Voronoi-Type Loss Functions

Updated 16 October 2025

Voronoi-type loss functions are mathematical tools that partition domains by minimizing loss or distance, extending classical Euclidean concepts.
They use diverse metrics such as Bregman divergences and Lie sphere geometry to construct efficient diagrams for clustering, regression, and decision-making.
Their applications span mesh generation, hashing, Bayesian optimization, and proper scoring in statistical inference and machine learning.

Voronoi-type loss functions are a class of mathematical constructs that generalize the assignment of regions in a domain according to minimal loss or distance, often extending the Euclidean notion of "closeness" to more complex functions or geometric structures. These loss functions partition space, probability simplices, or other abstract domains based on minimization criteria derived from divergences, distance functions, convex sets, or other regularization mechanisms. The paper and application of Voronoi-type loss functions span computational geometry, optimization, machine learning, hashing, regression, and statistical decision theory.

1. Mathematical Foundations and Variants

Voronoi-type loss functions arise from generalizations of classical Voronoi diagrams, wherein each point in space is assigned to a cell associated with a "^{^{^{^{1^{^{^{^"}}}}}}} by minimizing a loss function or divergence. The classical paradigm uses Euclidean distance, but advanced formulations include:

Bregman Divergence-based Losses: Bregman Voronoi diagrams replace squared Euclidean distance with a Bregman divergence $D_F(x||p)$ where $F$ is a strictly convex, differentiable generator (e.g., entropy, KL-divergence). Due to the non-symmetry of Bregman divergences, two types of diagrams are distinguished: first-type (minimize $D_F(x||p)$ ) and second-type (minimize $D_F(p||x)$ ). First-type diagrams can be represented as power diagrams via a lifting transformation $(x, F(x))$ and admit efficient algorithms (0709.2196).
Generalized Diagrams via Lie Sphere Geometry: The framework described in Lie sphere geometry encodes generalized Voronoi diagrams for a broad family of sites (points, spheres, half-spaces) through affine functions on a lifted quadric in projective space. Minimization diagrams of the form $f_i(x) = a_i\|x-q_i\|^2 + b_i\cdot x + c_i$ admit representations as intersections and projections of polyhedra in the lifted space, unifying classical, weighted, power, and Apollonius diagrams (Edwards et al., 17 Aug 2024).
2-site and Multi-site Loss Functions: Loss functions can be defined over unordered pairs (or larger tuples) of sites. Geometric distances such as circumscribing/containing circle radius, inscribed circle radius, view angle, and parameterized perimeter yield “loss” values associated with tuples and give rise to highly complex Voronoi diagrams, with combinatorial structures scaling as $O(n^{4+\epsilon})$ , but sometimes reduced for particular metrics (e.g., perimeter distance with $c \geq 0$ ) when only Delaunay neighbors generate active regions (Barequet et al., 2011).
Convex-analytic and Divergence-based Losses: Loss functions defined as subgradients of support functions of convex sets (superprediction sets) provide proper scoring rules for probability estimation, with ties to (anti)-norms and Fenchel-Young losses based on f-divergences (including KL, $\alpha$ -divergence, Tsallis, and others). These can induce tessellations or partitionings of the simplex with Voronoi-like boundaries according to induced decision regions (Williamson et al., 2022, Roulet et al., 30 Jan 2025).

2. Algorithmic Construction and Complexity

Voronoi-type loss functions underlie diagrams and optimization tasks that require efficient partitioning and minimization strategies:

Lifting and Polytope Intersection: For Bregman and affine-minimization diagrams, the lifting transformation embeds domain points into a higher-dimensional space (e.g., $(x, F(x))$ ) such that diagram cells correspond to intersections of half-spaces/polytopes (0709.2196, Edwards et al., 17 Aug 2024).
Explicit versus Implicit Construction: Full Voronoi diagrams have prohibitive complexity in high dimensions ( $\Theta(n^{\lceil d/2 \rceil})$ for $n$ sites, $d$ dimensions), but algorithms employing local computations (e.g., determining the minimal intersection point between queries and bisector hyperplanes) avoid explicit construction, as used in cellular regression (Sastry, 4 Oct 2025).
Candidate Generation for Optimization: In Bayesian optimization, Voronoi-type loss functions guide the sampling of candidate points on Voronoi cell boundaries (equidistant to two or more design points), using bisection search along directions to efficiently explore high-uncertainty regions, avoiding costly continuous acquisition optimization (Wycoff et al., 7 Feb 2024).
Power, Order-k, and Bag Diagrams: Extensions include weighted cells, order-k diagrams (regions associated with subsets of $k$ sites), and bag diagrams (combining multiple generating functions). After additional lifting, these can retain linearity and permit use of combinatorial geometry techniques for complexity analysis and algorithmic computation (0709.2196).

3. Connections to Statistical Decision Theory and Machine Learning

Voronoi-type loss functions have significant implications in statistical learning:

Properness and Calibration: Loss functions derived from convex sets (via superprediction sets) are automatically proper, meaning they are calibrated for probability estimation. The duality between losses and (anti)-norms ensures that the geometry of the loss reflects the risk landscape (Williamson et al., 2022).
Design and Tuning of Losses: The calculus of losses enables smooth interpolation and mixture (M-sum operations) of loss functions, allowing application-specific tailoring while preserving calibration and mixability. Polar loss constructions provide universal substitution functions for aggregation algorithms, crucial for online learning scenarios (Williamson et al., 2022).
f-Divergence Generated Losses: Novel convex loss functions can be systematically constructed using f-divergences through regularized Fenchel-Young loss frameworks, yielding softargmax operators and potentially inducing Voronoi partitioning in probability space. These frameworks generalize logistic/cross-entropy loss, can sparsify predictions, and empirically achieve competitive accuracy in classification and language modeling (notably, $\alpha$ -divergence with $\alpha=1.5$ ) (Roulet et al., 30 Jan 2025).

4. Applications in Geometry, Hashing, Regression, and Clustering

Voronoi-type loss functions appear in a diverse set of algorithmic applications:

Mesh Generation and Sampling: 2-site diagrams utilizing perimeter or inscribed radius functions provide quality measures for mesh elements. Observations that active pairs in certain diagrams coincide with Delaunay edges enable efficient mesh refinement and sampling (Barequet et al., 2011).
Binary Hashing: The Voronoi diagram encoded hashing (VDeH) paradigm partitions the data space using Voronoi cells defined by randomly sampled centers, constructing hash functions that maximize coverage, entropy, and bit independence. Encoded hashing provides mutually independent bits and achieves superior efficiency and retrieval accuracy compared to traditional learning-to-hash approaches. Properties of Voronoi tessellations naturally enforce regularization effects analogous to loss function constraints for hashing systems (Xu et al., 4 Aug 2025).
Regression over Scattered Data: Cellular learning utilizes seed vertices to infer Voronoi cells and composes local linear models according to cell-specific weights computed via distance to cell boundaries. Avoiding explicit high-dimensional Voronoi diagrams circumvents the curse of dimensionality, allowing competitive accuracy (e.g., $98.2\%$ on MNIST with $>700$ k degrees of freedom), with blending parameters controlling local influence (Sastry, 4 Oct 2025).
Clustering and Quantization: Centroidal Bregman Voronoi diagrams, computed via extensions of Lloyd’s algorithm, yield quantization and clustering algorithms where the Bregman centroid equals the mass centroid, rendering it divergence-independent. VC-dimension analysis (exactly $d+1$ for Bregman balls) provides statistical guarantees for k-NN and classification using divergence-induced regions (0709.2196).

5. Regularization, Optimality, and Error Bounds

Voronoi-type loss functions contribute to regularization and control of estimator properties in learning and inference:

Minimax Estimation and Total Variation: The Voronoigram estimator uses Voronoi cell-based partitions to regularize by discrete total variation, equivalent (under restrictions) to continuum TV for piecewise constant functions. This approach provides minimax rate-optimality for bounded variation function estimation, with theoretical guarantees derived from Poincaré-type inequalities and risk controls based on L1 and L2 deviations bounded by constants times total variation (Hu et al., 2022).
Finite Set Losses and Non-spurious Minimizers: Loss functions constructed for finite sets (sum-of-squares type) achieve zero loss exclusively on the set itself. For vertices of a simplex, such losses have no spurious local minima. Generalizations via linear transformations and polynomial lifts yield proper loss functions without spurious minimizers even for arbitrary finite sets, and robust numerical techniques enable recovery from noisy data via quadratic optimization (Nie et al., 2021).

6. Theoretical Unification and Further Directions

Unified frameworks, notably Lie sphere geometry, encompass the breadth of Voronoi-type constructs—a single mathematical language encoding minimization diagrams for points, spheres, hyperplanes, and their combinations. Such unification enables loss functions that mix different geometric criteria and allows computational geometry techniques (convex hulls, linear systems in higher-dimensional space) to be repurposed for statistical and learning tasks (Edwards et al., 17 Aug 2024).

Extensions include Voronoi-type summation formulas in number theory (connecting divisor sums and K-Bessel functions), potential for more arithmetic-structured loss functions in optimization, and algorithmic generalization to batch, multi-objective, or mixed-integer settings in Bayesian optimization (Banerjee et al., 2023, Wycoff et al., 7 Feb 2024).

7. Table: Principal Variants of Voronoi-type Loss Functions

Variant	Key Definition / Formula	Primary Application/Property
Bregman Voronoi Diagram	$D_F(x\|\|p),\,(x,F(x))$ lifting	Affine diagrams; clustering, VC-dimension (0709.2196)
Lie Sphere Geometry Diagram	$f_i(x) = a_i \\|x-q_i\\|^2 + b_i\cdot x + c_i$	Unified polytope computation (Edwards et al., 17 Aug 2024)
2-site Geometric Loss	Circumradius, perimeter, view angle	Mesh quality; high complexity (Barequet et al., 2011)
f-divergence Fenchel-Young Loss	$\ell_f(\theta, y; q) = \max_p[\langle\theta, p\rangle - D_f(p,q)] + D_f(y,q) - \langle\theta, y\rangle$	Sparsity; multiclass, language modeling (Roulet et al., 30 Jan 2025)
Voronoi-cell Hash Function	$h_i(x) = 1$ if $i$ is nearest; encoded bits	Full coverage; efficiency; independence (Xu et al., 4 Aug 2025)
Cellular Regression Weight	$\omega^{rel}_i(p) = 1 - \frac{\|\|p-q\|\|}{\alpha_i\|\|c_i - q\|\|}$	Piecewise-smooth regression (Sastry, 4 Oct 2025)

This table summarizes the principal formulas and application domains. Further technical details, complexity results, and precise geometric representations are elaborated in the cited sources and their supplementary sections.

Voronoi-type loss functions thus provide a powerful framework for geometric partitioning, statistical regularization, decision-making, and efficient algorithmic computation. Their versatility and mathematical depth enable meaningful applications across computational geometry, machine learning, statistical inference, and beyond.