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.

Jul 2, 2026

Geometric Signatures of Reasoning: A Spectral Perspective on Task Hardness

Chain-of-thought (CoT) reasoning enables large language models (LLMs) to solve complex problems by generating intermediate reasoning steps. While much attention has been paid to the length and content of these reasoning chains, far less is known about their internal geometry. We study the geometry of CoT trajectories in the hidden state space of transformer models, formalizing each reasoning chain as a discrete curve in Rd\mathbb{R}^d and characterizing it through spectral, positional, and kinematic geometric functionals. We introduce the effective dimension dρd_ρ as a measure of trajectory complexity and show theoretically that trajectories with flatter eigenvalue spectra correspond to harder tasks, as they explore more of the hidden dimensions. Lastly, we explore how kinematic features of the trajectory, mean position, positional dispersion, initial and current hidden states, mean velocity, mean speed, and speed dispersion, can be used to predict solution correctness before generation is complete, and may inform future early-stopping strategies. Experimentally, on mathematical reasoning problems from the MATH500 dataset, dρd_ρ achieves 0.930.93 AUC in distinguishing easy from hard problems, while kinematic features potentially can predict correctness from only the first 20%20\% of generated tokens. These correctness signatures transfer across questions of varying difficulty, establishing that the shape of a model's internal reasoning trajectory is a principled window into both task hardness and solution quality.

Aria Masoomi, Mahsa Bazzaz, Adel Javanmard +1

1Northeastern University · University of Southern California · 3Google Research

LLM EvaluationRepresentation Geometry in Language ModelsLLM Mathematical ReasoningLLM ReasoningReasoning Trace Analysis
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Jul 1, 2026

Ember: A Kilobyte-State Optimizer for Token Interfaces

Language models learn continuous programs over discrete symbols, with the embedding table and LM-head acting as the read/write interface between them. We show that this interface has gradient geometry distinct from dense hidden weights which can be exploited to improve the Pareto frontier across supervised finetuning, RL, and pretraining, while only utilizing kilobytes of optimizer state. We introduce Ember, a lightweight optimizer for embedding and LM-head matrices that utilizes O(V+D)\mathcal{O}(V + D) optimizer VRAM, instead of Adam's O(2VD)\mathcal{O}(2VD), and forgoes the need to shard both token table optimizer states. On 50k-step pretraining across two datasets and three paired seeds, Ember lowers val loss by 0.08-0.09 nats relative to tuned AdamW; we also provide empirical evidence that Ember scales across batch size and parameter count. Finally, we provide a distributed Ember implementation that merges cleanly with existing ZeRO/FSDP setups.

Kathan Shah

Preprint.

Language Model PretrainingLLM Fine-TuningMemory-Efficient OptimizationEfficient Language Model TrainingRepresentation Geometry in Language Models
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Jul 1, 2026

A Geometric Perspective on Composable Emotion Steering in Text-to-Speech Models

While prior work has explored emotion control in hybrid text-to-speech systems, the geometric properties of these modules, and their implications for steerability, remain poorly understood. We present the first comparative study of speech language model (SLM) and conditional flow-matching (CFM) modules as activation steering sites for mixed emotion speech synthesis. We first characterize emotion representations using linear probing and local intrinsic dimensionality (LID), and then evaluate single-site and joint steering for mixed-emotion synthesis. Our results show that SLM offers a clean, low-dimensional emotion-specific subspace with strong speaker--emotion disentanglement, while CFM exhibitspoor cross-speaker generalization due to speaker--emotion entanglement. Joint steering increases emotion intensity but degrades proportional control and speech quality on in-distribution data. These findings provide practical guidance for multi-site activation steering in hybrid TTS systems and highlight the importance of representation geometry in controllable speech generation.

Siyi Wang, James Bailey, Ting Dang

The University of Melbourne, Australia · Monash University, Australia

Language Model SteeringConditional Flow MatchingControllable Speech GenerationSpeech Language ModelsRepresentation Geometry in Language ModelsEmotional Speech Synthesis
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Jul 1, 2026

NeuroCogMap Reveals Cognitive Organization of Large Language Models

Understanding how complex cognitive functions are organized within artificial systems is central to interpreting large language models (LLMs) and relating them to biological cognition. Yet although LLMs exhibit broad cognitive-like behaviours, it remains unclear whether their internal representations form reproducible functional systems that explain behaviour, failure and links to human cognition. Here we present NeuroCogMap, a cognitive neuroscience-inspired framework that organizes internal features of LLMs into functional parcels and links them to interpretable functions, cognitive capabilities and a cognitive hierarchy. These parcels form a stable and semantically coherent organization that is partly conserved across models and functionally linked to model outputs. Within this organization, major LLM failures, including hallucination, bias, refusal failure and sycophancy, correspond to distinct disruptions in representational and behavioural-control systems, yielding internal signatures for mechanism-guided detection and targeted intervention. Beyond model behaviour, NeuroCogMap improves prediction of human cortical responses during naturalistic language comprehension, with the strongest correspondence in higher-order association cortex. At the cognitive level, its internal signatures expose latent strategies that guide refinements of classical models of human decision-making. Together, these findings establish NeuroCogMap as a system-level framework for mapping functional organization in artificial systems and for relating this organization to human cortical function and cognitive behaviour.

Zhongxiang Sun, Haolang Lu, Qiang Ma +11

Gaoling School of Artificial Intelligence, Renmin University of China, Beijing, China · Beijing University of Posts and Telecommunications, Beijing, China · The University of Hong Kong, Hong Kong, China +5

Cognitive ModelingLLM InterpretabilityRepresentation Geometry in Language Models
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Jul 1, 2026

DiscoLoop: Looping Discrete Embeddings and Continuous Hidden States for Multi-hop Reasoning

Large language models achieve strong performance on many reasoning tasks when allowed to externalize intermediate steps as Chain-of-Thought (CoT). However, many questions require the model to internalize the multi-step reasoning within a single forward pass before generating the answer. We study this challenge through two-hop reasoning, a representative task where the model must compose multiple pieces of parametric knowledge within a single forward pass. Standard non-recurrent Transformers suffer from a depth-local storage problem: facts learned in earlier layers are unavailable where second-hop retrieval happens. We found that Looped Transformers mitigate this issue by reusing the same memory, but still generalize imperfectly. We show that the remaining bottleneck is representational. In the two-hop reasoning task, the first loop often makes the correct bridge entity nearly perfectly decodable, yet the corresponding hidden state remains poorly aligned with the bridge token embedding. Surprisingly, an easy training-free realignment intervention nearly closes the generalization gap. Building upon this insight, we propose DiscoLoop, a looping architecture whose recurrence carries both a discrete embedding channel and a continuous hidden-state channel. DiscoLoop achieves near-perfect accuracy with substantially fewer training steps across symbolic and synthetic-language multi-hop reasoning tasks. When applied to real-world pretraining, DiscoLoop attains lower training loss and stronger benchmark performance than looped-transformer baselines, suggesting that the mixed-channel design transfers to practical language modeling.

Hengyu Fu, Tianyu Guo, Zixuan Wang +5

University of California, Berkeley

Recurrent TransformersMulti-Hop ReasoningRepresentation Geometry in Language Models
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Jun 30, 2026

SemRF: A Semantic Reference Frame for Residual-Stream Dynamics in Language Models

Residual-stream analysis asks how language-model computation evolves across depth, but intermediate decoding requires comparable readout coordinates across layers. If embedding anchors and unembedding readout disagree on the chosen span, apparent motion may reflect measurement drift rather than computation. We introduce Semantic Reference Frames (SemRF), an anchor-based formalism separating semantic measurement from residual dynamics. A SemRF fixes anchors and measures states against them. Pseudo-inverse tying gives exact synchronization; under restricted bi-invertibility, SemRF yields stable semantic-basis coordinates, distortion bounds, and near-identity changes. With the frame fixed, residual computation becomes a depthwise semantic trajectory. The anchors induce a semantic Voronoi diagram: distance, or evidence such as logits, assigns each layer to a coarse cell, while coordinates retain within-cell motion and margins. We define layerwise steps, contribution profiles, and imbalance diagnostics, then use the Voronoi trace to define a margin-relaxed tube. The canonical trace is the minimum-action path inside this tube; when nonempty with positive quadratic weight, it is unique and obeys a discrete spline equation away from active constraints. Excess action controls step, curvature, and profile mismatch. Low curvature implies piecewise-linear compressibility and local knowledge density: lower trace complexity means fewer semantic knots. Through the parameter-to-trajectory map, this gives a conditional link to parameter efficiency: among admissible settings fitting data, lower-action and lower-complexity traces use fewer semantic degrees of freedom. The guarantees require controlled interface error and small projection residual under explicit tube constraints.

Jian Gu, Aldeida Aleti, Chunyang Chen +1

Monash University · Technical University of Munich · Chongqing University

Mechanistic InterpretabilityRepresentation Geometry in Language Models
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Jun 29, 2026

Estimating Grammatical Gender Directions in Contextual Embeddings under Controlled and Natural Contexts

Contextual language models conflate grammatical gender and social semantic bias in gendered languages such as Spanish. Existing gender debiasing approaches only operate on static word embeddings leaving contextual representations unexplored for this two dimensional gender disentanglement. To address the this issue, we make the first attempt to disentangle grammatical gender from semantic contamination for contextual embeddings. We construct both controlled templates and natural Wikipedia contexts to build balanced datasets of inanimate nouns, and design a framework equipped with centroid, Support Vector Machine (SVM) and Linear Discriminant Analysis (LDA) gender direction estimators as well as contamination-aware weighting strategies. A set of dual-objective evaluation metrics is proposed to balance the suppression of grammatical gender leakage on inanimate nouns and the preservation of semantic gender distinctions for occupation terms. The results reveal that unweighted controlled contexts yield the purest grammatical gender direction, and the centroid estimator achieves better performance than discriminative baselines.

Huanping Xiao, Yingji Li

College of Foreign Languages and Cultures, Jilin University, Changchun, Jilin Province, China · College of Computer Science and Technology, Jilin University, Changchun, Jilin Province, China

Gender Bias in Language ModelsRepresentation DisentanglementLanguage Model Bias EvaluationRepresentation Geometry in Language Models
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Jun 28, 2026

Anisotropy Decides Cosine vs. Rank Metrics for Text Embeddings

The standard way to compare two text embeddings is cosine similarity. Scattered studies report that a different metric does better, but never pin down the geometric condition that decides when, or why. We settle both with a comprehensive empirical study: nineteen parameter-free similarity metrics on nineteen encoders, from compact sentence transformers up to seven-billion-parameter large language models, across seven datasets. The answer is geometric. When an encoder spreads its variance evenly across directions, cosine is the best parameter-free choice and no other metric helps by a usable margin. When the variance concentrates into a few dominant directions, a property known as anisotropy, rank-based and L1-type metrics beat cosine by a clear margin. The absolute gain is modest, but because cosine starts low on these encoders it is a sizable relative improvement, around twenty percent on average and largest where cosine is weakest. What decides this is the geometry of the embedding space, not how the model was trained: where the two disagree, the metric follows the geometry. One number, the fraction of variance held by the single most dominant dimension, predicts how much the alternatives help across all nineteen encoders, with a rank correlation of 0.86 and a linear correlation of 0.95. To test this as the cause rather than a correlate, we project out the dominant directions: cosine recovers and the advantage of the other metrics nearly vanishes, but only on the encoders that were anisotropic to begin with. The effect is directional, not magnitude based, since it survives normalizing every vector to unit length. Among parameter-free metrics, then, cosine is the right tool wherever an encoder is well spread, which includes the fine-tuned embedders commonly deployed for retrieval, and we give a one-number diagnostic for when it is not.

V. S. Raghu Parupudi

University of California, San Diego

Semantic Textual SimilarityNeural Representation GeometryText Embedding EvaluationRepresentation Geometry in Language Models
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Jun 28, 2026

Invariant Reasoning Directions in Latent Trajectories of Language Models

Latent reasoning models perform multi-step inference directly in hidden-state space, yet the structure of these latent reasoning trajectories remains poorly understood. We show that contrastive refinement signals between stronger and weaker reasoning trajectories exhibit a highly concentrated low-rank structure, while unconstrained latent updates remain sensitive to paraphrases, checkpoint choice, and trajectory perturbations. These observations suggest that latent reasoning trajectories contain stable invariant directions mixed with unstable instance-specific variation. We introduce Trajectory-Invariant Latent Refinement (TILR), a training-free intervention framework for identifying and manipulating stable reasoning directions in latent space. TILR first learns a low-rank invariant subspace from contrastive trajectory differences across inputs, then constrains latent interventions to this subspace while suppressing poorly aligned updates through an adaptive alignment gate. Across six reasoning benchmarks, we find that a small number of latent directions explain most variation between strong and weak reasoning trajectories. Interventions on these directions causally improve reasoning consistency and reduce trajectory instability under paraphrases and perturbations. TILR improves answer consistency under paraphrase by ~10% and reduces latent trajectory variance by up to 50%50\% while preserving reasoning accuracy. These results support a geometric view of latent reasoning in which transferable reasoning behavior emerges from stable low-dimensional structure within hidden-state trajectories.

Arun Vignesh Malarkkan, Manan Roy Choudhury, Utkarsh Byahut +3

Arizona State University, Tempe, Arizona.

Reasoning Consistency in Language ModelsImplicit Reasoning in Language ModelsCausal Interventions in Language ModelsLanguage Model RobustnessRepresentation Geometry in Language ModelsLatent Reasoning
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Jun 26, 2026

Search for Truth from Reasoning: A Dynamic Representation Editing Framework for Steering LLM Trajectories

Current approaches to enhance Large Language Model (LLM) reasoning, such as Chain-of-Thought and "Wait" prompts, primarily encourage models to think more, yet often fail to guide them toward Truth. While Representation Editing (RepE) offers a intrinsic control, its application to dynamic reasoning trajectories remains underexplored. In this work, we bridge this gap by investigating the geometry of truth within unfolding reasoning chains. We uncover three critical insights: (1) Truth is encoded at the sentence level and is entangled with latent reasoning patterns; (2) Effective intervention follows an Uncertainty Principle and a Decay Effect, requiring localization to early, high-entropy forks; (3) Naive steering vectors suffer from noise, risking collateral damage to correct trajectories. Based on these findings, we propose DynaSteer, a dynamic RepE framework. DynaSteer employs pattern clustering to disentangle reasoning manifolds and utilizes Fisher-LDA to project purified truth. By dynamically monitoring lookahead entropy, it selectively steers and rolls back trajectories only when necessary. Comprehensive experimental results on several MATH benchmark verify the effectiveness of DynaSteer, and experiments on out-of-domain coding tasks further confirm its generalization ability. Our code is publicly available at https://github.com/tianlwang/DynaSteer.

Tianlong Wang, Yuhang Wang, Weibin Liao +5

National Engineering Research Center of Software Engineering, Peking University · School of Computer Science, Peking University · Key Laboratory of High Confidence Software Technologies, Ministry of Education +1

Language Model SteeringRepresentation Geometry in Language ModelsLLM Reasoning
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Jun 25, 2026

The Geometry of Updates: Fisher Alignment at Vocabulary Scale

Training-free source selection for LLM families with shared vocabularies arises in scientific string domains such as SMILES, protein, and genomic sequences, where candidate corpora share a tokenizer but differ in prediction targets. This creates an activation-dark regime: representation-similarity metrics can be uninformative without assumptions about label-conditioned error geometry, while classical update-geometry metrics are computationally prohibitive at vocabulary scale. We show that, in a shared-output head setting, representation metrics (e.g., CKA) are non-identifiable for transfer; models can share identical representations yet have orthogonal head updates. The key identity is that head Fisher alignment is exactly a cosine between kernel mean embeddings in the joint activation-error space, exposing activation, error, and coupling factors rather than requiring a materialized Fisher matrix. FisherSketch estimates this cosine directly in a single streaming pass, making K=128,256 head Fisher alignment practical with a 16 KB task signature (m=4096) and a 192 KB per-task streaming state, small enough to store next to a model hash, but encoding transfer-relevant update structure. Beyond source selection, the same signatures and marginals provide a diagnostic instrument for studying whether LLM task similarity is driven by activations, errors, or their coupling; shared-parameter and internal-layer validations, together with Llama-3.1-8B verbalizer-shift experiments, show that FisherSketch remains informative when activation similarity cannot distinguish tasks.

John Sweeney

Sideplane AI

Task Transferability EstimationTransfer LearningRepresentation Geometry in Language Models
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Jun 25, 2026

Where Do Models Find Happiness? Emotion Vectors in Open-Source LLMs

Recent work identified emotion vectors in Claude Sonnet 4.5, which are internal representations that encode emotion concepts, causally influence behavior, and exhibit geometry mirroring human psychological structure. We test the generality of these findings in two open-weight models, Apertus-8B-Instruct-2509 and Gemma-4-E4B-it, extracting emotion contrast vectors across all layers, using two model-generated corpora. We recover valence geometry for both models, with peak PC1--valence correlations of r=0.76r = 0.76 and r=0.83r = 0.83, approaching the r=0.81r = 0.81 reported for Claude.Beyond replication, we observe notable differences in how valence representations emerge across model depth. In Gemma-4-E4B-it, valence is strongly encoded in early layers but collapses towards later layers, whereas Apertus-8B-Instruct-2509 exhibits the opposite pattern, with valence representations absent in early layers, but emerging at mid depths. Arousal encoding, in contrast, is sensitive to the extraction corpus: both models show stronger PC2--arousal alignment with Gemma-generated stories (rr up to 0.450.45) than Apertus-generated ones (r≤0.21r \leq 0.21), suggesting arousal-relevant cues are unevenly distributed across generated corpora. We open-source our experiment code and dataset for reproducible investigation of emotion representations across language model architectures.

Sinie van der Ben, Raphaël Baur, Yannick Metz +1

Department of Computer Science, ETH, Zurich, Switzerland.

LLM InterpretabilityValence-Arousal ModelingRepresentation Geometry in Language ModelsEmotion Representation in Language Models
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Jun 25, 2026

Structure Before Collapse: Transient semantic geometry in next-token prediction

Neural Collapse predicts that balanced one-hot classification pushes model representations to be equally far from each other; a symmetric configuration that depends only on the output label and ignores any semantic similarity in the inputs. This creates a puzzle: next-token prediction language models are trained predominantly (as context length increases) with one-hot labels: the same context is very unlikely to appear twice in training with different labels. However, they clearly learn latent structural features. That is, despite the one-hot training regime, a language model's contextual embeddings represent the fact that the next word in ''Mary broke the ___'' is likely to be filled by tokens in the latent classes of a) medium-sized, b) rigid, c) inanimate nouns. How does gradient descent find such categorical semantic structure when co-occurrence statistics collapse to one-hot sparsity, eliminating any shared next-tokens among different contexts? To investigate this tension we identify three synthetic controlled settings where inputs have latent semantic factors but are mapped to distinct one-hot labels. We find that semantic geometry emerges early in training, and that representations cluster by shared attributes despite receiving no explicit supervision to do so. This structure is transient: with sufficient capacity and time, the model eventually reaches the predicted symmetric state where all representations are equally separated. We study this phase transition through Gram matrix analysis and propose a preliminary modification to the commonly used unconstrained features model to capture the emergent semantic geometry.

Yize Zhao, Isabel Papadimitriou, Christos Thrampoulidis

Department of Electrical and Computer Engineering, The University of British Columbia · Department of Linguistics, The University of British Columbia

Autoregressive Language ModelingNext-Token PredictionNeural CollapseRepresentation Geometry in Language ModelsSemantic Representation Learning
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Jun 23, 2026

Reasoning as Attractor Dynamics: Latent Memory Retrieval via Gibbs-Weighted Energy Minimization

Large Language Models (LLMs) are traditionally viewed as autoregressive generators. However, from the perspective of collective computation, they function as high-dimensional Dense Associative Memories that store complex reasoning patterns as latent attractors. In this work, we investigate the energy landscape of mathematical reasoning. We posit that correct reasoning chains correspond to deep, wide attractor basins ("flat minima") in the model's output distribution, whereas hallucinations manifest as sharp, unstable local minima. To exploit this geometry, we introduce a retrieval mechanism based on a Gibbs measure of the trajectory's spectral entropy. By sampling multiple reasoning paths and weighting them by their inverse energy (P∝e−βEP \propto e^{-βE}), we approximate the equilibrium distribution of the associative memory, effectively ``relaxing'' the system into a robust solution. Empirically, this physics-inspired mechanism improves Microsoft Phi-3.5 performance on GSM8K by 5.38% (84.7% →\to 90.1%), demonstrating that inference is better modeled as a dynamic settling process into an attractor basin rather than greedy next-token prediction.

Kanishk Awadhiya

Independent Researcher

LLM InferenceRepresentation Geometry in Language ModelsLLM Reasoning
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Jun 23, 2026

Evidence for feature-specific error correction in LLMs

Understanding the features of large language models (LLMs) is a central goal of interpretability. LLMs are commonly assumed to use superposition to represent more features than they have dimensions. They may not only represent features in superposition but also perform computation in superposition. Theory predicts that computing in superposition requires error correction that privileges feature directions over generic ones, but this prediction has not been tested empirically. We propose an empirical test of error correction in LLMs based on activation perturbations. Perturbing residual-stream activations, we find that they are robust to small perturbations--forming activation plateaus consistent with error correction--but less robust along candidate feature directions ("pure" directions, constructed from contrastive prompt pairs) than along mixtures of two such directions, indicating that the pure directions are privileged. We quantify this privilegedness by modeling the perturbation effect as a function of the LpL^p-norm of its decomposition into feature components. For p=2p=2 the response is a quadratic form with at most as many nonzero eigenvalues as the residual-stream dimension, which cannot privilege the many feature directions superposition requires. p>2p>2 lifts this constraint and is consistent with feature-specific error correction. We find p>2p>2 for contrastive, MELBO, and SAE-decoder directions, and p≈2p\approx2 for random and PCA directions (controls). These results replicate across Gemma-2-9B, Qwen3-1.7B, Llama-3.1-8B, Mistral-7B-v0.3, Aya-Expanse-8B, and Yi-1.5-9B. We further validate our method on a toy model of error correction with known ground-truth features, recovering p>2p>2 for true feature directions, degrading toward 22 as we rotate away from them.

Francisco Ferreira da Silva, Stefan Heimersheim

Pivotal Research.

LLM InterpretabilityLanguage Model RobustnessRepresentation Geometry in Language Models
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Jun 23, 2026

Perfect Detection, Failed Control: The Geometry of Knowing vs. Steering in Language Models

A central aspiration of mechanistic interpretability is controllability: if we know where a behavior is represented in a model's activations, we should be able to modify it. This rests on a hidden premise -- that the direction which detects a behavior and the direction which controls it are the same, or close. We test this geometrically: what is the angle between the direction that best detects a behavior and the one that best causes it? If detection implies control the cosine is near 1; otherwise it quantifies a detection-intervention gap. On Gemma 2-2B-it, output format (clean JSON vs markdown fencing) collapses both roles onto one axis. Hallucination does not: the model detects fake entities with perfect linear separability (AUC = 1.000 from layer 5), yet that direction sits at cos = 0.12 (about 83 degrees) from the direction producing a refusal -- a small, reproducible alignment, far from the cos = 1 that "detection is control" would require. A detector built from activations, with no chosen tokens, likewise fails to align (cos = -0.06). The gap generalizes: across four models from three families and two scales (1B-9B), cos stays in [0.12, 0.20], identical before and after instruction tuning (0.1197 vs 0.1200), placing its origin in pretraining. A 15-degree rotation toward the refusal direction partially bridges it -- 73% and 60% refusal on two held-out fake-entity categories at 1.8% false positives. We then ask whether this cosine predicts steerability, and it does not: detection is a high-dimensional class, not a single direction, and what separates the steerable case is functional, not readable from a static angle. The cosine is a weight-computable signature of the dissociation between knowing and steering, not a predictor of it.

Cosimo Galeone, Anna Ettorre, Minsu Park +2

Alomana, Grottaglie, Italy

Mechanistic InterpretabilityCausal Interventions in Language ModelsRepresentation Geometry in Language ModelsActivation Steering
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Jun 22, 2026

Do LLM Embedding Spaces Recover Expert Structure?

Pretrained text embeddings are increasingly used as representational maps, yet high category separability does not imply that their geometry recovers expert-defined structure. We study this problem in mental-health-related language, where symptom relations provide an external reference and online communities introduce strong domain, affective, stylistic, and discourse confounds. Using 28 Reddit communities, we compare pretrained and supervised fine-tuned Qwen3 embedding spaces at two scales (0.6B and 4B). We construct category prototypes, evaluate their representational dissimilarity matrices against an expert symptom matrix with representational similarity analysis, and complement this global test with prototype-based typicality and multi-baseline confound controls. Pretrained embeddings show measurable alignment with expert structure within the mental-health subset; fine-tuning strengthens this alignment most at the finest category level; and larger scale improves both zero-shot alignment and supervision-induced gains. Residual alignment remains substantial after controlling for VAD, LIWC, lexical style, and topic-distribution structure. These results suggest that LLM embeddings can recover expert-relevant category geometry, but this recovery is level-dependent and should be tested against explicit confounds rather than inferred from classification alone.

Yixuan Zhu, Zhenke Duan, Fanghen Li

Zhongnan University of Economics and Law

LLM Fine-TuningMental HealthText Embedding EvaluationRepresentation Geometry in Language Models
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Jun 22, 2026

Abstract representational geometry supports inference in large language models

A defining feature of human intelligence is the ability to adapt to changing environments by inferring latent task structure from sparse observations. Neuroscientific research indicates that this capability relies on the hippocampus constructing abstract representations, expressed as low-dimensional, approximately orthogonal manifolds in neural state space. However, the internal mechanisms of large language models (LLMs) remain largely opaque, making it unclear whether they form comparable abstract representations or instead rely on task-specific statistical regularities when performing comparable reasoning tasks. Here we adapt a contextual reversal-learning paradigm to a text-based setting and compare humans and LLMs at both the Behavioural and representational levels. We report that although LLMs exhibit generalizable reasoning less frequently than humans, when such inference occurs, their internal states exhibit abstract geometric structures that resemble those reported in the hippocampus. Notably, this representational geometry is not uniformly distributed but is organized hierarchically across model depth: whereas lower layers show early, stable encoding of stimulus identity, higher layers form a hippocampal-like functional band enriched for abstract context geometry associated with inference. Furthermore, complementary intervention experiments mechanistically implicate geometry in reasoning: task-sequence language modelling induces geometric disentanglement, whereas geometric regularization of higher layers increases the emergence of generalizable inference. Together, these findings establish abstract representational geometry as a mechanistic principle supporting inference in large language models.

Yunan Zeng, Yuwang Wang

College of Future Information Technology, Fudan University, Shanghai, 200433, China · Beijing National Research Center for Information Science and Technology, Tsinghua University, Beijing, 100084, China

LLM InferenceLLM InterpretabilityRepresentation Geometry in Language Models
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Jun 21, 2026

Orthogonal Representation Editing: Decoupling Semantic Entanglement in Batch Knowledge Editing of LLMs

Knowledge editing aims to efficiently update factual information in Large Language Models (LLMs) without full retraining. However, existing methods still suffer from performance degradation in batch knowledge editing. We identify that semantic representation entanglement, such as overlapping concepts and shared syntactic patterns, accumulates interference in the representation space and reduces editing precision. To bridge this gap, in this paper, we propose Orthogonal Representation Editing (ORE), which performs edits in the hidden representation space of LLMs by constructing a general semantic subspace and enforcing orthogonal constraints on edit vectors, effectively decoupling semantic entanglement. Furthermore, we introduce a gated non-linear representation head to enable adaptive learning of editing locations and precise control over knowledge injection. Extensive experiments show that ORE outperforms existing methods and achieves superior performance in cross-lingual knowledge editing scenarios. We release our code at https://github.com/YVVH/ORE.

Wenhao Yu, Zhicong Lu, Bo Lv +4

School of Computer Science and Technology, Tianjin University · Kexin Technology · University of Chinese Academy of Sciences +2

Knowledge EditingModel EditingLanguage Model EditingRepresentation Geometry in Language Models
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Jun 21, 2026

ROMEVA: Geometry-Preserving Vocabulary Expansion for Roman Urdu Language Models

Multilingual Language Models like mBERT are widely used for low-resource NLP, yet their adaptation to morphologically inconsistent languages such as Roman Urdu remains underexplored. Roman Urdu spelling variation causes severe sub-word fragmentation, averaging 1.50 sub-words per token. We propose ROMEVA (Roman Urdu Embedding-preserving Vocabulary Adaptation), which combines sub-word-average initialization and a PCA-guided anchor loss to stabilize embeddings during vocabulary expansion. Using a 36,130-comment Roman Urdu corpus, we add 500 highly fragmented tokens to mBERT and compare naive fine-tuning, sub-word-aware fine-tuning, and ROMEVA. While ROMEVA most effectively preserves the pretrained embedding space, naive fine-tuning achieves the strongest downstream sentiment classification performance. These findings reveal a disconnect between embedding stability and downstream performance, suggesting that stronger adaptation may be preferable to strict embedding preservation in morphologically inconsistent languages.

Mahnoor Khan, Afsheen Asif, Milhan Afzal Khan +2

School of Electrical Engineering and Computer Science (SEECS), National University of Sciences and Technology (NUST), Islamabad, Pakistan · Department of Computer Science, University of Agriculture Faisalabad, Faisalabad, Pakistan

Multilingual Language ModelsWord EmbeddingsLow-Resource Language ProcessingRepresentation Geometry in Language ModelsLanguage Model Vocabulary Expansion
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Jun 18, 2026

From Texts to Scores: Tracing the Emergence of Essay Quality Representations in Large Language Models

Recent advances in Large Language Models (LLMs) have substantially transformed Automated Essay Scoring (AES), yet the internal mechanisms underlying LLM-based scoring remain poorly understood. In this work, we systematically analyze the hidden representations of eight LLMs across two English essay datasets (ASAP++, CSEE) and one Portuguese dataset (ENEM). Using linear probing, cross-prompt generalization, dimensionality reduction, and neuron-level analyses, we find consistent evidence that essay quality information is encoded in a linearly accessible form within LLM representations. These representations emerge progressively across layers, remain robust across prompting strategies, and partially transfer across essay prompts despite differences in scoring rubrics. In addition, nonlinear probes provide only marginal and inconsistent improvements over linear probes, suggesting that most essay quality information is already linearly decodable. We further identify individual ``essay scoring neurons'' whose activations strongly correlate with essay scores and whose behavior is sensitive to targeted intervention. Moreover, the layer-wise distribution of these neurons systematically shifts with essay length, with longer essays relying more heavily on deeper layers. Overall, our findings provide evidence that LLMs encode structured representations related to essay quality and offer new insights into the interpretability of LLM-based AES systems.

Jiaxu Zuo, Mu You, Kaixin Lan +5

NLP2CT Lab, Department of Computer and Information Science, University of Macau · Institute of International Language Services Studies, Macau Millennium College · School of Data Science and Information Engineering, Guizhou Minzu University

LLM InterpretabilityAutomated Essay ScoringRepresentation Geometry in Language ModelsLanguage Model Probing
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Jun 18, 2026

What Makes Effective Supervision in Latent Chain-of-Thought: An Information-Theoretic Analysis

Latent Chain-of-Thought (CoT) internalizes reasoning within continuous hidden states, offering a promising alternative to verbose discrete reasoning traces. However, robust latent reasoning remains difficult because outcome supervision provides weak learning signals and leaves latent trajectories prone to semantic drift. In this work, we analyze Latent CoT from an information-theoretic perspective and identify this failure as a dual collapse: gradient attenuation along the optimization path and representational drift in the latent space. We further decompose process supervision into two complementary dimensions: Trajectory Supervision, which injects dense stepwise reasoning signals, and Space Supervision, which preserves the semantic structure of the latent manifold. Our analysis shows that rigid geometric compression can collapse the reasoning space, whereas generative reconstruction provides a more flexible semantic anchor that better preserves information capacity. To measure these effects, we introduce the Unified Latent Probe (ULP), which quantifies the mutual information between latent trajectories and explicit reasoning steps. Experiments reveal a clear Information-Performance Binding: reasoning accuracy depends on the information fidelity preserved in the latent chain. These findings provide a principled framework for latent reasoning supervision and suggest shifting from geometric imitation toward mutual information maximization. Our code is available at this repository.

Xinghao Chen, Chak Tou Leong, Wenjin Guo +3

Ningbo Institute of Digital Twin, Eastern Institute of Technology, Ningbo, China · Department of Computing, The Hong Kong Polytechnic University, Hong Kong, China.

Continuous Latent ReasoningProcess SupervisionRepresentation Geometry in Language ModelsLatent CoT ReasoningCoT Supervision
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Jun 18, 2026

GEMS: Geometric Constraints Enable Multi-Semantic Superposition in LLMs

Activation steering controls model behavior by modifying intermediate hidden states at inference time without retraining. Existing methods handle only single-direction injection; when multiple semantic directions are superposed without constraints, the model collapses. We show that this collapse decomposes into two independently acting sources: distributional deviation, where additive perturbations accumulate in norm across layers and drive activations outside the training distribution, and directional interference, where non-orthogonal semantic vectors mutually dampen when superposed. These two sources define the design constraints that any training-free multi-directional intervention must address. As one instantiation of these principles, we propose GEMS, a training-free method that maps each source to a corresponding geometric constraint: norm-preserving weighted superposition and targeted attention-pathway injection for distributional deviation, and real-time orthogonalization for directional interference. On GSM8K, injecting three concurrent non-mathematical directions preserves accuracy at 98% (baseline 92%), while unconstrained addition collapses to 4%; on Wikitext-2, the same injection incurs only 2.2% PPL increase. Component ablation isolates the causal role of each constraint, and layer-level probes confirm that orthogonalized signals survive the FFN pathway and reach the output distribution with semantic specificity. Qualitative steering effects transfer across architectures from 3B to 31B.

Yu Deng

Causal Interventions in Language ModelsRepresentation Geometry in Language ModelsActivation Steering
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Jun 18, 2026

Code-Switching Reveals Language Anchoring in Multilingual LLMs

Multilingual Large Language Models (MLLMs) are increasingly expected to handle Code-Switched (CS) inputs, yet mixing languages frequently degrades performance relative to source- or target-language monolingual counterparts. To understand this degradation, we use grammar-forced CS as a controlled diagnostic setting for locating CS representations relative to their source and target counterparts. We introduce Anchor Bias, a geometric measure that quantifies language anchoring, whether a CS hidden state aligns closer to its source or target language counterpart. Across diverse MLLMs, Anchor Bias reveals a consistent grammar-frame effect: source-framed CS stays source-anchored, whereas target-framed CS shifts target-ward and shows larger Question Answering (QA) degradation. Motivated by this representational pattern, we propose CANVAS (Contextual Anchor-based Neural Vector Alignment Steering), an inference-time intervention that extracts a source-side canvas from the input and softly steers target-language hidden states toward the source anchor during prefill. CANVAS consistently recovers QA F1 across MLLMs and CS conditions, showing that internal anchoring signals provide an actionable target for mitigating CS inference failures.

Jeonghyun Park, Seunghyun Yoon, Yonghyun Jun +1

Chung-Ang University, Seoul, Korea · Adobe Research, USA

Multilingual Language ModelsCode-SwitchingCausal Interventions in Language ModelsRepresentation Geometry in Language Models
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Jun 17, 2026

Tracking Representation Dynamics in Large Language Models with Persistent Homology

Large language models are commonly aligned through supervised fine-tuning, yet little is known about how their internal representations evolve during this process. We study alignment dynamics using persistent homology by tracking the topology of activation spaces throughout fine-tuning. Across four transformer language models ranging from 1B to 7B parameters and three alignment objectives corresponding to helpful, harmless, and mixed training data, we find that the majority of topological reorganization occurs during the earliest stages of training. A dense checkpoint analysis reveals a transient peak in topological activity followed by rapid stabilization. We further show that different alignment objectives induce distinguishable topological trajectories, while instruction-tuned and pretrained models exhibit qualitatively different patterns of evolution. Our results suggest that persistent homology provides a complementary perspective on alignment, revealing representation-level changes that are not apparent from behavioral metrics alone.

Naman Malhotra, Jay Ambadkar, Abhinav Gupta +4

Department of Mathematics, Imperial College London, UK · Department of Computing, Imperial College London, UK

LLM AlignmentLLM Fine-TuningRepresentation Geometry in Language Models
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Jun 15, 2026

RepSelect: Robust LLM Unlearning via Representation Selectivity

Making large language models (LLMs) deeply forget specific knowledge and values without sacrificing general capabilities remains a central challenge in unlearning. Current methods are easily reversed by fine-tuning or few-shot prompting, suggesting their forgetting is only shallow. We identify the root cause. Existing methods target representations shared with both the retain set and the subspace recovered by a fine-tuning attacker, making unlearning both disruptive to general capabilities and easy to reverse. We propose RepSelect (Representation Selectivity), which isolates forget-set-specific representations by collapsing top principal components of weight gradients before each update, leaving general capabilities intact while limiting what fine-tuning can recover. We evaluate across two forget categories, biohazardous knowledge and abusive tendencies, and four model families spanning dense and Mixture-of-Experts architectures (Llama 3, Qwen 3.5, Gemma 4 E4B, DeepSeek V2 Lite). Compared to five popular baselines (GradDiff, NPO, SimNPO, RMU, UNDIAL), RepSelect achieves a 4-50x larger reduction in post-relearning answer accuracy than the strongest baseline, and is near-perfectly robust to few-shot prompting attacks. Targeting selective representations is thus an important step towards deep and robust LLM forgetting.

Filip Sondej, Yushi Yang, Adam Mahdi

Independent · University of Oxford

LLM SafetyRepresentation Geometry in Language ModelsMachine UnlearningLLM Unlearning
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Jun 15, 2026

The Value Axis: Language Models Encode Whether They're on the Right Track

We investigate whether language models internally track the value of their current trajectory, defined as the likelihood that their ongoing strategy will achieve their goals. Using synthetic, in-context reinforcement learning data, we construct a "value" axis for Qwen3-8B. We find that activations along this axis distinguish between high vs. low verbalized confidence, rollouts without and with backtracking, and correct vs. corrupted code. Steering towards high value causally suppresses self-correction and reduces explanatory verbosity, while steering towards low value induces backtracking and exploration. We demonstrate that direct preference optimization (DPO) can increase the internal value of rewarded behaviors (e.g. use a certain word), causing the model to act more confidently after exhibiting them. Finally, we apply the value axis to study in-the-wild settings. For example, we find that Qwen assigns low value to politically sensitive chat queries after post-training and that supervised fine-tuning increases internal confidence within the training domain. Our results suggest that language models linearly encode an estimate of expected goal success that modulates their confidence in pursuing a direction.

Nick Jiang, Isaac Kauvar, Jack Lindsey

1Stanford University · 2Anthropic

Confidence Estimation in Language ModelsLanguage Model CalibrationRepresentation Geometry in Language ModelsMetacognition in Language Models
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Jun 15, 2026

Contrastive-Difference CKA Reveals Concept-Specific Structural Alignment Across Language Model Architectures

Do different LLM architectures encode high-level concepts in structurally compatible ways? We systematically characterize a geometric-functional universality dissociation: across multiple concept domains and architectural families, moderate geometric convergence coexists with near-perfect functional transfer. Using contrastive-difference CKA (CKA_Delta), a training-free diagnostic that computes kernel alignment on per-sample contrastive differences, we isolate concept-specific convergence from generic similarity -- achieving significant discrimination where standard CKA cannot. The dissociation replicates across all six concept domains we test (five with p <= 0.017 geometric discrimination and safety as a converging-functional trend, p = 0.08), including two non-instruction concepts (code-vs-NL, reasoning-vs-recall) validated without system prompts; a single 70B--70B pair provides an observational note that universality may strengthen with scale, requiring replication with additional >=70B models. We position CKA_Delta as a practical regime classifier and architectural outlier detector (Gemma: d = 1.08, AUC = 0.79) rather than an absolute transfer-accuracy predictor, providing a training-free diagnostic for cross-architecture concept monitoring.

Xueping Gao

Alibaba Cloud

Kernel MethodsRepresentation Geometry in Language Models
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Jun 15, 2026

Data-Driven Decoding of Russell's Circumplex Model of Affect

Affective computing increasingly relies on deep learning to represent emotions, yet latent spaces often remain opaque, high-dimensional black boxes. This paper investigates whether Transformers' embeddings recover the geometric regularities of Russell's circumplex model. We unify two complementary experiments testing the hypothesis that, after training models on text and speech, their resulting latent spaces encode a topology consistent with valence-arousal and reproduce human-like neighborhood relations. Specifically, we evaluate deep representations extracted from Transformer-based text (RoBERTa) and speech (wav2vec 2.0) encoders, along with a multimodal Transformer fusion architecture, across naturalistic datasets like MSP-Podcast and controlled LLM-generated stimuli. Our analysis reveals that multimodal fusion of text and audio yields perfect topological alignment with Russell's primary emotion ordering. Furthermore, in a zero-shot setting using generic text embeddings, projected fine-grained emotion terms fall close to their established human-mapped coordinates. Our contribution is a novel, data-driven framework for validating emotion models, demonstrating that Russell's circumplex structure is intrinsically encoded in the embeddings of these modalities rather than being solely an artifact of human labeling, thereby bridging the gap between psychological theory and representation learning.

Amdjed Belaref, Samir Sadok, Zineb Noumir +1

Alten, France · CentraleSup´elec IETR UMR CNRS 6164, France

Multimodal EmbeddingRepresentation Geometry in Language ModelsAffective Computing
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Jun 13, 2026

High-Dimensional Random Projection for Activation Steering in Language Models

Activation steering has emerged as a key methodology for controlling the behavior of large language models (LLMs). Existing difference-in-means based methods, however, are fundamentally limited: they capture only mean differences between class activations and fail to recover discriminative signals that naturally exist in the nonlinear feature subspace under the superposition hypothesis. Motivated by that, we propose High-Dimensional Random-projection for Activation Steering (HiDRA), a training-free approach that integrates seamlessly with existing activation steering methods. By performing activation addition in the projected high-dimensional space, HiDRA can provably capture a better discriminative structure beyond the reach of linear methods. Experiments across diverse LLM families and benchmarks demonstrate that HiDRA consistently outperforms baseline counterparts, achieving stronger behavioral control without significant computational overhead.

Minh-Hieu Pham, Bach Do, Laziz Abdullaev +2

Hanoi University of Science and Technology

Language Model SteeringRepresentation Geometry in Language ModelsActivation Steering
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