Incrementally Constructed Power Diagram

An incrementally constructed power diagram is a spatial partitioning structure where the set of sites and their associated weights are expanded or refined over time, with each incremental update requiring local or global reconfiguration of the power diagram. Power diagrams generalize Voronoi diagrams by associating each site with a weight, yielding partitions that are critical in diverse fields such as computational geometry, information elicitation, neural network analysis, materials science, and geometric modeling.

1. Mathematical Foundations and Representations

A power diagram subdivides a domain $\mathbb{R}^d$ (or a restricted convex subset $D$) into cells determined by a collection of sites $\{s_i\}$ and real weights $\{w_i\}$. The canonical definition assigns each point $x \in \mathbb{R}^d$ to the cell of site $i$ that minimizes the power distance: $$cell(s_i) = \left\{ x : i = \arg\min_j \left( \|x - s_j\|^2 - w_j^2 \right) \right\}$$ Equivalently, using affine (linear) cell boundaries, each power cell can be defined as: $$cell(s_i) = \left\{ x : (s_j - s_i)^T x \leq \gamma_j - \gamma_i, \forall j \neq i \right\}$$ with $\gamma_i = \frac{1}{2} (s_i^T s_i - w_i)$.

Power diagrams are invariant under uniform scaling of the sites and offsets, which enables per-hyperplane positive rescaling, aiding in computational detection and incremental update strategies.

Incremental construction refers to the process of adding (or occasionally removing) sites or generators, thereby refactoring the partition as required by new geometric or semantic constraints.

2. Methodologies for Incremental Construction

2.1 Polyhedral Partition Detection and Adaptation

A formally addressed problem is recognizing whether a given polyhedral partition is, or remains after update, a power diagram (Borgwardt et al., 2017 ). The detection is cast as a linear program (LP), testing for $s_i$, $\gamma_i$, and scaling factors $\lambda_{ij}$ that satisfy

$$\begin{array}{lrclcl} \lambda_{ij} a_{ij} & = & s_i-s_j & & \forall i \leq k, \forall j \in J_i \ \lambda_{ij} \gamma_{ij} & = & \gamma_j - \gamma_i & & \forall i \leq k, \forall j \in J_i \ \lambda_{ij} & \geq & 1 & & \forall i \leq k, \forall j \in J_i \end{array}$$

Given a polyhedral cell complex, after any modification (addition of sites/cells), this LP can be solved, or updated, to verify whether a valid power diagram representation can be recovered.

This framework does not directly provide an incremental update algorithm, but its affine representation facilitates local recomputation: adding a new site involves updating cell adjacency and recomputing, which may require resolving all or part of the LP.

2.2 Subdivision via Operator Composition in Deep Networks

In the context of Max-Affine Spline Operator (MASO) neural networks, each MASO layer partitions input space by a power diagram (Balestriero et al., 2019 ). Layer-wise composition—adding new layers—can be interpreted as recursively subdividing the current power diagram: each MASO applies to each existing cell, inducing a product structure, and thus an exponentially refined, hierarchically nested sequence of power diagrams.

The process is constructive: each additional MASO layer (increment) subdivides regions induced by previous layers, with centroids and radii (or affine coefficients) determined recursively from previous parameters.

2.3 Analytical and Algorithmic Adaptation in Curved and Anisotropic Settings

For generalized power diagrams (e.g., the Generalized Balanced Power Diagram, GBPD), incremental construction is made possible by the explicit analytic representation of edges as conics and vertices as their intersections (Jung et al., 2023 ). Each new generator (defined by position, matrix, and weight) requires recomputing only local bisectors and updating adjacent cell boundaries; parametric edge representations streamline this update.

Though a full online incremental update algorithm is not detailed, the explicit forms and local geometric computation enable efficient adaptation and motivate further incremental algorithm development.

2.4 Adaptive Refinement via Isosurface Extraction

Recent work in isosurface extraction from signed distance fields presents a fully incremental power diagram construction (Wang et al., 11 Jun 2025 ). Here, sites (SDF samples and projections) are iteratively added in regions where mesh-surface discrepancy is highest, as measured by a gradient-based misalignment metric: $$\delta(\triangle) = \int_{\triangle} \left| \nabla \phi(\mathbf{x}) - \left( \nabla \phi(\mathbf{x}) \cdot \hat{\mathbf{n}} \right) \hat{\mathbf{n}} \right| dA$$ Each insertion leads only to localized updates of the regular (power) Delaunay tetrahedralization and the dual power diagram, directly updating the relevant mesh patch. This local update property is central for scalability, enabling repeated, rapid refinement of the surface extraction.

2.5 Restricted (Domain-Clipped) Diagrams and Topology Preservation

In 3D medial axis computation, the domain is explicitly restricted to a shape $S$ and cells are clipped accordingly (Wang et al., 27 Mar 2024 ). Medial spheres (sites) are incrementally inserted as needed to resolve local contractibility violations (failure of the nerve theorem's conditions for homotopy equivalence). Each insertion triggers only local recomputation within the RPD, ensuring the medial mesh adapts to preserve both geometric fidelity and topology.

The process relies on efficient, localized computation of the Euler characteristic (using a fractional method) of each restricted element, which allows for parallel amendment without global recomputation.

3. Computational and Structural Challenges

Incremental construction of power diagrams is fundamentally challenged by:

• Combinatorial Restructuring: Inserting a generator may add or remove neighbors, thus changing cell adjacency and requiring recomputation of affected boundaries and intersection points.
• Global vs. Local Impact: While some applications (deep networks, isosurface extraction, medial axis recovery) admit largely local updates, in polyhedral power diagram detection or complex topological domains, incremental addition may induce widespread reconfiguration.
• Numerical Stability and Scalability: As the number of sites grows, care must be taken to address degeneracies, maintain stability, and efficiently update underlying data structures.
• Handling of Non-Polyhedral Cells: Analytical or LP-based frameworks relying on halfspace or polyhedral descriptions cannot handle curved, non-convex, or disconnected cells, limiting their applicability in generalized or anisotropic settings.

4. Applications Across Domains

4.1 Information and Mechanism Design

Power diagram representations are essential in property elicitation, peer prediction, and mechanism design (Borgwardt et al., 2017 ). Here, incrementally constructed diagrams allow for stepwise refinement as new types, reports, or constraints arise, with LP detection ensuring truthful mechanisms remain possible.

4.2 Neural Network Expressivity

Incrementally constructed power diagrams, via MASO layers, underlie the expressive capacity of deep networks with piecewise-affine nonlinearity (Balestriero et al., 2019 ). The progressive subdivision clarifies the geometric role of depth and layer structure, enabling analysis of decision boundaries and their curvature.

4.3 Tessellation in Materials Science

Incremental and analytic construction of GBPDs is central for modeling polycrystalline grains and cellular structures with anisotropy and curved boundaries (Jung et al., 2023 ). It supports adaptation to real materials data as more generators or measurements are incorporated.

4.4 Adaptive Geometry Processing

Adaptive isosurface extraction from SDFs leverages incrementally constructed power diagrams to locally refine surface meshes, achieving high-fidelity approximation with minimal computation (Wang et al., 11 Jun 2025 ).

4.5 Medial Axis and Shape Representation

In medial axis extraction, incrementally constructed restricted power diagrams enable adaptive and topology-preserving meshing, critically supporting applications in CAD, organic modeling, and geometry analysis (Wang et al., 27 Mar 2024 ).

5. Comparative Analysis and Algorithmic Features

Incremental construction is facilitated in some contexts by underlying dual structures (e.g., regular Delaunay for power diagrams), explicit analytic formulas (enabling efficient recomputation), or by localized topological checks (enabling adaptive site insertion for topology correction).

6. Open Challenges and Future Directions

Several open problems and practical challenges remain central to incrementally constructed power diagrams:

• Generalization Beyond Polyhedral Cases: Efficient, incremental detection and update algorithms for diagrams with non-convex, non-polyhedral, or curved boundaries.
• Transformation Between Representations: Converting between vertex-based and halfspace-based descriptions efficiently to facilitate incremental updates.
• Online and Large-Scale Scalability: Developing numerically stable, parallelizable frameworks for incremental construction in high-dimensional or data-intensive settings.
• Algorithmic Adaptation to Arbitrary Domains: Supporting non-convex, disconnected, or dynamically changing domains with homotopy and geometric guarantees.
• Robustness to Degeneracies: Ensuring incremental update algorithms are robust to geometric degeneracies and can handle topological singularities.

Technological advances in fast linear programming, local mesh data structures, and analytic geometric computation continue to expand the domain of tractable and robust incremental power diagram construction, with applications across economics, data science, deep learning, materials science, and geometric processing.