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arxiv: 2510.26745 · v3 · pith:645CM2ASnew · submitted 2025-10-30 · 💻 cs.LG · cs.AI· cs.CL· stat.ML

Deep sequence models tend to memorize geometrically; it is unclear why

Shahriar Noroozizadeh , Vaishnavh Nagarajan , Elan Rosenfeld , Sanjiv Kumar This is my paper

Pith reviewed 2026-05-21 20:33 UTC · model grok-4.3

classification 💻 cs.LG cs.AIcs.CLstat.ML
keywords geometric memorysequence modelsembeddingsspectral biasNode2Veccomposition reasoningTransformer memoryknowledge representation
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The pith

Deep sequence models synthesize embeddings that encode global relationships between all entities, even without direct co-occurrence.

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

The paper identifies that deep sequence models store atomic facts not just as local associative lookups but as geometric memory in their embeddings. These embeddings capture novel relationships across the entire set of entities, allowing an ℓ-fold composition reasoning problem to be solved as a single navigation step. This geometric structure emerges counterintuitively, even when it is more complex than brute-force memorization of pairs. The authors trace it to a natural spectral bias, demonstrated through its similarity to Node2Vec, rather than standard training, architecture, or optimization forces.

Core claim

Deep sequence models synthesize embeddings encoding novel global relationships between all entities, including ones that do not co-occur in training. Such storage is powerful: for instance, it transforms a hard reasoning task involving an ℓ-fold composition into an easy-to-learn 1-step navigation task. The rise of such a geometry cannot be straightforwardly attributed to typical supervisory, architectural, or optimizational pressures. Instead, by analyzing a connection to Node2Vec, the geometry stems from a spectral bias that arises naturally despite the lack of various pressures.

What carries the argument

Geometric memory: embeddings that form a structure encoding global relationships, reducing multi-step composition to single-step navigation.

Load-bearing premise

The observed geometry stems from a spectral bias that arises naturally rather than from typical supervisory, architectural, or optimization pressures.

What would settle it

Training a sequence model on data where local co-occurrence statistics are preserved but global graph structure is removed, then checking whether the single-step navigation behavior disappears.

Figures

Figures reproduced from arXiv: 2510.26745 by Elan Rosenfeld, Sanjiv Kumar, Shahriar Noroozizadeh, Vaishnavh Nagarajan.

Figure 1
Figure 1. Figure 1: Associative vs. geometric memory of models trained on various graphs. There are two dramatically different ways to memorize a dataset of atomic facts. The common view is of associative memory: entities are embedded arbitrarily, and co-occurrences are stored in weight matrices. (left). §2.4: In practice, we find a geometric memory: the learned embeddings of a Transformer (middle) reflect global structure in… view at source ↗
Figure 2
Figure 2. Figure 2: Overview of in-context path-star task of B&N’24. Each training and test example corresponds to a fresh, randomly-labeled path-star graph (a tree graph where only the root node branches into d paths of length ℓ). For each example, the prefix specifies a randomized adjacency list (of edge bigrams) of the corresponding graph, followed by (vroot, vgoal). The target is the full path (vroot → vgoal) in that grap… view at source ↗
Figure 3
Figure 3. Figure 3: Overview of our in-weights path-star task. All examples are derived from a fixed path-star graph. Training involves two types of examples: (i) edge memorization examples; (ii) path-finding examples, where the prefix is some leaf, and the target is the full path. Test examples are path examples corresponding to a held-out set of leaves. 2.1 The in-weights path-star task Task definition. For our study, we co… view at source ↗
Figure 4
Figure 4. Figure 4: Success of Transformer in in-weights path-star task. (§2) (left) A next-token-trained Transformer achieves perfect or highly non-trivial accuracy on large path-star graphs Gd,ℓ. (middle) Learning order of tokens. The tokens of a path are not learned in the reverse order i.e., the model does not learn the right-to-left solution. Thus, gradients from the future tokens are not critical for success. (right) Su… view at source ↗
Figure 5
Figure 5. Figure 5: Failure of Transformer in in-context path-star task. (B&N’24) We report the failure of next-token Transformers in the in-context version of the path-star task, reproducing results from B&N’24. (left) Full path accuracy remains at chance level across different small graph sizes. (middle) Learning order of tokens with teacherless (multi-token-trained) objective shows a clear right-to-left learning cascade. (… view at source ↗
Figure 6
Figure 6. Figure 6: Evidence of global geometry of Transformer in path-star task. (a) In the heatmap, entry (i, j) is the cosine distance between the leaf embedding of path i (row) and the first-hop embedding of path j (col). The clear diagonal line implies that embeddings within each path are more aligned, reflecting global structure. (b) UMAP projection of token embeddings where each point is a node em￾bedding; color indica… view at source ↗
Figure 7
Figure 7. Figure 7: Associative memory can be discovered quickly by gradient descent, given a sufficiently wide model, and a sufficiently large learning rate: For the various tiny graphs described in §C.3, and for our TinyNN model (with frozen embedding/unembedding layers and one wide trainable weight matrix to prevent a geometry from taking over; see §B.2.2), we report memorization over timesteps of training with full-batch … view at source ↗
Figure 8
Figure 8. Figure 8: Geometric memorization takes much longer for gradient descent to discover: For the same architecture as in [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Spectral geometry arises in Node2Vec without low-rank pressure (Observation 3) for tiny Path-Star, Grid, Cycle, and Irregular Graphs (top to bottom). (a) The Fiedler-like vectors of the graphs encode global structure; this structure mirrors the Node2Vec embeddings shown in [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Embeddings of weight-untied Node2Vec do not show a clean geometry in the top directions. Corresponding multi-hop cosine similarities are in [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Embeddings of a weight-untied Transformer do not show a clean geometry in the top directions. Corresponding multi-hop cosine similarities are in [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Success of Transformer in in-weights path-star task. (left) A next-token-trained Trans￾former achieves perfect or highly non-trivial accuracy on large path-star graphs Gd,ℓ (Observation 4a). (middle) Learning order of tokens. The tokens of a path are not learned in the reverse order i.e., the model does not learn the right-to-left solution. Thus, gradients from the future tokens are not critical for succe… view at source ↗
Figure 13
Figure 13. Figure 13: Failure of Transformer in in-context path-star task. (B&N’24) We report the failure of next-token Transformers in the in-context version of the path-star task, reproducing results from B&N’24. (left) Full path accuracy remains at chance level across different small graph sizes. (middle) Learning order of tokens with teacherless (multi-token-trained) objective shows a clear right-to-left learning cascade. … view at source ↗
Figure 14
Figure 14. Figure 14: (left) Success of in-weights path-star task for Mamba. This figure is a counterpart to [PITH_FULL_IMAGE:figures/full_fig_p049_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Evidence of global geometry in path-star task for Mamba. This figure is a counterpart to Figs. 6 and 25 for the Mamba SSM architecture. Recall that entry (i, j) is the mean cosine distance between the leaf token in (an unseen) path i (row) and first/hardest token on (an unseen) path j (col). Each heatmap corresponds to a different training objective: Left: trained on edges and path-finding task (Dedge ∪ D… view at source ↗
Figure 16
Figure 16. Figure 16: UMAP projection of token embeddings of Mamba exhibits path-star topology. We corroborate the UMAP [109] observations from the Transformer ( [PITH_FULL_IMAGE:figures/full_fig_p050_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Transformer achieves non-trivial accuracy on the harder in-weights tree-star task. The tree-star task of §C.2 introduces decision-making at every step of the path, not just the first token. There are two variants of this task based on the test-train split. In the split on first token variant (top-left), we reserve some of the trees for generating training paths, and the rest for test paths. In the split o… view at source ↗
Figure 18
Figure 18. Figure 18: Tiny path-star: Geometries of various architectures on a smaller version of the path-star graph. See Observation 5. Associative Node2Vec Eigenvectors Transformer Neural Network Mamba SSM (Transformer with Frozen Embeddings) [PITH_FULL_IMAGE:figures/full_fig_p054_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Tiny grid: Geometries of various architectures on a small 4 × 4 grid graph. See Observation 5. 54 [PITH_FULL_IMAGE:figures/full_fig_p054_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Tiny cycle: Geometries of various architectures on a small cycle graph. See Observation 5. Associative Node2Vec Eigenvectors Transformer Neural Network Mamba SSM (Transformer with Frozen Embeddings) [PITH_FULL_IMAGE:figures/full_fig_p055_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Tiny irregular graph: Geometries of various architectures on a small irregular graph of two connected components, both asymmetric. See Observation 5. Note that unlike in the other graphs, we do not use a 1-layer model here, but a 3-layered one. 55 [PITH_FULL_IMAGE:figures/full_fig_p055_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: For various learning rates, geometric memorization takes much longer for gradient descent to discover: This is an extended version of [PITH_FULL_IMAGE:figures/full_fig_p056_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Node-node cosine similarity vs. adjacency matrix for the Transformer: For each graph, we plot the cosine similarity between all node embeddings and compare it against the adjacency matrix. Observe that the cosine similarities exhibit a richer structure than the adjacency matrix, reflecting some notion of multi-hop distance e.g., in the cycle graph, there is a gradual decrease in similarity as we walk towa… view at source ↗
Figure 24
Figure 24. Figure 24: Node-node cosine similarity vs. adjacency matrix for the Node2Vec model: Like in the Transformer ( [PITH_FULL_IMAGE:figures/full_fig_p057_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Evidence of global geometry in path-star task: Leaf-first-token cosine distance between node embeddings. We present again the heatmaps from [PITH_FULL_IMAGE:figures/full_fig_p058_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Evidence of global geometry in path-star task: Pathwise average cosine distance be￾tween node embeddings. While in [PITH_FULL_IMAGE:figures/full_fig_p059_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Zoomed in version of Fig [PITH_FULL_IMAGE:figures/full_fig_p059_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Mixed edge supervision enables forward path generation while forward-only fails due to reversal curse. Exact-match accuracy on held-out leaves for multiple path-star graphs (varying degree d and path length ℓ). As established, training on mixed edges Dedge yields high non-trivial forward accuracy across graphs. But training on forward-only D→ edge fails on both the forward and reverse tasks. This is indic… view at source ↗
Figure 29
Figure 29. Figure 29: Forward vs. reverse path generation: The figure contrasts the model’s performance on forward (start→leaf) and reverse (leaf→start) path generation tasks for path-star graphs learned either in-weights (left) or in-context (right). While both methods achieve perfect accuracy on the algorithmically simple reverse path task, their performance on the forward task differs dramatically. (left) The in-weights mod… view at source ↗
Figure 30
Figure 30. Figure 30: Embeddings of a Transformer with bi-directional vs. uni-directional edge memoriza￾tion. With our smaller graphs, we find a geometry arise regardless of whether the model is made to memorize both or only one direction of each edge. However, the geometry is weaker (e.g., for the grid graph) under uni-directional memorization. 62 [PITH_FULL_IMAGE:figures/full_fig_p062_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: shows that adding a short sequence of pause tokens after the prompt reliably boosts exact￾match accuracy across graphs, for a given amount of training time. Increasing the number of pause tokens increases speed of convergence. 0 5 10 15 20 25 Epochs [Thousands] 0 20 40 60 80 100 Full Path Accuracy[%] Number of Pause Tokens: 0 2 4 6 [PITH_FULL_IMAGE:figures/full_fig_p063_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: ). The fact that the composition task is learnable in this regime implies that the edge-pretrained model must have come with an adequate global geometry despite being trained only on local supervision (Refutation 3a). G5 × 10 3 , 5 G10 4 , 6 G10 4 , 10 0.1 100 Accuracy[%] 100 100 42 100 100 55 0.02 0.01 0.01 In-Weights Path-Star Chance Level Full Path Hardest Token [PITH_FULL_IMAGE:figures/full_fig_p063_… view at source ↗
Figure 33
Figure 33. Figure 33: Node-node cosine similarity vs. adjacency matrix for weight-untied Node2Vec. As discussed in §4.4, observe that the node-to-node cosine-similarity matrix appears to be a bland near-zero matrix, lacking any information about the multi-hop connectivity of the underlying graph. Node Embedding Node Embedding Node-Node Similarity Matrix Node Index Node Index Adjacency Matrix 1.0 0.5 0.0 0.5 1.0 Similarity 0.0 … view at source ↗
Figure 34
Figure 34. Figure 34: Node-node cosine similarity vs. adjacency matrix for weight-untied Transformer. As discussed in §4.4, observe that the node-to-node cosine-similarity matrix appears to be a bland near-zero matrix, lacking any information about the multi-hop connectivity of the underlying graph. 64 [PITH_FULL_IMAGE:figures/full_fig_p064_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: Learning dynamics per token. (a) In the in-context setting of G2,5 (trained with a multi-token, teacherless objective since standard next-token prediction fails), later tokens are learned first indicating strong reliance on future-token signals. (b) In the in-weights setting of G104,6 with next-token prediction, token accuracies rise largely in tandem (or in a somewhat confusing order); the first token is… view at source ↗
read the original abstract

Deep sequence models are said to store atomic facts predominantly in the form of associative memory: a brute-force lookup of co-occurring entities. We identify a dramatically different form of storage of atomic facts that we term as geometric memory. Here, the model has synthesized embeddings encoding novel global relationships between all entities, including ones that do not co-occur in training. Such storage is powerful: for instance, we show how it transforms a hard reasoning task involving an $\ell$-fold composition into an easy-to-learn $1$-step navigation task. From this phenomenon, we extract fundamental aspects of neural embedding geometries that are hard to explain. We argue that the rise of such a geometry, as against a lookup of local associations, cannot be straightforwardly attributed to typical supervisory, architectural, or optimizational pressures. Counterintuitively, a geometry is learned even when it is more complex than the brute-force lookup. Then, by analyzing a connection to Node2Vec, we demonstrate how the geometry stems from a spectral bias that -- in contrast to prevailing theories -- indeed arises naturally despite the lack of various pressures. This analysis also points out to practitioners a visible headroom to make Transformer memory more strongly geometric. We hope the geometric view of parametric memory encourages revisiting the default intuitions that guide researchers in areas like knowledge acquisition, capacity, discovery, and unlearning.

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 claims that deep sequence models memorize atomic facts geometrically rather than via associative lookup of co-occurrences. Embeddings are synthesized to encode novel global relationships between all entities (including non-co-occurring ones), transforming an ℓ-fold composition reasoning task into an easy 1-step navigation task. The authors argue this geometry cannot be straightforwardly attributed to typical supervisory, architectural or optimizational pressures, is counterintuitively more complex than brute-force lookup, and instead arises from a spectral bias demonstrated via a connection to Node2Vec; they point to headroom for making Transformer memory more strongly geometric and implications for knowledge acquisition, capacity, discovery and unlearning.

Significance. If the empirical observations and the spectral-bias explanation hold, the work would be significant for offering a distinct geometric view of parametric memory that challenges prevailing intuitions about associative storage. The Node2Vec link could bridge empirical findings with spectral graph methods, while the practical suggestion for enhancing geometric properties in Transformers would be useful for practitioners working on reasoning and knowledge representation.

major comments (2)
  1. [Node2Vec analysis] Section on Node2Vec connection: the central claim that the observed global geometry 'stems from a spectral bias that arises naturally' rests on the Node2Vec analysis. This connection is presented as explanatory, yet remains analogical; the manuscript does not isolate whether random-walk co-occurrence statistics alone produce the reported eigenstructure independently of the sequence model's layered back-propagation and next-token loss. Without such isolation or controls, the argument that the geometry cannot be attributed to standard training dynamics is not yet secured.
  2. [Empirical observations] Empirical sections describing non-co-occurring pairs and ℓ-fold to 1-step transformation: the claim that embeddings encode novel global relationships (including for entities that never co-occur) is load-bearing for the 'geometric memory' phenomenon. Full methods, data-selection controls, and ablations are needed to rule out post-hoc artifacts, as the current presentation leaves open whether the geometry is a general tendency or specific to the chosen setups.
minor comments (2)
  1. [Abstract] Abstract: the term 'geometric memory' is introduced without a concise formal characterization on first use, which would help readers grasp the distinction from associative memory immediately.
  2. [Throughout] Notation: ensure consistent use of symbols for embeddings and entity relationships across sections to avoid ambiguity when discussing global vs. local associations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which highlight opportunities to clarify the Node2Vec analysis and strengthen the empirical controls. We address each point below and will incorporate revisions to improve the manuscript.

read point-by-point responses

  1. Referee: [Node2Vec analysis] Section on Node2Vec connection: the central claim that the observed global geometry 'stems from a spectral bias that arises naturally' rests on the Node2Vec analysis. This connection is presented as explanatory, yet remains analogical; the manuscript does not isolate whether random-walk co-occurrence statistics alone produce the reported eigenstructure independently of the sequence model's layered back-propagation and next-token loss. Without such isolation or controls, the argument that the geometry cannot be attributed to standard training dynamics is not yet secured.

    Authors: We appreciate the referee's emphasis on securing the isolation. The manuscript presents the Node2Vec link to show that the eigenstructure follows from the co-occurrence statistics generated by next-token prediction on sequences, which implicitly perform random walks on the entity graph; this is not merely analogical but follows because the training objective directly optimizes for those statistics. We agree that an explicit control would make the separation from layered back-propagation clearer. In the revision we will add a non-neural baseline that factorizes the empirical co-occurrence matrix derived from the same data and verifies that the reported spectral properties are recovered without any neural architecture or gradient-based training. revision: yes

  2. Referee: [Empirical observations] Empirical sections describing non-co-occurring pairs and ℓ-fold to 1-step transformation: the claim that embeddings encode novel global relationships (including for entities that never co-occur) is load-bearing for the 'geometric memory' phenomenon. Full methods, data-selection controls, and ablations are needed to rule out post-hoc artifacts, as the current presentation leaves open whether the geometry is a general tendency or specific to the chosen setups.

    Authors: We agree that additional documentation and controls are warranted to establish generality. The manuscript already specifies the synthetic data generation process and the criterion used to identify non-co-occurring pairs, but the presentation can be made more self-contained. In the revision we will expand the methods section with explicit data-selection rules, include supplementary ablations that vary the proportion of non-co-occurring pairs and the graph structure, and report results on an alternative synthetic task to demonstrate that the ℓ-fold to 1-step transformation and the global geometry persist beyond the primary experimental setups. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation grounded in external Node2Vec connection and empirical observations

full rationale

The paper identifies geometric memory through direct empirical analysis of sequence model embeddings that encode non-co-occurring relations and simplify composition tasks. The explanation that this stems from a spectral bias is explicitly tied to an analysis of the connection with the independent Node2Vec algorithm, whose random-walk co-occurrence mechanism is external to the present work and does not rely on the paper's own fitted values or definitions. No step reduces the claimed geometry to a self-definition, a renamed prediction of the same data, or a load-bearing self-citation chain. The contrast with typical supervisory pressures is presented as an argument from the Node2Vec parallel rather than a tautological claim. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the assumption that the geometry is not explained by standard training pressures and that the Node2Vec connection captures the mechanism; no explicit free parameters or invented entities are stated in the abstract.

axioms (1)
  • domain assumption The rise of geometric embeddings cannot be attributed to typical supervisory, architectural, or optimizational pressures.
    Abstract states this as a key argument against straightforward explanations.
invented entities (1)
  • geometric memory no independent evidence
    purpose: A form of storage where embeddings encode global relationships between entities that do not co-occur.
    New term introduced to describe the observed phenomenon distinct from associative memory.

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Lean theorems connected to this paper

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  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    the geometry stems from a spectral bias that—in contrast to prevailing theories—indeed arises naturally despite the lack of various pressures... the converged solution... columns of embedding matrix V span the graph’s Fiedler-like vectors

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    global information readily available i.e., f(u)[v] is proportional to multi-hop distance... low-rank factorization of adjacency matrix

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    cs.CL 2026-05 conditional novelty 8.0

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