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QUESTION-025: Do correlated or hierarchical attributes in co-occurrence statistics still give linear attribute directions in an embedding, and with them a concept lattice that is not Boolean?

8 min read

Properties5
StatusOpen
Tagsrepresentation-learning, mathematics
Linepragmatic-transport
Refined byQUESTION-162 — In hyperbolic neural networks on tree-like data, does the advantage over Euclidean networks go with the Möbius bias translations and the distance-based output layer, the only operations through which curvature acts, or do networks without them keep it?, QUESTION-178 — Is word co-occurrence close to independent, multiplicative binary attributes in the analogy families where spectral embeddings complete parallelograms, and does adding an interaction between two attributes break their parallelogram?
BearingTHEORY-292 — In CNN image classifiers, concept directions fitted one concept at a time align for attributes that co-occur in the labels, yet a joint fit finds mutually orthogonal directions that still separate each concept with little loss of AUROC, so the alignment is a property of the separate fits and not forced by the representation, THEORY-196 — A finite tree embeds in the hyperbolic plane with worst-case distortion 1 + ε, but because a hyperbolic distance d takes about d bits to write, its coordinates need Θ((ℓ/ε) log deg_max) bits: logarithmic in the branching, linear in the longest path ℓ, so hyperbolic space is economical for shallow bushy hierarchies and not for deep ones, THEORY-201 — In next-token prediction on hierarchically generated sequences, the correlation between two tokens falls by about a factor m per level of their common ancestor, so a training set of P sequences resolves correlations only out to a distance t*(P) at which they meet sampling noise; learners come to represent the hidden symbols up to that depth, giving loss steps at about v m^(2ℓ−1) examples, and on character-level text the context at which the loss saturates grows as P^(1/(2β)) with β the measured correlation decay, THEORY-186 — A deep linear network trained from small weights learns the singular modes of its input–output correlations one at a time in order of strength, each in a sharp sigmoidal transition, while a shallow network learns them all together; the stages come from the product of layers and the data's spectrum, not from nonlinearity, THEORY-183 — When the co-occurrence of words on a latent continuum depends only on their separation, a spectral word embedding places them on the Fourier modes of that continuum, so the circles and rippled curves of months, years and places come from a symmetry of the corpus statistics, THEORY-698 — Thresholding linear attribute probes on a language model's embeddings yields a complete concept lattice for any incidence whatever, so the lattice is guaranteed by construction; what the Lattice Representation Hypothesis measured is per-attribute agreement with GPT-4o annotations at F1 70–83, and no lattice law, THEORY-478 — If the diffusion densities' Hellinger distances obey a Hölder bound with exponent θ in d_T / t^(1/β) and are bounded below once d_T exceeds a multiple of t^(1/β), then for 0 < α < θ/β summing scale-weighted hyperbolic distances between diffusion densities over dyadic times gives a distance two-sided equivalent, up to constants, to min{1, d_T^(αβ)} for pairs above the finest scale: it keeps the tree's neighbourhoods and not its global distances; whether a tree's heat kernel, or sampled data, meets the bounds, and with which θ, is not shown, THEORY-189 — Nested angular cones in the Poincaré ball, symmetric about the outward ray with an aperture depending only on the apex's radius, are transitive only if the aperture is at most π/2 and r sin ψ(r)/(1 − r²) does not increase outward; so no such cone can be defined near the origin, the widest admissible family is sin ψ = K(1 − r²)/r, and every point a cone contains lies farther from the origin than its apex, THEORY-117 — In Uselis, Dittadi and Oh's two-concept grids, ResNet-50s trained from scratch, with checkpoints chosen on the unseen pairs, reach unseen combinations as the share of combinations seen grows and not as up to four times more data of the same combinations is added; and when embeddings are exactly linearly factored, seen pairs that connect every value of both concepts, as two per value arranged in one cycle do, determine the embedding of every unseen pair, THEORY-185 — When each binary attribute of a word affects its co-occurrence independently and multiplicatively, the PMI is affine in the attributes with rank at most d + 1, so a spectral PMI embedding is a linear image of the attribute hypercube and parallelogram analogies hold exactly; over the full hypercube of words, each attribute's two values equally frequent, the raw co-occurrence ratio mixes in products of attributes, and keeps exact analogies only when the strengths are alike enough that every attribute mode outranks every product mode

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Why it is a question

Two accounts in the record meet at one premise and pull apart on the next step.

  • THEORY-185 (Korchinski, Karkada, Bahri & Wyart, LIT-863) derives the premise. If each binary attribute multiplies co-occurrence independently, the PMI is additive across attributes, and a spectral embedding is an affine image of each word’s ±1 attribute vector. Each attribute is then a linear direction, and thresholding it recovers the object–attribute incidence exactly.
  • The Lattice Representation Hypothesis (LIT-267, read in NOTE-240) assumes that premise, after Park et al., and reads the thresholded incidence as a formal context whose concept lattice the model is said to carry. CLAIM-119 holds that communicative categories form such a lattice: they share attributes and differ in a few constraints.

The derivation’s assumption is what the lattice needs to be false. With independent attributes every combination is possible, so the formal context is the full hypercube and its concept lattice is Boolean: no attribute implies another, and there is no hierarchy. Korchinski et al.’s relaxation to a random subset of words (their §8) adds no structured implications, and they leave hierarchical attributes as future work, naming the random hierarchy model of Cagnetta et al. Their sequel (LIT-860, THEORY-183) does the same for continuous attributes, and its Appendix D finds the binary and continuous subspaces orthogonal: separate, not nested.

So nothing the record holds says whether the linear directions survive the structure that makes a concept lattice worth having: attributes that imply each other (every poodle is a dog), exclude each other, or interact in co-occurrence rather than multiplying independently. If they do not, LIT-267’s first step fails exactly where its lattice becomes non-trivial.

What would count as an answer

  • A derivation. A generative model of co-occurrence with a stated implication structure among binary attributes (a tree, or a general formal context), with the spectrum of its PMI worked out: whether each attribute still has a direction, how far thresholding recovers the incidence, and whether the concept lattice of the recovered incidence equals the generating one.
  • A measurement. In real co-occurrence data, a test of whether the PMI is additive across the attributes of a known hierarchy (WordNet hypernymy, say), and of how a probe’s errors concentrate on attributes that imply one another. It would need attributes not annotated by another language model, which was NOTE-240’s first objection to LIT-267.

Either kind settles the question. A further demonstration that independent attributes give parallelograms does not.

What the record holds toward an answer

None of these answers it; together they narrow it.

  • A linear model with hierarchical features already gives tree-shaped directions. Saxe, McClelland & Ganguli 2019 (LIT-862, read in NOTE-666; THEORY-186) generate features by diffusion down a tree, which makes them imply one another. The item-similarity eigenvectors then respect the tree, as tree wavelets: one direction per branching, splitting one subtree from its sibling, with eigenvalues falling with depth. So in a linear network, implication among features yields linear directions, but they are directions for branch contrasts, not one per attribute. That is items against features in a supervised model, not word co-occurrence and PMI; whether the same holds for a co-occurrence embedding is still the question.
  • The hierarchy may live in distances rather than directions. Sala et al. (LIT-874, THEORY-196) embed trees in hyperbolic space so that every ancestor is nearer than any non-ancestor; Krioukov et al. (LIT-865, THEORY-188) put hierarchy in a radial coordinate. So a failure of linear directions would not show the hierarchy is absent from an embedding. But a space that can hold a hierarchy does not show that a trained one does: Yang et al. (LIT-871) find trained hyperbolic models order tree levels only 69–75% of the time.
  • A generative model with part–whole implication. Cagnetta et al.’s random hierarchy model (LIT-877, THEORY-195) has symbols that imply their parents, with the hierarchy visible in simple patch–class co-occurrence counts. It has no word attributes and no PMI, so a derivation built on it would first have to define attributes as ancestors.
  • Implication as containment. Ganea, Bécigneul & Hofmann’s hyperbolic entailment cones (LIT-866, THEORY-189) encode implication as nested regions, a third option beside half-spaces and distances. Cones can share descendants, so a partial order fits, but the overlap of two cones is not a cone: there are no geometric meets or joins, and so no lattice. Trained on the transitive reduction alone, the cones did not recover the closure.
  • Hyperbolic curvature is not by itself evidence of hierarchy. Bianconi & Rahmede (LIT-867, THEORY-193) concede the same grown complex fits a sphere with unequal link lengths. Zhou, Smith & Sharpee’s olfactory space (LIT-872, THEORY-191) and Zhang et al.’s hippocampus (LIT-876) reject only flat cubes, which leaves a sphere or an exponential spread of scales open, and neither recovers a hierarchy from the data. Zhou et al. do read linear axes for continuous attributes out of co-occurrence correlations in the fitted space.
  • What a coarse-graining keeps depends on what it must stay informative about. Koch-Janusz & Ringel (LIT-873, THEORY-194) and Larsson, Maity & Tsiotras (LIT-868) choose levels of a given hierarchy by mutual information; the second shows a level can carry nothing while the levels below it carry much, so a measurement should score every level.
  • In token co-occurrence, a hierarchy shows as partitions and fades with depth. Cagnetta & Wyart (LIT-883, THEORY-201) generate sequences from a random hierarchy and measure their token–token correlations, which fall by about a factor m per level of the common ancestor; so a finite corpus shows only the shallow levels. Tuples with one parent have identical co-occurrence rows, so the hierarchy shows as nested partitions rather than one direction per ancestor, and the paper notes that a parent is not a linear feature of its input tuple until after a nonlinear layer. That is the nearest the record comes to the derivation this question asks for, on part–whole rather than attribute hierarchies.
  • Hyperbolic half-spaces as a probe. Ganea, Bécigneul & Hofmann’s hyperbolic neural networks (LIT-879) separate WordNet subtrees with geodesic hyperplanes better than Euclidean ones do, in embeddings trained on hypernym edges. Probing a co-occurrence embedding with both kinds of half-space is one way to make the measurement asked for above.
  • A measuring tool. Lin et al.’s hyperbolic diffusion embedding (LIT-870) recovers a tree distance from multiscale densities, where a single Euclidean view on the same features loses most of it. Run on a word co-occurrence graph and scored against WordNet hypernymy, it is one way to make the measurement asked for above.
  • On the correlated branch, a joint fit separates what single fits entangle. Erogullari et al. (LIT-937, THEORY-292) find that concept directions fitted one at a time in CNN classifiers align for co-occurring attributes, while a joint fit finds orthogonal directions that still separate each. So correlated attributes still have linear separating directions there. The setting is labelled image attributes, not a co-occurrence embedding, and implications among the labels are not analysed.
  • Linearity is graded by exposure. Merullo et al. (LIT-972, THEORY-269) find that whether a factual relation is computed by an affine map tracks how often its subjects and objects co-occur in pretraining. The maps are per-relation maps between hidden states, not attribute directions, and the result is a correlation. A probe that finds no direction for a rare attribute may be finding low counts.
  • A rival premise: attributes that add, not multiply. Torii, Maeda and Hidaka (LIT-977, THEORY-271) get an exact parallelepiped in raw counts when aspects combine as a mixture over sentences, and show that any non-uniform sentence distribution breaks it. The incidence is the full hypercube, so its lattice is Boolean; it touches only the case of attributes that interact in co-occurrence.

Note of 2026-10-11: partial answers linked

THEORY-183, THEORY-185, THEORY-186, THEORY-195 and THEORY-201 were cited above as partial evidence toward an answer, with no typed link. They are now linked as partial answers (answered_by): two derive linear attribute directions, and three show hierarchical structure learned level by level. The question stays Open.

Note of 2026-10-12: a typed partial answer

THEORY-117 now carries an answered_by edge. In supervised models, features go from spurious, to decodable but entangled, to linearly factored as coverage of combinations grows. So correlated attributes at low coverage do not give factored directions.

Note of 2026-10-12: two prose citations typed

THEORY-196 and THEORY-189, which this question’s body already cites, now carry answered_by edges. THEORY-196 answers the branch on which hierarchy may live in distances rather than directions: a hyperbolic tree embedding is cheap for shallow, bushy trees and costs bits linear in depth. THEORY-189 answers the lattice half. Entailment cones encode a partial order, but the overlap of two cones is not a cone, so there are no meets or joins. Neither bears on linear directions.

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