Contribution
Supplies the compositional layer the Linear Representation Hypothesis lacks. That hypothesis says a concept is a direction, which accounts for binary features and for steering, and says nothing about how concepts relate to each other — inclusion, intersection, union, the structure that makes conceptual knowledge a hierarchy. This paper adds a threshold to each direction, which turns a concept into a half-space, and observes that the intersection lattice of half-spaces is exactly the structure Formal Concept Analysis studies. Meet and join then become operations one can perform on embeddings.
Key insight
A threshold is the whole difference between a feature and a concept. A direction alone gives you a scalar per datapoint — more or less of some attribute. Add a separating threshold and you have a region, and regions compose: intersect two and you get the concept satisfying both attributes; take the smallest region containing two and you get their generalisation. The algebra of concepts that symbolic AI built explicitly turns out to be available for free in a continuous embedding space, provided you are willing to say where each attribute stops.
Assumptions
- The Linear Representation Hypothesis holds — attributes are linear directions in embedding space. Everything is downstream of that and none of it is re-established here.
- Each attribute admits a separating threshold, which is what converts a direction into a half-space. Real embeddings do not separate cleanly, which is why soft versions are defined.
- The canonical form requires linearly independent attribute directions — a condition on the set of concepts chosen, not a property of the space.
- Join is approximated by the conic hull of the two concepts’ attribute directions rather than computed exactly; it is the least region subsuming both only under that approximation.
- WordNet sub-hierarchies supply both the concepts and the ground-truth structure.
Key results
- Half-space construction: a linear attribute direction with a separating threshold defines a concept region; a concept over several attributes is the intersection of their half-spaces.
- The regions form a complete lattice under inclusion — every pair has a greatest lower bound (meet) and a least upper bound (join).
- Meet is intersection of half-spaces; join is the conic hull spanned by the union of the two concepts’ attribute directions.
- A canonical form exists when attribute directions are linearly independent.
- Soft meet and join, via fuzzy projection profiles, giving a degree to which a point is subsumed rather than a boolean.
- WordNet evidence: LLM embeddings are reported to encode the concept lattices and their logical structure, supporting coherent generalisation (join) and refinement (meet).
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Linear directions with thresholds induce a concept lattice via half-space intersection | strong | a construction; it follows once the premises are granted |
| C2 | LLM embeddings actually encode such lattices | moderate | WordNet sub-hierarchies, which are hand-built to have this structure |
| C3 | Meet and join on embeddings correspond to conceptual refinement and generalisation | moderate | demonstrated on the same curated hierarchies |
| C4 | A canonical form exists under linear independence | strong | proved, under a stated condition |
Method
Take the Linear Representation Hypothesis as given. Attach a separating threshold to each attribute direction, making each attribute a half-space, and define a concept as the intersection of the half-spaces of its attributes.
Import Formal Concept Analysis’s apparatus: objects, attributes, the extent/intent duality, and the partial order by inclusion that makes the set of formal concepts a complete lattice. Show the geometric construction realises it.
Define meet as region intersection and join as the conic hull of the combined attribute directions, then soften both with fuzzy projection profiles so they apply to embeddings that do not sit cleanly on one side of a threshold.
Evaluate on WordNet sub-hierarchies: check whether LLM embeddings place concepts so that the geometric meet and join recover the ontology’s refinement and generalisation relations.
Concepts
- Linear Representation Hypothesis — semantic features and concepts are encoded as linear directions or subspaces. The premise. It was not in the record when this note was written, and is now LIT-606.
- Formal Concept Analysis — the lattice theory of objects and attributes, with extent (objects having the attributes) and intent (attributes shared by the objects) as duals.
- Half-space model — a concept as the intersection of the half-spaces defined by its attribute directions and thresholds.
- Meet / join — greatest lower bound and least upper bound in the lattice; here, intersection and conic hull.
Connections
Builds on the Linear Representation Hypothesis line, including the causal inner product formulation under which causally separable concepts are orthogonal — which is what makes the independence condition for the canonical form more than a convenience.
Imports Formal Concept Analysis wholesale, which is an old and well-developed symbolic framework, and the contribution is the bridge rather than either side.
Sits adjacent to the steering literature: if concepts are regions and regions compose, then a steering vector for a compound concept should be constructible rather than found. The paper does not take that step.
Recommendations
- R1 — When reasoning about compound concepts in embedding space, treat them as intersections of half-spaces rather than as sums of directions. Topic: interpretability. Status: experimental. Strength: moderate. Applies when: the attributes have thresholds that separate.
- R2 — Expect generalisation over concepts to need a conic hull rather than an average of directions. Topic: interpretability. Status: experimental. Strength: weak. Applies when: as above; the hull is itself an approximation here.
Bearing on the record
- Should produce a theory and no practice. The paper establishes a structure and demonstrates it on a curated ontology; nothing here says what to do differently when training or serving. The obvious candidate — build steering vectors by meet and join — is not evaluated as an intervention.
- Names the same absent trunk as LIT-458. The Linear Representation Hypothesis is this paper’s premise and the record holds no document for it, nor for steering vectors, nor for concept geometry. Filing the elaboration before the thing elaborated is backwards and should be said so in writing.
- First document in the record on concept geometry, in a topic whose fourteen existing notes are all tokenisation and positional encoding.
Limitations
- WordNet is the friendliest possible test. It is a hand-built ontology with exactly the clean hierarchical structure the hypothesis predicts. Whether the lattice holds for concepts nobody curated is the question and is not asked.
- Everything rests on the Linear Representation Hypothesis, which is assumed rather than tested, and which has its own live disputes.
- Thresholds are assumed to exist and separate; the soft version acknowledges they do not, and the softening is a definitional choice rather than a derived one.
- Join is an approximation. The conic hull is the least region under a construction, not the least region.
- No downstream task. The structure is demonstrated, not used.
Open questions
- Does the lattice appear for concepts a model learned and no ontology names? That is the severe test and it is available — cluster embeddings, look for lattice structure, check nothing curated it.
- Can meets and joins be used to steer? The construction is there and the intervention is not run.
- How do the thresholds behave across models? If they are model-specific the geometry is a per-model fact; if they transfer it is a stronger claim than the paper makes.