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LIT-995: Deep sequence models tend to memorize geometrically; it is unclear why

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Properties5
StatusActive — read 2026-10-14 (NOTE-800) in full; worth reading for QUESTION-060 as a case where neither the supervision nor what the model can represent fixes the learned variables. Sequence models trained only on a graph's edges embed it as a global geometry, although a lookup is expressible, found faster and as succinct. In a two-layer model the geometry is the top Laplacian eigenspace, reached by cross-entropy training alone. Read it knowing that every argument is empirical or informal by the authors' own account, that the spectral account covers only the two-layer model, and that initialization scale can tip deeper models into a lookup.
Version2
Tagsrepresentation-learning, learning-theory, anthology-candidate
Cited as “paper” byNOTE-800 — Noroozizadeh et al. — Deep sequence models tend to memorize geometrically
Cited as “source” byTHEORY-274 — Sequence models trained only on a graph's edge bigrams store it as a global embedding geometry, not as a lookup, although the lookup is expressible, found faster and no less succinct; in a wide two-layer weight-tied model the geometry is the top Laplacian eigenspace, reached by cross-entropy training without a rank bottleneck, regularization or multi-hop supervision

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Shahriar Noroozizadeh, Vaishnavh Nagarajan, Elan Rosenfeld and Sanjiv Kumar (2025), arXiv:2510.26745 (ICML 2026, per the arXiv comment).

Key takeaways

From the reading, NOTE-800.

  • In-weights reasoning without step-wise aid. On path-star graphs of up to 5×10⁴ nodes, a transformer and Mamba predict the first hop of unseen paths from first-token supervision alone, up to 100% against chance 1/d.
  • Geometry from local facts. Training on edges alone already clusters each path’s leaf and first node. On tiny graphs, the same architectures can store the edges as a lookup (frozen embeddings), find it in about 2 steps rather than 100, and pay no more than about twice the bits or norm.
  • The two-layer spectral bias. A wide weight-tied Node2Vec-style model under full-softmax cross-entropy ends with embeddings on the top eigenvectors of the symmetrized random-walk Laplacian, more skewed than the spectrum itself, with no bottleneck or regularizer. The account in App. F is informal and rests on an unproved observation.
  • Not pressure-free in every set-up. Under unit-normal initialization some runs end associative. Weight decay, dropout and higher learning rates strengthen the geometry. The large-graph runs used weight decay.

Standing in the record

Filed on 2026-10-12 for QUESTION-060, and read on 2026-10-14 in NOTE-800. THEORY-274 is filed from it as a partial answer that names the dynamics of training as a third determinant of the learned variables, beside the objective and cheap representability. The seeded reading holds, with two cautions. The spectral account covers only the two-layer model. The paper’s refutations target straightforward versions of each pressure, and the authors leave subtler ones, such as flatness, open. The paper’s suggestion to make transformer memory more geometric is practice and belongs to the anthology. The anthology-candidate flag stays.

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