-type (curved) diagram of $$D_F(p\,\|\,q) = F(p) - F(q) - \langle \nabla F(q), p-q\rangle$$7 under $$D_F(p\,\|\,q) = F(p) - F(q) - \langle \nabla F(q), p-q\rangle$$8 in gradient space, and vice versa (0709.2196).

3. Efficient Construction and Algorithmic Properties

The primal Bregman Voronoi diagram admits an efficient reduction to power diagrams via a "lifting" argument. For $$D_F(p\,\|\,q) = F(p) - F(q) - \langle \nabla F(q), p-q\rangle$$9 and $F: \mathbb{R}^d \to \mathbb{R}$0, one rewrites the domination constraint as

$F: \mathbb{R}^d \to \mathbb{R}$1

which is the power diagram (additively weighted Voronoi) condition. Therefore, all standard algorithms for constructing power diagrams, with $F: \mathbb{R}^d \to \mathbb{R}$2 time complexity, can be leveraged for Bregman Voronoi diagrams (0709.2196).

Higher-order variants are defined:

• $F: \mathbb{R}^d \to \mathbb{R}$3-order Bregman Voronoi: Regions are partitioned by the $F: \mathbb{R}^d \to \mathbb{R}$4 nearest prototypes (with reduction to $F: \mathbb{R}^d \to \mathbb{R}$5 centroids).
• $F: \mathbb{R}^d \to \mathbb{R}$6-bag Bregman Voronoi: Uses site-specific generators at each site $F: \mathbb{R}^d \to \mathbb{R}$7; $F: \mathbb{R}^d \to \mathbb{R}$8, resulting in an affine subdivision in $F: \mathbb{R}^d \to \mathbb{R}$9.

The dual structure, a Bregman triangulation, is obtained by projecting the lower hull of the lifted points $D_F(p\,\|\,q)$0, yielding Delaunay-type triangulations with geometric properties linked to the underlying divergence.

4. Integration with Prototype-Based Machine Learning

Loss functions associated with Bregman Voronoi diagrams instantiate prototype assignment via

$D_F(p\,\|\,q)$1

where $D_F(p\,\|\,q)$2 is a training set, and $D_F(p\,\|\,q)$3 is the set of prototypes. Partitioning $D_F(p\,\|\,q)$4 into cells $D_F(p\,\|\,q)$5, each $D_F(p\,\|\,q)$6 is assigned to its nearest prototype under $D_F(p\,\|\,q)$7, and the assignment loss can be minimized by a Lloyd-type fixed-point iteration. This underpins:

• $D_F(p\,\|\,q)$8-means clustering: $D_F(p\,\|\,q)$9 recovers the classical setting.
• KL-means (exponential-family clustering): Here, $p=q$0 is the cumulant (log-partition) function, and $p=q$1 becomes the reversed Kullback-Leibler divergence $p=q$2.
• Gaussian mixtures via relative-entropy cells: For Gaussian parameter space $p=q$3, the KL divergence between normal distributions is a Bregman divergence; Voronoi cells partition parameter space for assignment based on sample-covariance estimates (0709.2196).

Prototype-based classification uses Bregman divergences as distortion measures in nearest-neighbor assignments. The VC-dimension of Bregman balls in $p=q$4 is $p=q$5, matching that of half-spaces, implying identical generalization bounds to their Euclidean-analogous classifiers.

5. Bregman-Centric Tessellations and Quantization

Lloyd’s fixed-point iteration converges to a (local) minimum of $p=q$6, thus producing a “Bregman-centroidal Voronoi tessellation.” Such tessellations are critical for vector quantization, data compression, and codebook design in information theory. Their construction leverages the affine geometry of first-type diagrams, retaining computational efficiency and geometric properties (fatness, empty-ball, locality) advantageous for $p=q$7-means seeding, $p=q$8-net construction, and sample adaptivity (0709.2196).

Bregman triangulations, as regular triangulations of the equivalent power diagram in gradient space, can be pulled back via $p=q$9 to form geodesic triangulations whose edges are gradient-geodesics with linear dual-coordinate parametrization. These support rapid geometric queries essential in high-dimensional clustering and classification pipelines.

6. Extension: Voronoi-Structured TV-Regularization and Connections

While classical Voronoi diagrams, including those induced by Bregman divergences, are fundamentally proximity partitions, they also serve as discretization frameworks for regularization in statistical estimation. The "Voronoigram" estimator fits a piecewise-constant function over Voronoi cells and regularizes by the discretized total variation along shared cell boundaries: $Q=\{q_1,\dots, q_n\}$0 where $Q=\{q_1,\dots, q_n\}$1 records adjacent cells and $Q=\{q_1,\dots, q_n\}$2 measures the $Q=\{q_1,\dots, q_n\}$3-dimensional shared-face between $Q=\{q_1,\dots, q_n\}$4 and $Q=\{q_1,\dots, q_n\}$5. As the number of sample sites increases, the weighted Voronoi-TV converges (in probability) to the continuum total variation $Q=\{q_1,\dots, q_n\}$6, in contrast to density-weighted asymptotics for $Q=\{q_1,\dots, q_n\}$7-neighborhood or $Q=\{q_1,\dots, q_n\}$8-NN graph TV (Hu et al., 2022). This suggests that Voronoi-based regularization is density-free and naturally extrapolates to continuum TV for piecewise constant functions, while alternative random-graph TV penalties intrinsically encode local density.

The Voronoigram estimator is minimax rate optimal (up to log factors) for estimating bounded variation functions, with $Q=\{q_1,\dots, q_n\}$9 rates up to logarithmic factors, and is tuning-free with respect to graph construction parameters.

7. Summary and Unifying Perspective

Bregman Voronoi diagrams generalize the classic Voronoi partition to arbitrary Bregman divergences, enveloping a wide array of distance measures—squared Euclidean, Mahalanobis, KL divergence, and other entropic forms. Their geometric structure is combinatorially equivalent, via Legendre duality, to dual diagrams in gradient space, and their affine first-type cells enable efficient computation by reduction to power diagrams. These diagrams provide the foundation for prototype-based loss functions and clustering algorithms, underlie advanced quantization and tessellation techniques, and possess extensions to higher-order, bag, and geodesic constructions. In statistical estimation, Voronoi-based discretizations underpin TV-regularized estimators with optimal theoretical guarantees. The geometric and algorithmic properties of Bregman Voronoi diagrams thus render them central to modern computational geometry, clustering, and information-theoretic learning (0709.2196, Hu et al., 2022).