Technical overview

How a semantic map is made

Every concept pair contributes to the geometry. The map is a two-dimensional, best-fit representation of those distances—not a claim that a high-dimensional semantic space can be reproduced perfectly on a flat screen.

The pipeline

From words to coordinates

One deterministic geometry is calculated for the complete set.

01

Context

Disambiguate each label within its set.
02

Embedding

Encode every described concept as a vector.
03

Set-centering

Expose differences internal to the set.
04

All pairs

Build the full angular-distance matrix.
05

Classical MDS

Project the matrix onto two dimensions.

The central principle

Every pair matters

For n concepts, the system calculates all n(n−1)/2 unique pairwise distances. No nearest-neighbor sampling or sparse edge list is substituted for the full matrix used by the layout.

Five concepts therefore produce ten constraints; twenty concepts produce 190. Classical MDS considers them together when deriving one shared coordinate system.

The lines illustrate the complete distance matrix. The published map does not draw all of them as visible edges.

Steps 1–4

From meaning to distance

The labels alone can be ambiguous. A short, private helper description selects the intended sense in the context of the whole map before an embedding is generated.

Normalize and center the set

Embeddings are L2-normalized. For four or more concepts, their centroid is subtracted and each residual is normalized again. This removes the broad meaning shared by the set and emphasizes its internal contrasts. Sets with two or three concepts use the normalized embeddings directly.

v̄ = (1/n) Σ v_i

r_i = (v_i − v̄) / ‖v_i − v̄‖₂

Convert similarity to a metric distance

Cosine similarity is calculated between every pair of relative vectors. The angular transform maps identical directions to 0, orthogonal directions to 0.5, and opposite directions to 1.

s_ij = r_i^Tr_j

d_ij = acos(s_ij) / π

Step 5

Classical MDS in two dimensions

Classical multidimensional scaling turns the distance matrix into an inner-product matrix, then retains its two strongest positive directions.

J = I − 11^T/n

B = −½ JD^∘2J

X₂ = Q₂Λ₂^½

If the distance geometry is exactly representable in two dimensions, these coordinates preserve it exactly up to rotation, reflection, and scale. Usually it is not. In that case, the two leading positive eigenvalue axes give the classical MDS rank-2 representation of the centered geometry.

Orientation is made deterministic and the coordinates are uniformly scaled for display. Left/right and up/down have no independent semantic meaning; relative positions and distances do.

How fidelity is measured

The map reports projection stress. It compares every target distance d_ij with its two-dimensional Euclidean counterpart δ_ij, after fitting one global scale factor.

stress = √[Σ(d_ij − aδ_ij)² / Σd_ij²]

Lower is better. A value of zero means all pairwise distances agree up to a common scale.

The honest reading

The placement uses every distance, but it cannot guarantee that every displayed distance equals its high-dimensional target. Improving one relationship in 2D can worsen another; the projection is a global compromise.

A second diagnostic compares each concept’s up to three closest semantic neighbors with its closest neighbors in the projection. Together, stress and neighbor preservation describe global and local fidelity.

Geometry and presentation

What changes the map—and what does not

Visible edges are filtered

Lines use raw cosine similarity, a threshold, and a per-node top-k limit so the graph remains readable. They are not the sparse input to MDS; the full distance matrix is.

Clusters reuse the full matrix

Average-linkage clustering evaluates candidate partitions with a silhouette score on the same angular distances. Weak structure is left as one cluster.

An anchor is only a lens

An optional anchor ranks and colors nodes by similarity to a reference concept. It does not enter the distance matrix and never changes node coordinates.

Labels may move; nodes do not

The collision-aware label layer can displace text or add a leader line. The colored dot stays fixed at its calculated semantic coordinate.

Limits

How to interpret the result

It is model-dependent. Embedding models and contextual descriptions shape the source geometry.
It is set-relative. Centering emphasizes distinctions inside one map, so coordinates should not be compared as absolute locations across different maps.
Axes are not named dimensions. A horizontal or vertical direction has no intrinsic interpretation unless supported by the concepts around it.
The projection is descriptive. Proximity reflects modeled semantic association, not causality, truth, importance, or moral value.

Current implementation

Layout model: set_centered_angular_mds_v1
Layout distance: Angular distance · acos/π
Projection: Classical MDS · 2D
Clustering: Average linkage + silhouette