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arxiv: 2605.12756 · v1 · submitted 2026-05-12 · 🧮 math.OC · cs.AI· stat.ML

Recognition: 2 theorem links

· Lean Theorem

Uncovering Symmetry Transfer in Large Language Models via Layer-Peeled Optimization

Hangfeng He, Weijie Su, Zhehang Du

Authors on Pith no claims yet

Pith reviewed 2026-05-14 20:07 UTC · model grok-4.3

classification 🧮 math.OC cs.AIstat.ML
keywords symmetry transferlayer-peeled optimizationcirculant matricesequiangular tight frameslarge language modelsnext-token predictiongeometric structurecyclic symmetry
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The pith

Symmetries in next-token targets transfer exactly to circulant logit matrices and equiangular structures in LLM weights and embeddings.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates whether cross-entropy training for next-token prediction in large language models creates specific geometric patterns in the learned weights. It analyzes a simplified constrained optimization problem that treats the output projection matrix and final context embeddings as variables. When the target tokens show cyclic-shift symmetry, such as the days of the week, the optimal logit matrix is proven to be exactly circulant, with the Gram matrices of the projections and embeddings also forming circulant geometries. For targets invariant under permutations or two-transitive group actions, the optimal output projection becomes a simplex equiangular tight frame while inheriting the input symmetries. Real open-source models are shown to display these structures naturally.

Core claim

Symmetries in the target next-token distributions transfer to the global minimizers of the layer-peeled model in a group-theoretic sense: cyclic-shift symmetries make the optimal logit matrix exactly circulant and the relevant Gram matrices circulant, while exchangeable targets under the symmetric group make the optimal output projection matrix a simplex equiangular tight frame, with the logit matrix and context embeddings inheriting the permutation symmetries.

What carries the argument

The constrained layer-peeled optimization program that optimizes the output projection matrix and last-layer context embeddings to minimize cross-entropy loss as a surrogate for full LLM training.

If this is right

  • Optimal output projections form simplex equiangular tight frames when targets are exchangeable under the symmetric group.
  • Context embeddings and logits inherit the exact permutation symmetries of the input data distribution.
  • The reduction of the nonconvex factorized problem to a convex logit-level characterization allows exact proofs for the cyclic and permutation cases.
  • Open-source LLMs exhibit these circulant and equiangular geometries without any explicit symmetry-promoting regularization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This transfer mechanism may underlie LLMs' ability to handle calendar or periodic reasoning tasks without special architectural provisions.
  • Fine-tuning on data that breaks cyclic symmetry could be tested by measuring how far the logit matrix deviates from circulant form.
  • The same symmetry analysis might extend to other group actions that appear in language data, such as spatial or temporal invariances.

Load-bearing premise

The constrained layer-peeled optimization serves as an accurate surrogate for the behavior of actual large language model training.

What would settle it

Measure the logit matrix entries for a cyclic set such as days of the week in a trained LLM and check whether off-diagonal entries that should be identical under cyclic shift differ by more than a small numerical tolerance.

Figures

Figures reproduced from arXiv: 2605.12756 by Hangfeng He, Weijie Su, Zhehang Du.

Figure 1
Figure 1. Figure 1: This diagram shows the common form behind both symmetry transfers I and II. Let [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Normalized output projection Gram matrices for Mistral-7B-Instruct-v0.3 (Jiang et al., [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Output projection Grams for kitchen utensils. [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Output projection Grams for weekdays. 5.2 Context Embeddings and Logits Permutation. For the permutation template “Among three primary colors, <TOKEN1> is a mix of <TOKEN2> and ”, the six contexts permute (<TOKEN1>, <TOKEN2>) as follows: (orange, red), (orange, yellow), (green, yellow), (green, blue), (purple, red), (purple, blue) [PITH_FULL_IMAGE:figures/full_fig_p028_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Predicted next-token probability matrices for color-mix prompts. Row labels denote [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Context embedding Grams for color-mix prompts. [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Predicted next-token probability matrices for cyclic-shift weekday prompts. Row labels [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Context embedding Grams for cyclic-shift weekday prompts. [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Output projection Grams for kitchen utensils. [PITH_FULL_IMAGE:figures/full_fig_p062_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Output projection Grams for modes of transportation. [PITH_FULL_IMAGE:figures/full_fig_p062_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Output projection Grams for modes of transportation. [PITH_FULL_IMAGE:figures/full_fig_p063_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Output projection Grams for weekdays. E.2 Context Embeddings and Logits Permutation. “Among three primary colors, <TOKEN1> is a mix of <TOKEN2> and ”. The six contexts permute (<TOKEN1>, <TOKEN2>) as (orange, red), (orange, yellow), (green, yellow), (green, blue), (purple, red), and (purple, blue) [PITH_FULL_IMAGE:figures/full_fig_p063_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Predicted next-token probability matrices for color-mix prompts. [PITH_FULL_IMAGE:figures/full_fig_p064_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Context embedding Grams for color-mix prompts. [PITH_FULL_IMAGE:figures/full_fig_p064_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Predicted next-token probability matrices for cyclic-shift weekday prompts. [PITH_FULL_IMAGE:figures/full_fig_p064_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Context embedding Grams for cyclic-shift weekday prompts. [PITH_FULL_IMAGE:figures/full_fig_p065_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Direct sum (S2 × S3); full orbit n = 12. In the second experiment, three blocks of sizes 2, 3, 2 are used with y = h (y (1)) ⊤ | (y (2)) ⊤ | (y (3)) ⊤ i⊤ =  1 16 , 1 3 | 1 4 , 3 16 , 0 | 1 8 , 1 24⊤ , the acting group is S2 × S3 × S2, the full orbit gives n = 24, and Y = [τ ◦ y]τ=(τ1,τ2,τ3)∈S2×S3×S2 (see [PITH_FULL_IMAGE:figures/full_fig_p067_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Direct sum (S2 × S3 × S2); full orbit n = 24. In the numerical solutions, WW⊤ has diagonal blocks with a two-level within-block pattern and off-diagonal blocks with constant entries. Concretely, for the partition m = m1 + · · · + mr, WW⊤ =      A1 κ121m1 1 ⊤ m2 · · · κ1r1m1 1 ⊤ mr κ211m2 1 ⊤ m1 A2 · · · κ2r1m2 1 ⊤ mr . . . . . . . . . . . . κr11mr 1 ⊤ m1 κr21mr 1 ⊤ m2 · · · Ar      , Ai = αiImi … view at source ↗
Figure 19
Figure 19. Figure 19: Direct product (S2 × S3); full orbit n = 12. The second configuration reshapes the same entries as P =    1 12 2 12 1 6 1 12 1 10 4 10    , y = vec (P), The acting group is S3 × S2, and n = 12 (see [PITH_FULL_IMAGE:figures/full_fig_p069_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Direct product (S3 × S2); full orbit n = 12. The third configuration uses P =    0 1 9 1 18 1 18 1 9 1 6 1 12 1 6 1 4    , y = vec (P), The acting group is S3 × S3, and n = 36 (see [PITH_FULL_IMAGE:figures/full_fig_p070_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Direct product (S3 × S3); full orbit n = 36. In the computed solution, we partition WW⊤ into a × a blocks, each of size ℓ × ℓ, following the row-major a × ℓ grid convention. Every diagonal block equals αIℓ + β (Jℓ − Iℓ), and every off-diagonal block equals α ′ Iℓ + β ′ (Jℓ − Iℓ). Moreover, all diagonal blocks are identical to one another, and all off-diagonal blocks are identical to one another. F.3 Wreat… view at source ↗
Figure 22
Figure 22. Figure 22: Wreath ( [PITH_FULL_IMAGE:figures/full_fig_p072_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Wreath (S3 ≀ S2); full orbit n = 72. In the computed solution, we partition WW⊤ into s × s blocks. Every diagonal block equals αIs + β (Js − Is), and every off-diagonal block equals α ′ Is + β ′ (Js − Is). Moreover, all diagonal blocks are identical to one another, and all off-diagonal blocks are identical to one another. Thus the observed block pattern coincides with the one described above for the direc… view at source ↗
read the original abstract

Large language models (LLMs) are pretrained by minimizing the cross-entropy loss for next-token prediction. In this paper, we study whether this optimization strategy can induce geometric structure in the learned model weights and context embeddings. We approach this problem by analyzing a constrained layer-peeled optimization program, which serves as a mathematically tractable surrogate for LLMs by treating the output projection matrix and last-layer context embeddings as optimization variables. Our analysis of this nonconvex optimization program demonstrates that symmetries in the target next-token distributions are transferred to the global minimizers of the layer-peeled model in a precise group-theoretic sense. Specifically, we prove that when the target tokens exhibit a cyclic-shift symmetry (such as the seven days of the week or the twelve months of the year), the optimal logit matrix is exactly circulant, and the Gram matrices of both the output projections and the context embeddings form circulant geometries as well. Next, for exchangeable target distributions invariant under the symmetric group and, more generally, under two-transitive group actions, we show that the global optimal output projection matrix forms a simplex equiangular tight frame, while the optimal logit matrix and context embeddings inherit the permutation symmetries present in the input data. A key technical step is to reduce the constrained nonconvex factorized problem to an explicit logit-level convex characterization for cyclic symmetry and to a symmetry-based lower bound for permutation symmetry, together with a sharp characterization of the optimal factorization. Finally, we empirically demonstrate that open-source LLMs naturally exhibit symmetries consistent with our theoretical predictions, despite being trained without any explicit regularization promoting such geometric structure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper analyzes a constrained layer-peeled optimization program as a surrogate for LLM pretraining under cross-entropy loss. It proves that cyclic-shift symmetries in target tokens induce exactly circulant optimal logit matrices and circulant Gram matrices for output projections and context embeddings; for exchangeable targets under symmetric-group or two-transitive actions, the optimal output projection is a simplex equiangular tight frame while logits and embeddings inherit the input permutation symmetries. The proofs reduce the nonconvex factorized program to a convex logit-level characterization (cyclic case) or symmetry-based lower bound (permutation case) with a sharp factorization characterization. Empirical checks on open-source LLMs confirm the predicted geometric structures appear without explicit regularization.

Significance. If the surrogate analysis holds and the empirical patterns are robust, the work supplies a group-theoretic mechanism explaining why certain geometric regularities emerge in LLM weights and embeddings from standard next-token training. The explicit convex reductions and the demonstration that real models exhibit the predicted circulant and ETF structures are concrete strengths that could guide future theoretical and architectural work on symmetry in transformers.

major comments (2)
  1. [§1] §1 and §2: the central claim that the optimization strategy induces geometric structure in LLMs rests on the layer-peeled surrogate treating W and H as free variables; no perturbation analysis, error bound, or lifting argument is supplied showing that the proven circulant or ETF properties survive when these quantities are instead outputs of the full transformer stack with gradients flowing through earlier layers.
  2. [§4.1] §4.1, Theorem 1: the reduction of the nonconvex program to an explicit logit-level convex characterization for cyclic symmetry is load-bearing for the exact-circulant claim, yet the manuscript does not verify that the optimal factorization recovered from the convex solution remains feasible and unique under the original nonconvex constraints when the target distribution is only approximately cyclic.
minor comments (2)
  1. [§3] Notation for the cyclic group action and the definition of circulant matrices could be introduced with a small concrete example (e.g., 7-day cycle) at the beginning of §3 to improve readability.
  2. [§5] The empirical section would benefit from a quantitative metric (e.g., Frobenius distance to the nearest circulant matrix) rather than qualitative visual inspection alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The comments highlight important aspects of the surrogate model's scope and the robustness of the cyclic characterization. We address each major comment point by point below, indicating planned revisions to strengthen the presentation while preserving the core contributions.

read point-by-point responses

  1. Referee: §1 and §2: the central claim that the optimization strategy induces geometric structure in LLMs rests on the layer-peeled surrogate treating W and H as free variables; no perturbation analysis, error bound, or lifting argument is supplied showing that the proven circulant or ETF properties survive when these quantities are instead outputs of the full transformer stack with gradients flowing through earlier layers.

    Authors: We agree that the layer-peeled formulation is a deliberate surrogate that isolates the final-layer optimization under cross-entropy loss, treating the output projection and context embeddings as free variables. A rigorous perturbation or lifting analysis connecting the surrogate minimizers to the full transformer stack would indeed strengthen the transfer claim, but such an analysis lies outside the present scope because it would require controlling gradient flow through all preceding layers and attention mechanisms. In the revised manuscript we will expand the discussion in §2 to explicitly state the surrogate assumptions, their relation to standard LLM training, and the empirical evidence already provided in §5 that the predicted circulant and ETF structures appear in real open-source models. This addition clarifies the intended scope without overstating the theoretical reach. revision: partial

  2. Referee: §4.1, Theorem 1: the reduction of the nonconvex program to an explicit logit-level convex characterization for cyclic symmetry is load-bearing for the exact-circulant claim, yet the manuscript does not verify that the optimal factorization recovered from the convex solution remains feasible and unique under the original nonconvex constraints when the target distribution is only approximately cyclic.

    Authors: Theorem 1 establishes an exact convex reduction and sharp factorization characterization only for precisely cyclic target distributions. For approximately cyclic targets the exact circulant property is expected to degrade gracefully. To address the concern we will add a short numerical study (new paragraph in §4.1 together with an appendix figure) that perturbs the target distribution by small random shifts away from exact cyclicity, solves the convex logit-level program, recovers the factorization, and checks the residual violation of the original nonconvex constraints. The experiment will confirm that the recovered factors remain nearly feasible and that the circulant deviation scales with the perturbation size, thereby supporting robustness of the exact result. revision: yes

Circularity Check

0 steps flagged

No circularity: symmetry claims derived from explicit surrogate program and group theory

full rationale

The paper defines a constrained layer-peeled optimization program as a surrogate, then applies group theory and convex reduction techniques to prove that cyclic-shift symmetry in targets forces circulant structure in the optimal logit matrix and Gram matrices of W and H. The derivation chain consists of the stated nonconvex program, its reduction to logit-level convex characterization, and symmetry-based lower bounds; none of these steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations. Empirical checks on open-source LLMs are presented separately as validation and do not enter the proof. The central claims therefore remain independent of their inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the validity of the layer-peeled surrogate as an approximation to full LLM pretraining, which is a domain assumption without independent empirical validation in the abstract. No free parameters or invented entities are introduced.

axioms (1)
  • domain assumption The constrained layer-peeled optimization program is a valid surrogate for LLMs
    Invoked to make the nonconvex program tractable and to analyze symmetry transfer to global minimizers.

pith-pipeline@v0.9.0 · 5596 in / 1409 out tokens · 73295 ms · 2026-05-14T20:07:32.146525+00:00 · methodology

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Works this paper leans on

98 extracted references · 11 canonical work pages · 5 internal anchors

  1. [1]

    Expositiones Mathematicae , volume =

    Carlsson, Marcus , title =. Expositiones Mathematicae , volume =. 2021 , doi =

    2021

  2. [2]

    and Johnson, Charles R

    Horn, Roger A. and Johnson, Charles R. , title =. 2012 , isbn =

    2012

  3. [3]

    1977 , publisher =

    Serre, Jean-Pierre , title =. 1977 , publisher =. doi:10.1007/978-1-4684-9458-7 , isbn =

    work page doi:10.1007/978-1-4684-9458-7 1977

  4. [4]

    and Kaiser,

    Vaswani, Ashish and Shazeer, Noam and Parmar, Niki and Uszkoreit, Jakob and Jones, Llion and Gomez, Aidan N. and Kaiser,. Attention is All you Need , booktitle =. 2017 , publisher =

    2017

  5. [5]

    Papyan, Vardan and Han, X. Y. and Donoho, David L. , title =. Proceedings of the National Academy of Sciences , volume =. 2020 , doi =

    2020

  6. [6]

    , title =

    Ji, Wenlong and Lu, Yiping and Zhang, Yiliang and Deng, Zhun and Su, Weijie J. , title =. International Conference on Learning Representations , year =

  7. [7]

    A Geometric Analysis of Neural Collapse with Unconstrained Features , volume =

    Zhu, Zhihui and Ding, Tianyu and Zhou, Jinxin and Li, Xiao and You, Chong and Sulam, Jeremias and Qu, Qing , booktitle =. A Geometric Analysis of Neural Collapse with Unconstrained Features , volume =

  8. [8]

    OpenAI Technical Report , year=

    Improving Language Understanding by Generative Pre-Training , author=. OpenAI Technical Report , year=

  9. [9]

    Bubeck, C

    Bubeck, S. Early science acceleration experiments with. arXiv preprint arXiv:2511.16072 , year=

    work page arXiv

  10. [10]

    First Conference on Language Modeling , year =

    Zhao, Yize and Behnia, Tina and Vakilian, Vala and Thrampoulidis, Christos , title =. First Conference on Language Modeling , year =

  11. [11]

    The First Workshop on the Interplay of Model Behavior and Model Internals , year =

    On the Geometry of Semantics in Next-token Prediction , author =. The First Workshop on the Interplay of Model Behavior and Model Internals , year =

  12. [12]

    and Parshall, Hans and Pi, Jianzong , title =

    Mixon, Dustin G. and Parshall, Hans and Pi, Jianzong , title =. Sampling Theory, Signal Processing, and Data Analysis , volume =. 2022 , publisher =. doi:10.1007/s43670-022-00027-5 , issn =

    work page doi:10.1007/s43670-022-00027-5 2022

  13. [13]

    , title =

    Fang, Cong and He, Hangfeng and Long, Qi and Su, Weijie J. , title =. Proceedings of the National Academy of Sciences , volume =. 2021 , doi =

    2021

  14. [14]

    2002 , month =

    Fazel, Maryam , title =. 2002 , month =

    2002

  15. [15]

    The Prevalence of Neural Collapse in Neural Multivariate Regression , booktitle =

    Andriopoulos, George and Dong, Zixuan and Guo, Li and Zhao, Zifan and Ross, Keith , editor =. The Prevalence of Neural Collapse in Neural Multivariate Regression , booktitle =. 2024 , publisher =

    2024

  16. [16]

    Imbalance Trouble: Revisiting Neural-Collapse Geometry , booktitle =

    Thrampoulidis, Christos and Kini, Ganesh Ramachandra and Vakilian, Vala and Behnia, Tina , editor =. Imbalance Trouble: Revisiting Neural-Collapse Geometry , booktitle =. 2022 , publisher =

    2022

  17. [17]

    , editor =

    Rangamani, Akshay and Lindegaard, Marius and Galanti, Tomer and Poggio, Tomaso A. , editor =. Feature learning in deep classifiers through Intermediate Neural Collapse , booktitle =. 2023 , month =

    2023

  18. [18]

    , title =

    He, Hangfeng and Su, Weijie J. , title =. Proceedings of the National Academy of Sciences , volume =. 2023 , doi =

    2023

  19. [19]

    Han, X. Y. and Papyan, Vardan and Donoho, David L. , title =. International Conference on Learning Representations , year =

  20. [20]

    On the Optimization Landscape of Neural Collapse under

    Zhou, Jinxin and Li, Xiao and Ding, Tianyu and You, Chong and Qu, Qing and Zhu, Zhihui , editor =. On the Optimization Landscape of Neural Collapse under. Proceedings of the 39th International Conference on Machine Learning , series =. 2022 , month =

    2022

  21. [21]

    Are All Losses Created Equal: A Neural Collapse Perspective , booktitle =

    Zhou, Jinxin and You, Chong and Li, Xiao and Liu, Kangning and Liu, Sheng and Qu, Qing and Zhu, Zhihui , editor =. Are All Losses Created Equal: A Neural Collapse Perspective , booktitle =. 2022 , publisher =

    2022

  22. [22]

    Neural Collapse in Deep Linear Networks: From Balanced to Imbalanced Data , booktitle =

    Dang, Hien and Huu, Tho Tran and Osher, Stanley and Tran, Hung The and Ho, Nhat and Nguyen, Tan Minh , editor =. Neural Collapse in Deep Linear Networks: From Balanced to Imbalanced Data , booktitle =. 2023 , month =

    2023

  23. [23]

    Neural Networks , volume =

    Hornik, Kurt and Stinchcombe, Maxwell and White, Halbert , title =. Neural Networks , volume =. 1989 , doi =

    1989

  24. [24]

    Linguistic Collapse: Neural Collapse in (Large) Language Models , booktitle =

    Wu, Robert and Papyan, Vardan , editor =. Linguistic Collapse: Neural Collapse in (Large) Language Models , booktitle =. 2024 , publisher =

    2024

  25. [25]

    , title =

    He, Hangfeng and Su, Weijie J. , title =. Physical Review E , volume =. 2025 , month =. doi:10.1103/5rn3-49lc , numpages =

    work page doi:10.1103/5rn3-49lc 2025

  26. [26]

    Wide Neural Networks Trained with Weight Decay Provably Exhibit Neural Collapse , booktitle =

    Jacot, Arthur and S. Wide Neural Networks Trained with Weight Decay Provably Exhibit Neural Collapse , booktitle =

  27. [27]

    , title =

    Garrod, Connall and Keating, Jonathan P. , title =. The Thirty-ninth Annual Conference on Neural Information Processing Systems , year =

  28. [28]

    , editor =

    Zaheer, Manzil and Kottur, Satwik and Ravanbakhsh, Siamak and Poczos, Barnabas and Salakhutdinov, Ruslan and Smola, Alexander J. , editor =. Deep Sets , booktitle =. 2017 , publisher =

    2017

  29. [29]

    Group Equivariant Convolutional Networks , booktitle =

    Cohen, Taco and Welling, Max , editor =. Group Equivariant Convolutional Networks , booktitle =. 2016 , month =

    2016

  30. [30]

    International Conference on Learning Representations , year =

    Sun, Zhiqing and Deng, Zhi-Hong and Nie, Jian-Yun and Tang, Jian , title =. International Conference on Learning Representations , year =

  31. [31]

    , title =

    Strohmer, Thomas and Heath, Robert W. , title =. Applied and Computational Harmonic Analysis , volume =. 2003 , doi =

    2003

  32. [32]

    and Furnas, George W

    Deerwester, Scott and Dumais, Susan T. and Furnas, George W. and Landauer, Thomas K. and Harshman, Richard , title =. Journal of the American Society for Information Science , volume =. 1990 , doi =

    1990

  33. [33]

    A Neural Probabilistic Language Model , journal =

    Bengio, Yoshua and Ducharme, R. A Neural Probabilistic Language Model , journal =

  34. [34]

    and Dean, Jeff , editor =

    Mikolov, Tomas and Sutskever, Ilya and Chen, Kai and Corrado, Greg S. and Dean, Jeff , editor =. Distributed Representations of Words and Phrases and their Compositionality , booktitle =. 2013 , publisher =

    2013

  35. [35]

    Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (

    Pennington, Jeffrey and Socher, Richard and Manning, Christopher , editor =. Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (. 2014 , month =

    2014

  36. [36]

    Neural Word Embedding as Implicit Matrix Factorization , booktitle =

    Levy, Omer and Goldberg, Yoav , editor =. Neural Word Embedding as Implicit Matrix Factorization , booktitle =. 2014 , publisher =

    2014

  37. [37]

    A Latent Variable Model Approach to

    Arora, Sanjeev and Li, Yuanzhi and Liang, Yingyu and Ma, Tengyu and Risteski, Andrej , editor =. A Latent Variable Model Approach to. Transactions of the Association for Computational Linguistics , volume =. 2016 , publisher =

    2016

  38. [38]

    The Linear Representation Hypothesis and the Geometry of Large Language Models , booktitle =

    Park, Kiho and Choe, Yo Joong and Veitch, Victor , editor =. The Linear Representation Hypothesis and the Geometry of Large Language Models , booktitle =. 2024 , month =

    2024

  39. [39]

    The Thirteenth International Conference on Learning Representations , year =

    The Geometry of Categorical and Hierarchical Concepts in Large Language Models , author =. The Thirteenth International Conference on Learning Representations , year =

  40. [40]

    gpt-oss-120b & gpt-oss-20b Model Card

    2025 , doi =. 2508.10925 , archivePrefix =

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  41. [41]

    and Wu, Tianyi and Wuttke, Daniel and Zhou-Zheng, Christian , title =

    Peng, Bo and Zhang, Ruichong and Goldstein, Daniel and Alcaide, Eric and Du, Xingjian and Hou, Haowen and Lin, Jiaju and Liu, Jiaxing and Lu, Janna and Merrill, William and Song, Guangyu and Tan, Kaifeng and Utpala, Saiteja and Wilce, Nathan and Wind, Johan S. and Wu, Tianyi and Wuttke, Daniel and Zhou-Zheng, Christian , title =. Second Conference on Lang...

  42. [42]

    Jiang, Albert Q. and Sablayrolles, Alexandre and Mensch, Arthur and Bamford, Chris and Chaplot, Devendra Singh and de las Casas, Diego and Bressand, Florian and Lengyel, Gianna and Lample, Guillaume and Saulnier, Lucile and Renard Lavaud, L. Mistral. 2023 , doi =. 2310.06825 , archivePrefix =

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  43. [43]

    2019 , publisher =

    Radford, Alec and Wu, Jeffrey and Child, Rewon and Luan, David and Amodei, Dario and Sutskever, Ilya , title =. 2019 , publisher =

    2019

  44. [44]

    The Llama 3 Herd of Models

    Grattafiori, Aaron and Dubey, Abhimanyu and Jauhri, Abhinav and Pandey, Abhinav and Kadian, Abhishek and Al-Dahle, Ahmad and Letman, Aiesha and Mathur, Akhil and others , title =. 2024 , doi =. 2407.21783 , archivePrefix =

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  45. [45]

    Gemma 2: Improving Open Language Models at a Practical Size

    2024 , doi =. 2408.00118 , archivePrefix =

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  46. [46]

    Extended Unconstrained Features Model for Exploring Deep Neural Collapse , booktitle =

    Tirer, Tom and Bruna, Joan , editor =. Extended Unconstrained Features Model for Exploring Deep Neural Collapse , booktitle =. 2022 , month =

    2022

  47. [47]

    Journal of Machine Learning Research , volume =

    Diamond, Steven and Boyd, Stephen , title =. Journal of Machine Learning Research , volume =

  48. [48]

    How Contextual are Contextualized Word Representations? Comparing the Geometry of

    Ethayarajh, Kawin , editor =. How Contextual are Contextualized Word Representations? Comparing the Geometry of. Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP) , pages =. 2019 , month =

    2019

  49. [49]

    International Conference on Learning Representations , year =

    Mu, Jiaqi and Viswanath, Pramod , title =. International Conference on Learning Representations , year =

  50. [50]

    2022 , eprint =

    Elhage, Nelson and Hume, Tristan and Olsson, Catherine and Schiefer, Nicholas and Henighan, Tom and Kravec, Shauna and Hatfield-Dodds, Zac and Lasenby, Robert and Drain, Dawn and Chen, Carol and others , title =. 2022 , eprint =

    2022

  51. [51]

    The Fourteenth International Conference on Learning Representations , year =

    From Data Statistics to Feature Geometry: How Correlations Shape Superposition , author =. The Fourteenth International Conference on Learning Representations , year =

  52. [52]

    Representation Engineering: A Top-Down Approach to AI Transparency

    Zou, Andy and Phan, Long and Chen, Sarah and Campbell, James and Guo, Phillip and Ren, Richard and Pan, Alexander and Yin, Xuwang and others , title =. 2025 , doi =. 2310.01405 , archivePrefix =

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  53. [53]

    Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing , pages =

    Zhong, Zexuan and Wu, Zhengxuan and Manning, Christopher and Potts, Christopher and Chen, Danqi , editor =. Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing , pages =. 2023 , month =

    2023

  54. [54]

    Advances in Neural Information Processing Systems , volume =

    Belrose, Nora and Schneider-Joseph, David and Ravfogel, Shauli and Cotterell, Ryan and Raff, Edward and Biderman, Stella , editor =. Advances in Neural Information Processing Systems , volume =. 2023 , publisher =

    2023

  55. [55]

    The Eleventh International Conference on Learning Representations , year =

    Burns, Collin and Ye, Haotian and Klein, Dan and Steinhardt, Jacob , title =. The Eleventh International Conference on Learning Representations , year =

  56. [56]

    The Internal State of an

    Azaria, Amos and Mitchell, Tom , editor =. The Internal State of an. Findings of the Association for Computational Linguistics: EMNLP 2023 , pages =. 2023 , month =

    2023

  57. [57]

    Journal of Control and Decision , volume =

    Agrawal, Akshay and Verschueren, Robin and Diamond, Steven and Boyd, Stephen , title =. Journal of Control and Decision , volume =. 2018 , publisher =

    2018

  58. [58]

    Implicit Optimization Bias of Next-token Prediction in Linear Models , booktitle =

    Thrampoulidis, Christos , editor =. Implicit Optimization Bias of Next-token Prediction in Linear Models , booktitle =. 2024 , publisher =

    2024

  59. [59]

    The Fourteenth International Conference on Learning Representations , year =

    Sequences of Logits Reveal the Low Rank Structure of Language Models , author =. The Fourteenth International Conference on Learning Representations , year =

  60. [60]

    Lost in backpropagation: The lm head is a gradient bottleneck

    Nathan Godey and Yoav Artzi , year =. Lost in Backpropagation: The. 2603.10145 , archivePrefix =

    work page arXiv

  61. [61]

    and Nava, Andres and Wyart, Matthieu and Bahri, Yasaman , title =

    Karkada, Dhruva and Korchinski, Daniel J. and Nava, Andres and Wyart, Matthieu and Bahri, Yasaman , title =. 2026 , doi =. 2602.15029 , archivePrefix =

    work page arXiv 2026

  62. [62]

    2025 , eprint =

    The Origins of Representation Manifolds in Large Language Models , author =. 2025 , eprint =

    2025

  63. [63]

    and Liao, Isaac and Gurnee, Wes and Tegmark, Max , title =

    Engels, Joshua and Michaud, Eric J. and Liao, Isaac and Gurnee, Wes and Tegmark, Max , title =. The Thirteenth International Conference on Learning Representations , year =

  64. [64]

    Successor Heads: Recurring, Interpretable Attention Heads In The Wild , booktitle =

    Gould, Rhys and Ong, Euan and Ogden, George and Conmy, Arthur , editor =. Successor Heads: Recurring, Interpretable Attention Heads In The Wild , booktitle =

  65. [65]

    The Eleventh International Conference on Learning Representations , year =

    Nanda, Neel and Chan, Lawrence and Lieberum, Tom and Smith, Jess and Steinhardt, Jacob , title =. The Eleventh International Conference on Learning Representations , year =

  66. [66]

    A Toy Model of Universality: Reverse Engineering how Networks Learn Group Operations , booktitle =

    Chughtai, Bilal and Chan, Lawrence and Nanda, Neel , editor =. A Toy Model of Universality: Reverse Engineering how Networks Learn Group Operations , booktitle =. 2023 , month =

    2023

  67. [67]

    Transformer Circuits Thread , year =

    Elhage, Nelson and Nanda, Neel and Olsson, Catherine and Henighan, Tom and Joseph, Nicholas and Mann, Ben and Askell, Amanda and Bai, Yuntao and Chen, Anna and Conerly, Tom and DasSarma, Nova and Drain, Dawn and Ganguli, Deep and Hatfield-Dodds, Zac and Hernandez, Danny and Jones, Andy and Kernion, Jackson and Lovitt, Liane and Ndousse, Kamal and Amodei, ...

  68. [68]

    Analyzing Transformers in Embedding Space , booktitle =

    Dar, Guy and Geva, Mor and Gupta, Ankit and Berant, Jonathan , editor =. Analyzing Transformers in Embedding Space , booktitle =. 2023 , month = jul, publisher =

    2023

  69. [69]

    Transformer Feed-Forward Layers Build Predictions by Promoting Concepts in the Vocabulary Space , booktitle =

    Geva, Mor and Caciularu, Avi and Wang, Kevin and Goldberg, Yoav , editor =. Transformer Feed-Forward Layers Build Predictions by Promoting Concepts in the Vocabulary Space , booktitle =. 2022 , month = dec, publisher =

    2022

  70. [70]

    Transformer Feed-Forward Layers Are Key-Value Memories , booktitle =

    Geva, Mor and Schuster, Roei and Berant, Jonathan and Levy, Omer , editor =. Transformer Feed-Forward Layers Are Key-Value Memories , booktitle =. 2021 , month = nov, publisher =

    2021

  71. [71]

    How does

    Hanna, Michael and Liu, Ollie and Variengien, Alexandre , editor =. How does. Advances in Neural Information Processing Systems , volume =. 2023 , publisher =

    2023

  72. [72]

    2025 , eprint =

    Belrose, Nora and Ostrovsky, Igor and McKinney, Lev and Furman, Zach and Smith, Logan and Halawi, Danny and Biderman, Stella and Steinhardt, Jacob , title =. 2025 , eprint =

    2025

  73. [73]

    , title =

    Guo, Li and Andriopoulos, George and Zhao, Zifan and Dong, Zixuan and Ling, Shuyang and Ross, Keith W. , title =. Transactions on Machine Learning Research , issn =

  74. [74]

    The Twelfth International Conference on Learning Representations , year =

    Fisher, Quinn LeBlanc and Meng, Haoming and Papyan, Vardan , title =. The Twelfth International Conference on Learning Representations , year =

  75. [75]

    Burer, Samuel and Monteiro, Renato D. C. , title =. Mathematical Programming , volume =. 2003 , month = feb, day =

    2003

  76. [76]

    Burer, Samuel and Monteiro, Renato D. C. , title =. Mathematical Programming , volume =. 2005 , month = jul, day =

    2005

  77. [77]

    The non-convex

    Boumal, Nicolas and Voroninski, Vlad and Bandeira, Afonso , editor =. The non-convex. Advances in Neural Information Processing Systems , volume =. 2016 , publisher =

    2016

  78. [78]

    Rank Optimality for the

    Waldspurger, Ir. Rank Optimality for the. SIAM Journal on Optimization , volume =. 2020 , doi =

    2020

  79. [79]

    , title =

    Recht, Benjamin and Fazel, Maryam and Parrilo, Pablo A. , title =. SIAM Review , volume =. 2010 , doi =

    2010

  80. [80]

    IEEE Transactions on Information Theory , volume =

    Sun, Ruoyu and Luo, Zhi-Quan , title =. IEEE Transactions on Information Theory , volume =. 2016 , keywords =

    2016

Showing first 80 references.