Representation Geometry in Language Models

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Overview

This field studies the shapes and relationships formed by token embeddings and hidden activations, and how those structures change across layers, contexts, and training. Researchers use linear probes, representational similarity analysis, spectral and manifold methods, sparse autoencoders, and causal interventions such as activation patching and steering. These tools are applied to questions about reasoning, multilingual transfer, retrieval, safety, and reliability, using settings that range from controlled synthetic tasks to benchmarks such as GSM8K and MTEB. A central open tension is that information being decodable or geometrically similar does not guarantee that a model uses it causally; useful structure may also be local, nonlinear, or sensitive to measurement choices.

Latest papers

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417 papers78 papers in the last four weeks, up 34% on the four weeks before. 0.5% of all new papers.

Apr 8, 2026

Geometric Properties of the Voronoi Tessellation in Latent Semantic Manifolds of Large Language Models

Language models operate on discrete tokens but compute in continuous vector spaces, inducing a Voronoi tessellation over the representation manifold. We study this tessellation empirically on Qwen3.5-4B-Base, making two contributions. First, using float32 margin recomputation to resolve bfloat16 quantization artifacts, we validate Mabrok's (2026) linear scaling law of the expressibility gap with R2R^2 = 0.9997 - the strongest confirmation to date - and identify a mid-layer geometric ambiguity regime where margin geometry is anti-correlated with cross-entropy (layers 24-28, ρρ = -0.29) before crystallizing into alignment at the final layer (ρρ = 0.836). Second, we show that the Voronoi tessellation of a converged model is reshapable through margin refinement procedures (MRP): short post-hoc optimization runs that widen token-decision margins without retraining. We compare direct margin maximization against Fisher information distance maximization across a dose-response sweep. Both methods find the same ceiling of ~16,300 correctable positions per 256K evaluated, but differ critically in collateral damage. Margin maximization damage escalates with intervention strength until corrections are overwhelmed. Fisher damage remains constant at ~5,300 positions across the validated range (λλ = 0.15-0.6), achieving +28% median margin improvement at λλ = 0.6 with invariant downstream benchmarks - a geometric reorganization that compresses the expressibility gap while preserving its scaling law. However, frequency and token-class audits reveal that gains concentrate in high-frequency structural tokens (84% of net corrections at λλ = 0.6), with content and entity-like contributions shrinking at higher λλ. Fisher MRP is therefore a viable geometric polishing tool whose practical ceiling is set not by aggregate damage but by the uniformity of token-level benefit.

Marshall Brett

MARS Labs

Representation Geometry in Language ModelsInformation Geometry
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Apr 8, 2026

When Safety Geometry Collapses: Fine-Tuning Vulnerabilities in Agentic Guard Models

A guard model fine-tuned on entirely benign data can lose all safety alignment -- not through adversarial manipulation, but through standard domain specialization. We demonstrate this failure across three purpose-built safety classifiers -- LlamaGuard, WildGuard, and Granite Guardian -- deployed as protection layers in agentic AI pipelines, and show that it originates in the destruction of latent safety geometry: the structured harmful -- benign representational boundary that guides classification. We extract per-layer safety subspaces via SVD on class-conditional activation differences and track how this boundary evolves under benign fine-tuning. Granite Guardian undergoes complete collapse -- refusal rate drops from 85% to 0%, CKA falls to zero, and 100% of outputs become ambiguous -- a severity exceeding prior findings on general-purpose LLMs, explained by the specialization hypothesis: concentrated safety representations are efficient but catastrophically brittle. To mitigate this, we propose Fisher-Weighted Safety Subspace Regularization (FW-SSR), a training-time penalty combining (i) curvature-aware direction weights derived from diagonal Fisher information and (ii) an adaptive λtλ_t that scales with task-safety gradient conflict. FW-SSR recovers 75% refusal on Granite Guardian (CKA = 0.983) and reduces WildGuard's Attack Success Rate to 3.6% -- below the unmodified baseline -- by actively sharpening the safety subspace rather than merely anchoring it. Across all three models, structural representational geometry (CKA, Fisher score) predicts safety behavior more reliably than absolute displacement metrics, establishing geometry-based monitoring as a necessary component of guard model evaluation in agentic deployments.

Ismail Hossain, Sai Puppala, Jannatul Ferdaus +4

Department of Computer Science, University of Texas at El Paso, TX, USA 79902 · School of Computing, Southern Illinois University Carbondale, IL, USA 62901 · Hanyang University, Seoul, South Korea +1

LLM GuardrailsLLM Safety AlignmentLanguage Model Safety EvaluationRepresentation Geometry in Language Models
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Apr 8, 2026

Latent Structure of Affective Representations in Large Language Models

The geometric structure of latent representations in large language models (LLMs) is an active area of research, driven in part by its implications for model transparency and AI safety. Existing literature has focused mainly on general geometric and topological properties of the learnt representations, but due to a lack of ground-truth latent geometry, validating the findings of such approaches is challenging. Emotion processing provides an intriguing testbed for probing representational geometry, as emotions exhibit both categorical organization and continuous affective dimensions, which are well-established in the psychology literature. Moreover, understanding such representations carries safety relevance. In this work, we investigate the latent structure of affective representations in LLMs using geometric data analysis tools. We present three main findings. First, we show that LLMs learn coherent latent representations of affective emotions that align with widely used valence--arousal models from psychology. Second, we find that these representations exhibit nonlinear geometric structure that can nonetheless be well-approximated linearly, providing empirical support for the linear representation hypothesis commonly assumed in model transparency methods. Third, we demonstrate that the learned latent representation space can be leveraged to quantify uncertainty in emotion processing tasks. Our findings suggest that LLMs acquire affective representations with geometric structure paralleling established models of human emotion, with practical implications for model interpretability and safety.

Benjamin J. Choi, Melanie Weber

Harvard University

Emotion Representation in Language ModelsRepresentation Geometry in Language ModelsLLM Interpretability
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Apr 7, 2026

The Illusion of Superposition? A Principled Analysis of Latent Thinking in Language Models

Latent reasoning via continuous chain-of-thoughts (Latent CoT) has emerged as a promising alternative to discrete CoT reasoning. Operating in continuous space increases expressivity and has been hypothesized to enable superposition: the ability to maintain multiple candidate solutions simultaneously within a single representation. Despite theoretical arguments, it remains unclear whether language models actually leverage superposition when reasoning using latent CoTs. We investigate this question across three regimes: a training-free regime that constructs latent thoughts as convex combinations of token embeddings, a fine-tuned regime where a base model is adapted to produce latent thoughts, and a from-scratch regime where a model is trained entirely with latent thoughts to solve a given task. Using Logit Lens and entity-level probing to analyze internal representations, we find that only models trained from scratch exhibit signs of using superposition. In the training-free and fine-tuned regimes, we find that the superposition either collapses or is not used at all, with models discovering shortcut solutions instead. We argue that this is due to two complementary phenomena: i) pretraining on natural language data biases models to commit to a token in the last layers ii) capacity has a huge effect on which solutions a model favors. Together, our results offer a unified explanation for when and why superposition arises in continuous chain-of-thought reasoning, and identify the conditions under which it collapses.

Michael Rizvi-Martel, Guillaume Rabusseau, Marius Mosbach

Mila – Quebec AI Institute · Universit´e de Montr´eal · McGill University

Continuous Latent ReasoningLLM InterpretabilityRepresentation Geometry in Language Models
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Apr 7, 2026

Toward Consistent World Models with Multi-Token Prediction and Latent Semantic Enhancement

Whether Large Language Models (LLMs) develop coherent internal world models remains a core debate. While conventional Next-Token Prediction (NTP) focuses on one-step-ahead supervision, Multi-Token Prediction (MTP) has shown promise in learning more structured representations. In this work, we provide a theoretical perspective analyzing the gradient inductive bias of MTP, supported by empirical evidence, showing that MTP promotes the convergence toward internal belief states by inducing representational contractivity via gradient coupling. However, we reveal that standard MTP often suffers from structural hallucinations, where discrete token supervision encourages illegal shortcuts in latent space that violate environmental constraints. To address this, we propose a novel method Latent Semantic Enhancement MTP (LSE-MTP), which anchors predictions to ground-truth hidden state trajectories. Experiments on synthetic graphs and real-world Manhattan Taxi Ride show that LSE-MTP effectively bridges the gap between discrete tokens and continuous state representations, enhancing representation alignment, reducing structural hallucinations, and improving robustness to perturbations.

Qimin Zhong, Hao Liao, Haiming Qin +4

Shenzhen University · Microsoft Research Asia

Multi-Token PredictionLatent World ModelsHallucination in Language ModelsRepresentation Geometry in Language Models
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Apr 7, 2026

LAG-XAI: A Lie-Inspired Affine Geometric Framework for Interpretable Paraphrasing in Transformer Latent Spaces

Modern Transformer-based language models achieve strong performance in natural language processing tasks, yet their latent semantic spaces remain largely uninterpretable black boxes. This paper introduces LAG-XAI (Lie Affine Geometry for Explainable AI), a novel geometric framework that models paraphrasing not as discrete word substitutions, but as a structured affine transformation within the embedding space. By conceptualizing paraphrasing as a continuous geometric flow on a semantic manifold, we propose a computationally efficient mean-field approximation, inspired by local Lie group actions. This allows us to decompose paraphrase transitions into geometrically interpretable components: rotation, deformation, and translation. Experiments on the noisy PIT-2015 Twitter corpus, encoded with Sentence-BERT, reveal a "linear transparency" phenomenon. The proposed affine operator achieves an AUC of 0.7713. By normalizing against random chance (AUC 0.5), the model captures approximately 80% of the non-linear baseline's effective classification capacity (AUC 0.8405), offering explicit parametric interpretability in exchange for a marginal drop in absolute accuracy. The model identifies fundamental geometric invariants, including a stable matrix reconfiguration angle (~27.84°) and near-zero deformation, indicating local isometry. Cross-domain generalization is confirmed via direct cross-corpus validation on an independent TURL dataset. Furthermore, the practical utility of LAG-XAI is demonstrated in LLM hallucination detection: using a "cheap geometric check," the model automatically detected 95.3% of factual distortions on the HaluEval dataset by registering deviations beyond the permissible semantic corridor. This approach provides a mathematically grounded, resource-efficient path toward the mechanistic interpretability of Transformers.

Olexander Mazurets, Olexander Barmak, Leonid Bedratyuk +1

Department of Computer Science, Khmelnytskyi National University, Khmelnytskyi, 29016, Ukraine · Department of Theoretical Cybernetics, Taras Shevchenko National University of Kyiv, Kyiv, 03187, Ukraine · Laboratory of Communicative Information Technologies, V.M. Glushkov Institute of Cybernetics of NAS of Ukraine, Kyiv, 03187, Ukraine

Transformer InterpretabilityRepresentation Geometry in Language Models
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Apr 7, 2026

LLM Reasoning as Trajectories: Step-Specific Representation Geometry and Correctness Signals

This work characterizes large language models' chain-of-thought generation as a structured trajectory through representation space. We show that mathematical reasoning traverses functionally ordered, step-specific subspaces that become increasingly separable with layer depth. This structure already exists in base models, while reasoning training primarily accelerates convergence toward termination-related subspaces rather than introducing new representational organization. While early reasoning steps follow similar trajectories, correct and incorrect solutions diverge systematically at late stages. This late-stage divergence enables mid-reasoning prediction of final-answer correctness with ROC-AUC up to 0.87. Furthermore, we introduce trajectory-based steering, an inference-time intervention framework that enables reasoning correction and length control based on derived ideal trajectories. Together, these results establish reasoning trajectories as a geometric lens for interpreting, predicting, and controlling LLM reasoning behavior.

Lihao Sun, Hang Dong, Bo Qiao +3

Microsoft

LLM ReasoningRepresentation Geometry in Language ModelsLanguage Model Steering
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Apr 7, 2026

Turbulence-like 5/3 spectral scaling in contextual representations of language as a complex system

Natural language is a complex system that exhibits robust statistical regularities. Here, we represent text as a trajectory in a high-dimensional embedding space generated by transformer-based language models, and quantify scale-dependent fluctuations along the token sequence using an embedding-step signal. Across multiple languages and corpora, the resulting power spectrum exhibits a robust power law with an exponent close to 5/35/3 over an extended frequency range. This scaling is observed consistently in contextual embeddings from both human-written and AI-generated text, but is absent in static word embeddings and is disrupted by randomization of token order. These results show that the observed scaling reflects multiscale, context-dependent organization rather than lexical statistics alone. By analogy with the Kolmogorov spectrum in turbulence, our findings suggest that semantic information is integrated in a scale-free, self-similar manner across linguistic scales, and provide a quantitative, model-agnostic benchmark for studying complex structure in language representations.

Zhongxin Yang, Chun Bao, Yuanwei Bin +2

College of Engineering, Peking University, Beijing, 100871, China · Shenzhen Tenfong Technology Co., Ltd., Shenzhen 518000, China · Mechanical Engineering, Pennsylvania State University, State College, PA, 16802, USA

Representation Geometry in Language Models
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Apr 6, 2026

On the Geometry of Positional Encodings in Transformers

Neural language models process sequences of words, but the mathematical operations inside them are insensitive to the order in which words appear. Positional encodings are the component added to remedy this. Despite their importance, positional encodings have been designed largely by trial and error, without a mathematical theory of what they ought to do. This paper develops such a theory. Four results are established. First, any Transformer without a positional signal cannot solve any task sensitive to word order (Necessity Theorem). Second, training assigns distinct vector representations to distinct sequence positions at every global minimiser, under mild and verifiable conditions (Positional Separation Theorem). Third, the best achievable approximation to an information-optimal encoding is constructed via classical multidimensional scaling (MDS) on the Hellinger distance between positional distributions; the quality of any encoding is measured by a single number, the stress (Proposition 5, Algorithm 1). Fourth, the optimal encoding has effective rank r = rank(B) <= n-1 and can be represented with r(n+d) parameters instead of nd (minimal parametrisation result). Appendix A develops a proof of the Monotonicity Conjecture within the Neural Tangent Kernel (NTK) regime for masked language modelling (MLM) losses, sequence classification losses, and general losses satisfying a positional sufficiency condition, through five lemmas. Experiments on SST-2 and IMDB with BERT-base confirm the theoretical predictions and reveal that Attention with Linear Biases (ALiBi) achieves much lower stress than the sinusoidal encoding and Rotary Position Embedding (RoPE), consistent with a rank-1 interpretation of the MDS encoding under approximate shift-equivariance.

Giansalvo Cirrincione

Laboratoire LTI, Université de Picardie Jules Verne Chemin du Thil, 80025 Amiens, France

Multidimensional ScalingPositional EncodingRepresentation Geometry in Language Models
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Apr 6, 2026

Multilingual Language Models Encode Script Over Linguistic Structure

Multilingual language models (LMs) organize representations for typologically and orthographically diverse languages into a shared parameter space, yet the nature of this internal organization remains elusive. In this work, we investigate which linguistic properties - abstract language identity or surface-form cues - shape multilingual representations. To do so, we analyze language-associated units across different model families and scales using the Language Activation Probability Entropy (LAPE) metric, and further decompose activations with Sparse Autoencoders. We find that these units are strongly conditioned on orthography: romanization induces near-disjoint representations that align with neither native-script inputs nor English, while word-order shuffling has limited effect on unit identity. Probing shows that typological structure becomes increasingly accessible in deeper layers, while causal interventions indicate that generation is most sensitive to units that are invariant to surface-form perturbations rather than to units identified by typological alignment alone. Overall, our results suggest that multilingual LMs organize representations around surface form, with linguistic abstraction emerging gradually without collapsing into a unified interlingua.

Aastha A K Verma, Anwoy Chatterjee, Mehak Gupta +1

Indian Institute of Technology Delhi, New Delhi, India · Indian Institute of Technology Delhi, Abu Dhabi, UAE

Multilingual Language ModelsCross-Lingual Representation AlignmentRepresentation Geometry in Language Models
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Apr 6, 2026

Under Pressure: Emotional Framing Induces Measurable Behavioral Shifts and Structured Internal Geometry in Small Language Models

I study whether emotionally framed evaluation follow-ups change both the behavior and the calm-relative internal representations of small, locally deployed language models. Our main benchmark uses Qwen 3.5 0.8B on four impossible-constraint coding tasks and eight follow-up framings: calm, pressure, urgency, approval, shame, curiosity, encouragement, and threat. In the 0.8B eight-condition sweep (160 conversations), pressure produces the strongest shortcut markers (11/20 runs) and the clearest overfit pattern (3/20), while calm and curiosity preserve explicit honesty more often (7/20 and 6/20). For all seven non-baseline conditions, the corresponding calm-relative direction vectors peak at the final transformer layer. An exploratory PCA of the layer-23 direction vectors reveals a dominant first component (59.5% explained variance) aligned with a hand-labeled positive/negative split (cosine alignment 0.951); approval and urgency are nearly identical internally (cosine 0.957), whereas curiosity points away from urgency (-0.252). In a separate calm-vs.-pressure rerun used for scale comparison, Qwen 3.5 2B shows higher honest rates under calm framing and directionally consistent activation steering on a small 4-prompt A/B probe, whereas the 0.8B steering result reverses. I interpret these results as evidence for measurable prompt-sensitive control directions in small open models, while stopping short of claiming intrinsic emotional states.

Rana Muhammad Usman

Independent Researcher

Framing Effects in Language ModelsEmotion Representation in Language ModelsRepresentation Geometry in Language ModelsActivation SteeringLLM Evaluation
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Apr 6, 2026

Hallucination Basins: A Dynamic Framework for Understanding and Controlling LLM Hallucinations

Large language models (LLMs) hallucinate: they produce fluent outputs that are factually incorrect. We present a geometric dynamical systems framework in which hallucinations arise from task-dependent basin structure in latent space. Using autoregressive hidden-state trajectories across multiple open-source models and benchmarks, we find that separability is strongly task-dependent rather than universal: factoid settings can show clearer basin separation, whereas summarization and misconception-heavy settings are typically less stable and often overlap. We formalize this behavior with task-complexity and multi-basin theorems, characterize basin emergence in L-layer transformers, and show that geometry-aware steering can reduce hallucination probability without retraining.

Kalyan Cherukuri, Lav R. Varshney

Illinois Mathematics and Science Academy, Illinois, USA · AI Innovation Institute, Stony Brook University, New York, USA

Hallucination in Language ModelsLLM Hallucination MitigationRepresentation Geometry in Language Models
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Apr 6, 2026

Same Geometry, Opposite Noise: Transformer Magnitude Representations Lack Scalar Variability

Scalar variability -- the finding that representational noise scales proportionally with magnitude, producing a constant coefficient of variation -- is a hallmark of biological magnitude systems. We tested whether transformer language models exhibit this property by analysing the dispersion of hidden-state representations across carrier sentences for 26 numerical magnitudes in three 7-8B parameter models (Llama-3-8B-Instruct, Mistral-7B-Instruct-v0.3, Llama-3-8B-Base; data from Cacioli, 2026). We found the opposite: representational variability decreased with magnitude along the magnitude axis (scaling exponent alpha approx -0.19; 0/16 primary layers with alpha > 0, all three models). The negative sign was consistent in full-dimensional space (alpha approx -0.04) and after sentence-identity correction (alpha approx -0.007). The anti-scalar pattern was 3-5x stronger along the magnitude axis than orthogonal dimensions, and corpus frequency strongly predicted per-magnitude variability (rho = .84). These results demonstrate that distributional learning alone is insufficient to produce scalar variability: transformers reproduce log-compressive magnitude geometry but not the constant-CV noise signature observed in biological systems.

Jon-Paul Cacioli

Independent Researcher, Melbourne, Australia

Representation Geometry in Language Models
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Apr 6, 2026

On the Invariants of Softmax Attention

Softmax attention maps every query--key interaction into a probability distribution, but the underlying structure remains largely unexplored. We define the energy field, the row-centered attention logit, and show that it exhibits invariant properties across models, architectures, and inputs. Two classes of invariants emerge. Mechanism-level invariants follow from the algebraic structure of softmax attention. They include a per-row zero-sum constraint, a rank bound determined by the head dimension, and spectral signatures that follow from them. Model-level regularities are not required by the mechanism, yet hold in every autoregressive language model we test, spanning several architecture families. The energy field distributes its variance over key positions without concentrating at a few. This delocalization traces to a property of the key matrix we call key incoherence. These invariants have practical consequences. The rank bound confines the energy field to a low-dimensional subspace. Key incoherence yields a per-head training monitor. All results are verified at multiple context lengths and input texts.

Wonsuk Lee

Seoul National University & SK Hynix America

Softmax AttentionAttention Head AnalysisRepresentation Geometry in Language Models
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Apr 3, 2026

Measuring Representation Robustness in Large Language Models for Geometry

Large language models (LLMs) are increasingly evaluated on mathematical reasoning, yet their robustness to equivalent problem representations remains poorly understood. In geometry, identical problems can be expressed in Euclidean, coordinate, or vector forms, but existing benchmarks report accuracy on fixed formats, implicitly assuming representation invariance and masking failures caused by representational changes alone. We propose GeoRepEval, a representation-aware evaluation framework that measures correctness, invariance, and consistency at the problem level across parallel formulations, combining strict answer matching, bootstrap confidence intervals, paired McNemar tests, representation-flip analyses, and regression controls for surface complexity. We prove that our Invariance@3 metric decomposes accuracy into robust and fragile components and is bounded by the weakest representation. Evaluating eleven LLMs on 158 curated high-school geometry problems (474 instances), we find accuracy gaps of up to 14 percentage points induced solely by representation choice. Vector formulations emerge as a consistent failure point, with Invariance@3 as low as 0.044 even after controlling for length and symbolic complexity. A convert-then-solve prompting intervention improves vector accuracy by up to 52 percentage points for high-capacity models, suggesting that failures reflect representation sensitivity rather than inability; however, low-capacity models show no gains, indicating deeper limitations. These results suggest that current models rely on representation-specific heuristics rather than abstract geometric reasoning. All datasets, prompts, and scripts are released at https://github.com/vedjaw/GeoRepEval.

Vedant Jawandhia, Yash Sinha, Murari Mandal +2

Department of Computer Science and Information Systems, BITS Pilani · School of Computer Science, KIIT University · Department of Mathematics, BITS Pilani

LLM EvaluationRepresentation Geometry in Language ModelsInvariant Representation Learning
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Apr 3, 2026

The Spectral Geometry of Thought: Phase Transitions, Instruction Reversal, Token-Level Dynamics, and Perfect Correctness Prediction in How Transformers Reason

We discover that large language models exhibit spectral phase transitions in their hidden activation spaces when engaging in reasoning versus factual recall. Through systematic spectral analysis across 11 models spanning 5 architecture families (Qwen, Pythia, Phi, Llama, DeepSeek-R1), we identify seven core phenomena: (1)Reasoning Spectral Compression -- 9/11 models show significantly lower αα for reasoning (p<0.05p < 0.05), with larger effects in stronger models; (2)Instruction Tuning Spectral Reversal -- base models show reasoning α<α< factual αα, while instruction-tuned models reverse this relationship; (3)Architecture-Dependent Generation Taxonomy -- prompt-to-response shifts partition into expansion, compression, and equilibrium regimes; (4)Spectral Scaling Law -- αreasoning∝−0.074ln⁡Nα_\text{reasoning} \propto -0.074 \ln N across 4 Qwen base models (R2=0.46R^2 = 0.46); (5)Token-Level Spectral Cascade -- per-token alpha tracking reveals local synchronization that decays exponentially with layer distance, and is weaker for reasoning than factual tasks; (6)Reasoning Step Spectral Punctuation -- phase-transition signatures align with reasoning step boundaries; and (7)~Spectral Correctness Prediction -- spectral αα alone achieves AUC =1.000= 1.000 (Qwen2.5-7B, late layers) and mean AUC =0.893= 0.893 across 6 models in predicting correctness before the final answer is generated. Together, these findings establish a comprehensive spectral theory of reasoning in transformers, revealing that the geometry of thought is universal in direction, architecture-specific in dynamics, and predictive of outcome.

Yi Liu

Representation Geometry in Language Models
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Apr 3, 2026

Finding Belief Geometries with Sparse Autoencoders

Understanding the geometric structure of internal representations is a central goal of mechanistic interpretability. Prior work has shown that transformers trained on sequences generated by hidden Markov models encode probabilistic belief states as simplex-shaped geometries in their residual stream, with vertices corresponding to latent generative states. Whether large language models trained on naturalistic text develop analogous geometric representations remains an open question. We introduce a pipeline for discovering candidate simplex-structured subspaces in transformer representations, combining sparse autoencoders (SAEs), kk-subspace clustering of SAE features, and simplex fitting using AANet. We validate the pipeline on a transformer trained on a multipartite hidden Markov model with known belief-state geometry. Applied to Gemma-2-9B, we identify 13 priority clusters exhibiting candidate simplex geometry (K≥3K \geq 3). A key challenge is distinguishing genuine belief-state encoding from tiling artifacts: latents can span a simplex-shaped subspace without the mixture coordinates carrying predictive signal beyond any individual feature. We therefore adopt barycentric prediction as our primary discriminating test. Among the 13 priority clusters, 3 exhibit a highly significant advantage on near-vertex samples (Wilcoxon p<10−14p < 10^{-14}) and 4 on simplex-interior samples. Together 5 distinct real clusters pass at least one split, while no null cluster passes either. One cluster, 768_596, additionally achieves the highest causal steering score in the dataset. This is the only case where passive prediction and active intervention converge. We present these findings as preliminary evidence that genuine belief-like geometry exists in Gemma-2-9B's representation space, and identify the structured evaluation that would be required to confirm this interpretation.

Matthew Levinson

Independent Researcher

Representation Geometry in Language ModelsTransformer Interpretability
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Apr 2, 2026

On the Geometric Structure of Layer Updates in Deep Language Models

We study the geometric structure of layer updates in deep language models. Rather than analyzing what information is encoded in intermediate representations, we ask how representations change from one layer to the next. We show that layerwise updates admit a decomposition into a dominant tokenwise component and a residual that is not captured by restricted tokenwise function classes. Across multiple architectures, including Transformers and state-space models, we find that the full layer update is almost perfectly aligned with the tokenwise component, while the residual exhibits substantially weaker alignment, larger angular deviation, and significantly lower projection onto the dominant tokenwise subspace. This indicates that the residual is not merely a small correction, but a geometrically distinct component of the transformation. This geometric separation has functional consequences: approximation error under the restricted tokenwise model is strongly associated with output perturbation, with Spearman correlations often exceeding 0.7 and reaching up to 0.95 in larger models. Together, these results suggest that most layerwise updates behave like structured reparameterizations along a dominant direction, while functionally significant computation is concentrated in a geometrically distinct residual component. Our framework provides a simple, architecture-agnostic method for probing the geometric and functional structure of layer updates in modern language models.

Jun-Sik Yoo

Institute of Basic Science, Korea University, 145 Anam-ro, Seoul, South Korea 02841.

Representation Geometry in Language ModelsLLM Interpretability
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Apr 2, 2026

Grounded Token Initialization for New Vocabulary in LMs for Generative Recommendation

Language models (LMs) are increasingly extended with new learnable vocabulary tokens for domain-specific tasks, such as Semantic-ID tokens in generative recommendation. The standard practice initializes these new tokens as the mean of existing vocabulary embeddings, then relies on supervised fine-tuning to learn their representations. We present a systematic analysis of this strategy: through spectral and geometric diagnostics, we show that mean initialization collapses all new tokens into a degenerate subspace, erasing inter-token distinctions that subsequent fine-tuning struggles to fully recover. These findings suggest that token initialization is a key bottleneck when extending LMs with new vocabularies. Motivated by this diagnosis, we propose the Grounded Token Initialization Hypothesis: linguistically grounding novel tokens in the pretrained embedding space before fine-tuning better enables the model to leverage its general-purpose knowledge for novel-token domains. We operationalize this hypothesis as GTI (Grounded Token Initialization), a lightweight grounding stage that, prior to fine-tuning, maps new tokens to distinct, semantically meaningful locations in the pretrained embedding space using only paired linguistic supervision. Despite its simplicity, GTI outperforms both mean initialization and existing auxiliary-task adaptation methods in the majority of evaluation settings across multiple generative recommendation benchmarks, including industry-scale and public datasets. Further analyses show that grounded embeddings produce richer inter-token structure that persists through fine-tuning, corroborating the hypothesis that initialization quality is a key bottleneck in vocabulary extension.

Daiwei Chen, Zhoutong Fu, Chengming Jiang +12

University of Wisconsin-Madison · LinkedIn Corporation · Northeastern University +1

Language Model Vocabulary ExpansionLarge Language Model-Based RecommendationGenerative RecommendationRepresentation Geometry in Language Models
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Apr 1, 2026

In-context superposition: human-like working memory interference in large language models

Intelligent systems must maintain and manipulate task-relevant information online to adapt to dynamic environments. This capacity, known as working memory, is fundamental to human reasoning. Yet, human working memory is strikingly limited, maintaining only three to four items in a brain with billions of neurons. Surprisingly, large language models (LLMs), despite different substrates and direct access to prior context through attention, exhibit similar working memory limitations. Why should such different systems face analogous constraints? We propose that working memory limitations reflect a general trade-off of shared representations: representational compression and reuse support efficient learning and generalization, but also cause simultaneously active representations to interfere. We show a two-layer transformer trained on a working memory task can solve it perfectly, but diverse trained LLMs exhibit human-like limitations: performance declines with memory load, while retrieval is biased by recency and stimulus statistics. Mirroring humans, working memory performance in LLMs is also associated with broader model capability. Mechanistically, we show that LLMs encode multiple memories in entangled representations --- a condition we call in-context superposition --- and progressively suppress competing content while aligning the target with the readout. Moreover, a causal intervention that suppresses interfering information improves performance. Together, these findings suggest that working memory capacity reflects the ability to select task-relevant information under interference, a computational challenge shared by biological and artificial systems.

Hua-Dong Xiong, Li Ji-An, Jiaqi Huang +3

School of Psychological and Brain Sciences, Georgia Tech · Department of Psychology, New York University · Department of Cognitive Science, Indiana University Bloomington +3

Working Memory in Language ModelsLLM InterpretabilityRepresentation Geometry in Language Models
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Mar 31, 2026

What Cosine Similarity of Label Representations Can and Cannot Tell us

Cosine similarity is often used to measure the similarity of vector representations of neural network models. However, the cosine similarity of representations is not guaranteed to tell us anything about model probabilities. In this paper we show that for a softmax classifier, be it an image classifier or an autoregressive language model, the cosine similarity between label representations (called unembeddings in the paper) does not give any information on the probabilities assigned by the model. Specifically, we prove that given two unembeddings, it is possible to create another model which assigns the same probabilities for all inputs, but where the cosine similarity between the representations is now either 1 or -1. We also show that for a sigmoid classifier (where each input can be assigned multiple labels), all pairwise cosine similarities between the unembeddings define the set of possible label combinations. However, for softmax classifiers (where each input is assigned a ranking of the labels from most to least likely), we need all pairwise cosine similarities between all differences of unembeddings to know which rankings the model can predict. We conclude that it is misleading to interpret the cosine similarity between unembeddings without reference to the classifier that produced them.

Beatrix M. G. Nielsen, Andreas Grivas

IT University of Copenhagen · School of Mathematics, University of Edinburgh

Representation Geometry in Language Models
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Mar 30, 2026

Categorical Perception in Large Language Model Hidden States: Structural Warping at Digit-Count Boundaries

Categorical perception (CP) -- enhanced discriminability at category boundaries -- is among the most studied phenomena in perceptual psychology. This paper reports that analogous geometric warping occurs in the hidden-state representations of large language models (LLMs) processing Arabic numerals. Using representational similarity analysis across six models from five architecture families, the study finds that a CP-additive model (log-distance plus a boundary boost) fits the representational geometry better than a purely continuous model at 100% of primary layers in every model tested. The effect is specific to structurally defined boundaries (digit-count transitions at 10 and 100), absent at non-boundary control positions, and absent in the temperature domain where linguistic categories (hot/cold) lack a tokenisation discontinuity. Two qualitatively distinct signatures emerge: "classic CP" (Gemma, Qwen), where models both categorise explicitly and show geometric warping, and "structural CP" (Llama, Mistral, Phi), where geometry warps at the boundary but models cannot report the category distinction. This dissociation is stable across boundaries and is a property of the architecture, not the stimulus. Structural input-format discontinuities are sufficient to produce categorical perception geometry in LLMs, independently of explicit semantic category knowledge.

Jon-Paul Cacioli

Independent Researcher, Melbourne, Australia

Tokenization in Language ModelsRepresentation Geometry in Language ModelsLanguage Modeling
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Mar 27, 2026

Weight Tying Biases Token Embeddings Towards the Output Space

Weight tying, i.e. sharing parameters between input and output embedding matrices, is common practice in language model design, yet its impact on the learned embedding space remains poorly understood. In this paper, we show that tied embedding matrices align more closely with output (unembedding) matrices than with input embeddings of comparable untied models, indicating that the shared matrix is shaped primarily for output prediction rather than input representation. This unembedding bias arises because output gradients dominate early in training. Using tuned lens analysis, we show this negatively affects early-layer computations, which contribute less effectively to the residual stream. Scaling input gradients during training reduces this bias, providing causal evidence for the role of gradient imbalance. This is mechanistic evidence that weight tying optimizes the embedding matrix for output prediction, compromising its role in input representation. These results help explain why weight tying can harm performance at scale and have implications for training smaller LLMs, where the embedding matrix contributes substantially to total parameter count.

Antonio Lopardo, Avyukth Harish, Catherine Arnett +1

EleutherAI · University of California Berkeley

Language ModelingRepresentation Geometry in Language ModelsLLM Interpretability
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Mar 26, 2026

How Pruning Reshapes Features: Sparse Autoencoder Analysis of Weight-Pruned Language Models

Weight pruning is a standard technique for compressing large language models, yet its effect on learned internal representations remains poorly understood. We present the first systematic study of how unstructured pruning reshapes the feature geometry of language models, using Sparse Autoencoders (SAEs) as interpretability probes. Across three model families (Gemma 3 1B, Gemma 2 2B, Llama 3.2 1B), two pruning methods (magnitude and Wanda), and six sparsity levels (0--60%), we investigate five research questions spanning seed stability, feature survival, SAE transferability, feature fragility, and causal relevance. Our most striking finding is that rare SAE features--those with low firing rates--survive pruning far better than frequent ones, with within-condition Spearman correlations of rho = -1.0 in 11 of 17 experimental conditions. This counter-intuitive result suggests that pruning acts as implicit feature selection, preferentially destroying high-frequency generic features while preserving specialized rare ones. We further show that Wanda pruning preserves feature structure up to 3.7x better than magnitude pruning, that pre-trained SAEs remain viable on Wanda-pruned models up to 50% sparsity, and that geometric feature survival does not predict causal importance--a dissociation with implications for interpretability under compression.

Hector Borobia, Elies Seguí-Mas, Guillermina Tormo-Carbó

VRAIN – Valencian Research Institute for Artificial Intelligence, Universitat Politècnica de València, Valencia, Spain · Department of Economics and Social Sciences, Universitat Politècnica de València, Valencia, Spain · Department of Business Organisation, Universitat Politècnica de València, Valencia, Spain

LLM PruningRepresentation Geometry in Language ModelsLLM Compression
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Mar 25, 2026

A visual observation on the geometry of UMAP projections of the difference vectors of antonym and synonym word pair embeddings

Antonyms, or opposites, are sometimes defined as word pairs that have all of the same contextually relevant properties but one. Seeing how transformer models seem to encode concepts as directions, this begs the question if one can detect antonymity'' in the geometry of the embedding vectors of word pairs, especially based on their difference vectors. Such geometrical studies are then naturally contrasted by comparing antonymic pairs to their opposites; synonyms. This paper started as an exploratory project on the complexity of the systems needed to detect the geometry of the embedding vectors of antonymic word pairs. What we now report is a curious swirl'' that appears across embedding models in a somewhat specific projection configuration.

Rami Luisto

Faculty of Information Technology, University of Jyväskylä, Jyväskylä; HUS Helsinki University Hospital, Helsinki; Digital Workforce Services, Helsinki.

Representation Geometry in Language ModelsWord Embeddings
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Mar 24, 2026

Steering Code LLMs with Activation Directions for Language and Library Control

Code LLMs often default to particular programming languages and libraries under neutral prompts. We investigate whether these preferences are encoded as approximately linear directions in activation space that can be manipulated at inference time. Using a difference-in-means method, we estimate layer-wise steering vectors for five language/library pairs and add them to model hidden states during generation. Across three open-weight code LLMs, these interventions substantially increase generation toward the target ecosystem under neutral prompts and often remain effective even when prompts explicitly request the opposite choice. Steering strength varies by model and target, with common ecosystems easier to induce than rarer alternatives, and overly strong interventions can reduce output quality. Overall, our results suggest that code-style preferences in LLMs are partly represented by compact, steerable structure in activation space.

Md Mahbubur Rahman, Arjun Guha, Harshitha Menon

Iowa State University · Northeastern University · Lawrence Livermore National Laboratory

Language Model SteeringCode Language ModelsActivation SteeringCode GenerationRepresentation Geometry in Language Models
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Mar 24, 2026

The Geometric Price of Discrete Logic: Context-driven Manifold Dynamics of Number Representations

Large language models (LLMs) generalize smoothly across continuous semantic spaces, yet strict logical reasoning demands the formation of discrete decision boundaries. Prevailing theories relying on linear isometric projections fail to resolve this fundamental tension. In this work, we argue that task context operates as a non-isometric dynamical operator that enforces a necessary "topological distortion." By applying Gram-Schmidt decomposition to residual-stream activations , we reveal a dual-modulation mechanism driving this process: a class-agnostic topological preservation that anchors global structure to prevent semantic collapse, and a specific algebraic divergence that directionally tears apart cross-class concepts to forge logical boundaries. We validate this geometric evolution across a gradient of tasks, from simple mapping to complex primality testing. Crucially, targeted specific vector ablation establishes a strict causal binding between this topology and model function: algebraically erasing the divergence component collapses parity classification accuracy from 100% to chance levels (38.57%). Furthermore, we uncover a three-phase layer-wise geometric dynamic and demonstrate that under social pressure prompts, models fail to generate sufficient divergence. This results in a "manifold entanglement" that geometrically explains sycophancy and hallucination. Ultimately, our findings revise the linear-isometric presumption, demonstrating that the emergence of discrete logic in LLMs is purchased at an irreducible cost of topological deformation.

Long Zhang, Dai-jun Lin, Wei-neng Chen

School of Computer Science and Engineering, South China University of Technology, Guangzhou City, Guangdong Province, China

Representation Geometry in Language ModelsLLM InterpretabilityNumerical Reasoning in Language Models
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Mar 24, 2026

Improving LLM Final Representations with Inter-Layer Geometry

The standard in LLM-based prediction is to use the final-layer representation as the input to a downstream predictor. However, intermediate layers may encode complementary task-relevant signals. Existing approaches therefore either search for the best layer for each task or apply expensive attention-based mechanisms to learn inter-layer aggregation. In this work, we first show that such complexity is unnecessary: a lightweight Graph Neural Network over a fully connected graph of LLM layers is more efficient and achieves significantly stronger predictive performance than existing approaches. We then introduce the Cayley-Encoder, which further improves both efficiency and predictive performance by replacing the fully connected graph with a Cayley graph over SL(2, Zn). These Cayley graphs provide a mathematically grounded topology that is sparse, regular by construction, and has low diameter. This enables effective communication across layers while constraining the aggregation structure and reducing the risk of GNN overfitting. In an evaluation of Cayley-Encoder across 13 tasks and 9 LLMs, Cayley-Encoder consistently outperforms baselines, achieving improvements of up to 40 percentage points in accuracy, while introducing at most 0.1% additional parameters relative to the LLM size. We further show that Cayley-Encoder is effective in few-shot regimes. Finally, we show that Cayley-Encoder outperforms LoRA fine-tuning while operating on the frozen LLM. We conclude with an explainability analysis showing that multiple layers contribute meaningfully to the final prediction, supporting our hypothesis.

Tom Ulanovski, Eyal Blyachman, Maya Bechler-Speicher

Blavatnik School of Computer Science, Tel Aviv University · Tel Aviv University · Meta AI

Graph Neural NetworksRepresentation Geometry in Language Models
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Mar 23, 2026

Riding Brainwaves in LLM Space: Understanding Activation Patterns Using Individual Neural Signatures

Consumer-grade EEG is entering everyday devices, from earbuds to headbands, raising the question of whether language models can be adapted to individual neural responses. We test this by asking whether frozen LLM representations encode person-specific EEG signals, directions in activation space that predict one person's brain activity but not another's. Using word-level EEG from 30 participants reading naturalistic sentences (ZuCo corpus), we train a separate linear probe for each person, mapping hidden states from a frozen Qwen 2.5 7B to that individual's EEG power. Person-specific probes outperform a single population probe on every EEG feature tested; for high-gamma power, the person-specific probe achieves rho = 0.183, a ninefold improvement over the population probe (rho = 0.020, p < 10^-4). A negative control, fixation count, shows no person-specific advantage (p = 0.360); fixation count reflects word length and frequency rather than individual cognition. The individual directions are temporally stable (split-half cosine = 0.824), non-transferable across people (self rho = 0.369 vs. other rho = 0.143, p < 10^-19), and distinct from the shared population signal: person-specific probes retain predictive power after the population component is removed. The person-specific signal concentrates in the model's deep layers, rising consistently with depth and peaking at Layer 24 of 28. The results are consistent across architectures (LLaMA 3.1 8B) and survive word-level confound controls. Frozen language models contain stable, person-specific neural directions in their deep layers, providing a geometric foundation for EEG-driven personalization.

Ajan Subramanian, Sumukh Bettadapura, Rohan Sathish

EEG DecodingLanguage Model ProbingRepresentation Geometry in Language Models
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Mar 22, 2026

Pooling and Semantic Shift: The Fundamental Challenges in Long Text Embedding and Retrieval

Transformer-based embedding models frequently exhibit geometric pathologies, such as anisotropy and length-induced representation collapse, which can degrade downstream retrieval performance. While prior work often attributes these issues directly to text length or attention mechanisms, we argue that the fundamental drivers are instead the inherent pooling operations coupled with internal semantic shift. In this paper, we establish a unified theoretical framework proving that contextual pooling intrinsically causes embedding collapse. Specifically, we mathematically prove that pooling semantically diverse sentences inevitably leads to micro-level semantic dilution, and strictly reduces the Mean Pairwise Distance of the vector space, guaranteeing macro-level spatial concentration. Grounded in these geometric insights, we formally define semantic shift to capture the natural semantic evolution and dispersion within a text. Through carefully controlled experiments across diverse models and corpora, we disentangle text length from semantic content. We demonstrate that semantic shift is the primary predictor of severe embedding concentration. Crucially, our retrieval evaluations reveal that anisotropy is fundamentally harmful only when induced by strong semantic shifts, reconciling conflicting observations in prior literature and offering a principled explanation for the long-context challenges faced by modern embedding models.

Hang Gao, Wujiang Xu, Kai Mei +1

Rutgers University

Text EmbeddingsText Embedding ModelsRepresentation Geometry in Language ModelsLong-Context RetrievalRepresentation Collapse
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