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Laguerre-Voronoi partition (weighted power diagram of a linear readout layer)

OBJECTgeometric-objectsubsetK:zeroimplicitpolyhedral complexintermediatehow it's classified →

The weighted Voronoi (Laguerre / power-diagram) partition of representation space induced by a linear readout layer's own weights and biases: each output unit $j$ becomes a center $c_j$ with a weight $\nu_j$ derived from its bias, and the cell $\mathcal{C}_j=\{z: \|z-c_j\|^2-\nu_j \le \|z-c_i\|^2-\nu_i\ \forall i\}$ is a convex polyhedron algebraically identical to the layer's own argmax decision region. A 'concept' is redefined as an entire cell (a region), not a point or a direction, and a 'category' as a union of cells.

Replicationcomputed from the corpus — never hand-assigned
1 paper1 architecture class1 domain4 model families · one study
Filled = two or more values reported by papers that share no author — replication. Outlined = two or more values, but all from a single study — breadth, not replication. Grey = a single value. Derived from paper authorship and each model's architecture class, domain and family; it updates itself when a paper is added.

Definition

For a linear readout layer with weight matrix WW and bias bb (e.g. an LLM’s unembedding layer, or any linear classifier head), define per-unit centers and weights

cj=Wj2,νj=bj+∥Wj∥24c_j = \frac{W_j}{2}, \qquad \nu_j = b_j + \frac{\|W_j\|^2}{4}

The Laguerre-Voronoi diagram (equivalently, a power diagram) is the partition of representation space into cells

Cj={z∈Rd:∥z−cj∥2−νj≤∥z−ci∥2−νi  ∀i}\mathcal{C}_j = \{z \in \mathbb{R}^d : \|z-c_j\|^2-\nu_j \le \|z-c_i\|^2-\nu_i\ \ \forall i\}

Expanding the squared-distance terms shows this is algebraically identical to the layer’s own linear argmax rule, arg⁡max⁡j(z⋅Wj+bj)\arg\max_j(z\cdot W_j+b_j) — the diagram is not a new computation, but a re-description of an ordinary linear/softmax readout as a tessellation of convex polyhedral regions, each with its own center and weight, rather than as KK separate hyperplane-vs-hyperplane comparisons.

Relative to the decision boundary and the linear region arrangement

Decision boundary (as a codimension-1 hypersurface) catalogs a single codimension-1 level set — one binary (or one-vs-rest) hypersurface. Linear region arrangement (polyhedral tessellation of input space) catalogs the full combinatorial tessellation induced by every piecewise-linear unit at every layer of a network, not just its final readout. This node sits between them: it is the entire KK-way tessellation (like the region arrangement), but induced by a single linear+bias layer’s centers and weights specifically (like the decision boundary is tied to one layer’s output), rather than by breakpoints accumulated across depth. Each cell Cj\mathcal{C}_j is a convex polyhedron, exactly analogous to a linear region, but the partition’s generating data (weighted centers cj,νjc_j,\nu_j) is directly interpretable per-vocabulary-item/per-class, rather than combinatorial per-unit activation patterns.

Key evidence

Ma & Wolfinger (2026) prove (their Theorem 4.1) that any LLM’s linear unembedding layer analytically induces exactly this partition over representation space, one cell per vocabulary token, and use it to redefine a “concept” as an entire Laguerre-Voronoi cell (rather than a point, direction, or Gaussian cluster) and a “category” as a union of cells — explicitly generalizing beyond the point/direction-based definitions this map’s linear-direction and linear-subspace nodes otherwise use. On six real pretrained LLMs (Phi-2, Gemma-2-2B/9B, Gemma-3-270M, Pythia-70M, Llama-3.1-8B), a linear probe over this cell-based definition of a concept achieves lower error than a point/cluster-based “Categorical Geometry” baseline and a “Lattice Geometry” baseline on Phi-2 (e.g. Animal vs. Plant: 0.111 vs. 0.121 vs. 0.145; Event vs. Cognition: 0.180 vs. 0.213 vs. 0.192), and an unsupervised hierarchy/domination-score built from cell weights reaches 58.7-88.4% accuracy on Phi-2 versus a near-random 42-56% for the Lattice baseline. A diagnostic “Geometric Lens” readout of which cell an intermediate-layer hidden vector occupies recovers the factually correct answer token under adversarial in-context interference more often than Logit Lens or Patchscopes across Phi-2, Gemma-2-2B/9B and Llama-3.1-8B (e.g. Phi-2: 0.56/0.31/0.49 across three city-swap datasets, versus near-zero for greedy top-1 decoding). No causal intervention (patching, ablation, steering) is performed — validation is via diagnostic readout accuracy, not an edit to the network’s computation. See ma-wolfinger-2026-a-linear-unembedding-layers-weights-analytically-induce-a-laguerre-voronoi-cell-partition-of-representation-space-and-cell-based-concept-boundaries-outperform-point-based-baselines.

Key papers

  • Ma, C. & Wolfinger, R. (2026). Laguerre Geometry for Interpreting Large Language Models. arXiv:2607.10578 — origin of the Laguerre-Voronoi / power-diagram redefinition of “concept” as a cell and “category” as a union of cells, and the Geometric Lens diagnostic.

Found in (1 observation · 4 families)

Phi

Laguerre Geometry for Interpreting Large Language Models (2026)measured

A linear unembedding layer's weights analytically induce a Laguerre-Voronoi cell partition of representation space, and cell-based concept boundaries outperform point-based baselines

Details

Ma & Wolfinger (2026) prove that any real LLM's linear unembedding layer (weight W, bias b) analytically induces a Laguerre-Voronoi diagram (weighted power diagram) over representation space -- each vocabulary token j becomes a center c_j=W_j/2 with weight nu_j=b_j+||W_j||^2/4, and the space partitions into convex cells algebraically identical to the layer's own argmax rule -- redefining a "concept" as an entire cell (not a point/direction) and a "category" as a union of cells [ma-wolfinger-2026-laguerre-geometry-for-interpreting-llms] On six real pretrained LLMs (Phi-2, Gemma-2-2B/9B, Gemma-3-270M, Pythia-70M, Llama-3.1-8B), a linear probe over this cell-based concept definition reaches lower error than point/cluster-based "Categorical Geometry" and "Lattice Geometry" baselines on Phi-2 (e.g. Animal vs. Plant: 0.111 vs. 0.121 vs. 0.145), and an unsupervised hierarchy/domination-score built from cell weights reaches 58.7-88.4% accuracy on Phi-2 versus a near-random 42-56% for the Lattice baseline [ma-wolfinger-2026-laguerre-geometry-for-interpreting-llms] A diagnostic "Geometric Lens" readout of which cell an intermediate-layer hidden vector occupies recovers the factually correct answer token under adversarial in-context interference more often than Logit Lens or Patchscopes across Phi-2, Gemma-2-2B/9B and Llama-3.1-8B (e.g. Phi-2: 0.56/0.31/0.49 across three city-swap datasets, versus near-zero for greedy top-1 decoding); no causal intervention (patching/ablation/steering) is performed [ma-wolfinger-2026-laguerre-geometry-for-interpreting-llms]

models: Phi-2 · method: Linear probing

Gemma

Laguerre Geometry for Interpreting Large Language Models (2026)measured

A linear unembedding layer's weights analytically induce a Laguerre-Voronoi cell partition of representation space, and cell-based concept boundaries outperform point-based baselines

Details

Ma & Wolfinger (2026) prove that any real LLM's linear unembedding layer (weight W, bias b) analytically induces a Laguerre-Voronoi diagram (weighted power diagram) over representation space -- each vocabulary token j becomes a center c_j=W_j/2 with weight nu_j=b_j+||W_j||^2/4, and the space partitions into convex cells algebraically identical to the layer's own argmax rule -- redefining a "concept" as an entire cell (not a point/direction) and a "category" as a union of cells [ma-wolfinger-2026-laguerre-geometry-for-interpreting-llms] On six real pretrained LLMs (Phi-2, Gemma-2-2B/9B, Gemma-3-270M, Pythia-70M, Llama-3.1-8B), a linear probe over this cell-based concept definition reaches lower error than point/cluster-based "Categorical Geometry" and "Lattice Geometry" baselines on Phi-2 (e.g. Animal vs. Plant: 0.111 vs. 0.121 vs. 0.145), and an unsupervised hierarchy/domination-score built from cell weights reaches 58.7-88.4% accuracy on Phi-2 versus a near-random 42-56% for the Lattice baseline [ma-wolfinger-2026-laguerre-geometry-for-interpreting-llms] A diagnostic "Geometric Lens" readout of which cell an intermediate-layer hidden vector occupies recovers the factually correct answer token under adversarial in-context interference more often than Logit Lens or Patchscopes across Phi-2, Gemma-2-2B/9B and Llama-3.1-8B (e.g. Phi-2: 0.56/0.31/0.49 across three city-swap datasets, versus near-zero for greedy top-1 decoding); no causal intervention (patching/ablation/steering) is performed [ma-wolfinger-2026-laguerre-geometry-for-interpreting-llms]

models: Gemma-2-2B, Gemma-2-9B, Gemma-3-270M-it · method: Linear probing

Pythia

Laguerre Geometry for Interpreting Large Language Models (2026)measured

A linear unembedding layer's weights analytically induce a Laguerre-Voronoi cell partition of representation space, and cell-based concept boundaries outperform point-based baselines

Details

Ma & Wolfinger (2026) prove that any real LLM's linear unembedding layer (weight W, bias b) analytically induces a Laguerre-Voronoi diagram (weighted power diagram) over representation space -- each vocabulary token j becomes a center c_j=W_j/2 with weight nu_j=b_j+||W_j||^2/4, and the space partitions into convex cells algebraically identical to the layer's own argmax rule -- redefining a "concept" as an entire cell (not a point/direction) and a "category" as a union of cells [ma-wolfinger-2026-laguerre-geometry-for-interpreting-llms] On six real pretrained LLMs (Phi-2, Gemma-2-2B/9B, Gemma-3-270M, Pythia-70M, Llama-3.1-8B), a linear probe over this cell-based concept definition reaches lower error than point/cluster-based "Categorical Geometry" and "Lattice Geometry" baselines on Phi-2 (e.g. Animal vs. Plant: 0.111 vs. 0.121 vs. 0.145), and an unsupervised hierarchy/domination-score built from cell weights reaches 58.7-88.4% accuracy on Phi-2 versus a near-random 42-56% for the Lattice baseline [ma-wolfinger-2026-laguerre-geometry-for-interpreting-llms] A diagnostic "Geometric Lens" readout of which cell an intermediate-layer hidden vector occupies recovers the factually correct answer token under adversarial in-context interference more often than Logit Lens or Patchscopes across Phi-2, Gemma-2-2B/9B and Llama-3.1-8B (e.g. Phi-2: 0.56/0.31/0.49 across three city-swap datasets, versus near-zero for greedy top-1 decoding); no causal intervention (patching/ablation/steering) is performed [ma-wolfinger-2026-laguerre-geometry-for-interpreting-llms]

models: Pythia-70M · method: Linear probing

Llama

Laguerre Geometry for Interpreting Large Language Models (2026)measured

A linear unembedding layer's weights analytically induce a Laguerre-Voronoi cell partition of representation space, and cell-based concept boundaries outperform point-based baselines

Details

Ma & Wolfinger (2026) prove that any real LLM's linear unembedding layer (weight W, bias b) analytically induces a Laguerre-Voronoi diagram (weighted power diagram) over representation space -- each vocabulary token j becomes a center c_j=W_j/2 with weight nu_j=b_j+||W_j||^2/4, and the space partitions into convex cells algebraically identical to the layer's own argmax rule -- redefining a "concept" as an entire cell (not a point/direction) and a "category" as a union of cells [ma-wolfinger-2026-laguerre-geometry-for-interpreting-llms] On six real pretrained LLMs (Phi-2, Gemma-2-2B/9B, Gemma-3-270M, Pythia-70M, Llama-3.1-8B), a linear probe over this cell-based concept definition reaches lower error than point/cluster-based "Categorical Geometry" and "Lattice Geometry" baselines on Phi-2 (e.g. Animal vs. Plant: 0.111 vs. 0.121 vs. 0.145), and an unsupervised hierarchy/domination-score built from cell weights reaches 58.7-88.4% accuracy on Phi-2 versus a near-random 42-56% for the Lattice baseline [ma-wolfinger-2026-laguerre-geometry-for-interpreting-llms] A diagnostic "Geometric Lens" readout of which cell an intermediate-layer hidden vector occupies recovers the factually correct answer token under adversarial in-context interference more often than Logit Lens or Patchscopes across Phi-2, Gemma-2-2B/9B and Llama-3.1-8B (e.g. Phi-2: 0.56/0.31/0.49 across three city-swap datasets, versus near-zero for greedy top-1 decoding); no causal intervention (patching/ablation/steering) is performed [ma-wolfinger-2026-laguerre-geometry-for-interpreting-llms]

models: Llama-3.1-8B · method: Linear probing