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LIT-950: All or None: Identifiable Linear Properties of Next-token Predictors in Language Modeling

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Properties5
StatusActive — read 2026-10-14 (NOTE-746) in full; worth reading as the exact characterisation of which softmax next-token predictors induce the same distribution: an invertible linear map on the subspaces M and N that enter the logits, with arbitrary distortion off them. Its all-or-none theorem for relational linearity, linear probing and linear subspaces is the record's clearest proved case of structure identified when the full latent model is not. Read it knowing that the result needs exact equality of distributions over every sequence, a shared vocabulary and the conditions Γ ⊆ N and Γ_q ⊆ M, that parallelism survives only inside N, and that there are no experiments. For QUESTION-027 it is an instance of the unrestricted, already answered form, since the whole distribution is observed and no decision lies outside the probes.
Version3
Tagsrepresentation-learning, anthology-candidate
Cited as “paper” byNOTE-746 — Marconato et al. — All or None: identifiable linear properties of next-token predictors
Cited as “source” byTHEORY-234 — Two softmax next-token predictors induce the same distribution exactly when their embeddings and unembeddings are related by an invertible linear map on the subspaces that enter the logits, with arbitrary distortion off them, so a linear relational property is shared by all such models or none when it lies in those subspaces, and can fail to be shared otherwise

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Emanuele Marconato, Sébastien Lachapelle, Sebastian Weichwald, Luigi Gresele (2024), AISTATS 2025; arXiv:2410.23501.

Key takeaways

From the reading, NOTE-746.

  • Identifiability (Theorem 5). For softmax predictors p(y | x) ∝ exp(f(x)ᵀ g(y)) of any dimensions, p_{f,g} = p_{f̃,g̃} ⇔ (f, g) ∼EL (f̃, g̃). This extended linear equivalence is an invertible linear map between the projections onto M := Im(P_F P_G) and N := ker(P_F P_G)^⊥. Components outside M and N can be distorted arbitrarily, even non-linearly (Appendix B.5).
  • Effective complexity. dim(M) is fixed by the distribution. Roeder et al.’s linear identifiability is the special case of equal dimensions plus diversity (Corollary 6).
  • Parallelism is a property of the law. Difference vectors are parallel in N exactly when their token pairs have proportional log-odds across all contexts (Lemma 8). Only parallelism inside N is shared by equivalent models (Theorem 15).
  • All or none (Theorem 14). Relational linearity of a query q on Γ, and the relational linear probing and linear subspaces it implies, holds in every equivalent model or in none when Γ ⊆ N and Γ_q ⊆ M. Without the second condition an equivalent model can break it (Appendix D.1).
  • What it does not explain (§6). The parallelism that PCA shows in trained models is not shared across the equivalence class. The authors conclude that training or architecture selects among equivalent models.
  • Scope. The results hold for exact equality of distributions over all sequences and a shared vocabulary. There are no experiments.

Standing in the record

Held for QUESTION-027, which asks for a signature identified from the probes when the full latent model is not. The paper proves such a case for next-token predictors. The linear properties that live in M and N are constant on each class of distribution-equivalent models, while the representations themselves are not identified. But its probes are the whole conditional distribution, so it is an instance of the unrestricted form that QUESTION-027 records as already answered. It is a case of gauge-invariant functionals (CLAIM-148) and supports CLAIM-209’s unrestricted sentence only. It also refines CLAIM-148’s gauge: without diversity, equivalent models are related linearly only on M and N. Its §6 warns that agreement between trained models need not reflect identified structure. That bears on reading cross-model agreement such as THEORY-175’s and LIT-936’s. Its results are filed as THEORY-234, which extends THEORY-018.

Picozzi (LIT-936) cites it. The follow-up on approximate equivalence by Nielsen, Marconato, Dittadi and Gresele is LIT-253, held unread. The paper carries no instruction for machine-learning practice. An anthology topic on representation learning could hold it, so it keeps the anthology-candidate flag, and it is held here for its own question under ADR-013.

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