Dimension d = O(m^{-2} log n) nearly achieves the optimal margin m^rd(+∞, A) for retrieval embeddings, with matching lower bounds showing d = O(k log(n/k)) suffices and is necessary for m = Θ(k^{-1/2}) on k-sparse query matrices.
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The Linear Representation Hypothesis and the Geometry of Large Language Models
Canonical reference. 88% of citing Pith papers cite this work as background.
49
Pith papers citing it
Background
88% of classified citations
abstract
Informally, the 'linear representation hypothesis' is the idea that high-level concepts are represented linearly as directions in some representation space. In this paper, we address two closely related questions: What does "linear representation" actually mean? And, how do we make sense of geometric notions (e.g., cosine similarity or projection) in the representation space? To answer these, we use the language of counterfactuals to give two formalizations of "linear representation", one in the output (word) representation space, and one in the input (sentence) space. We then prove these connect to linear probing and model steering, respectively. To make sense of geometric notions, we use the formalization to identify a particular (non-Euclidean) inner product that respects language structure in a sense we make precise. Using this causal inner product, we show how to unify all notions of linear representation. In particular, this allows the construction of probes and steering vectors using counterfactual pairs. Experiments with LLaMA-2 demonstrate the existence of linear representations of concepts, the connection to interpretation and control, and the fundamental role of the choice of inner product.
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background 8representative citing papers
SLAM achieves 100% detection on Gemma-2 models with only 1-2 point quality cost by causally steering SAE-identified residual-stream directions for linguistic structure.
Function vectors steer LLMs successfully where the logit lens fails to decode the target answer, showing the two properties come apart.
Subliminal learning is a LoRA artifact that disappears with full finetuning, depends on context tokens like system prompts, and localizes to overlapping finetuning-evaluation tokens.
VerifySteer selectively steers hidden states at paragraph boundaries using latent correctness signals to control verifier strictness and outperform baselines on ProcessBench and Hard2Verify with lower compute.
Manifold steering along activation geometry induces behavioral trajectories matching the natural manifold of outputs, while linear steering produces off-manifold unnatural behaviors.
Transformers store count information internally but cannot read it out as digits due to near-orthogonal alignment with output-head rows; updating digit rows or applying LoRA to attention layers improves constrained and unconstrained counting respectively.
Large language models encode relational bindings via a cell-based representation: a low-dimensional linear subspace in which each cell corresponds to an entity-relation index pair and attributes are retrieved from the matching cell.
Local linearity of LLM layers enables LQR-based closed-loop activation steering with theoretical tracking guarantees.
Refusal in language models is mediated by a single direction in residual stream activations that can be erased to disable safety or added to elicit refusal.
Activation Addition steers language models by adding contrastive activation vectors from prompt pairs to control high-level properties like sentiment and toxicity at inference time without training.
LLMs show hippocampal-like abstract representational geometry in higher layers during inference, with layer-wise hierarchy and interventions linking geometry to reasoning success.
Steering LLM residual streams with random sparse vectors creates detectable self-recognition fingerprints that enable over 98% accurate attribution of generated text to specific models without degrading output quality.
A monolingually trained linear probe on intermediate LLM representations predicts answer correctness zero-shot across typologically diverse languages, with confidence signals concentrated in middle layers.
MAGS learns low-dimensional subspaces from correct versus incorrect reasoning traces and applies targeted projection corrections to attention heads when they deviate from the correctness manifold during inference.
LFD discovers predictive text features via LLM contrastive proposals, cross-LLM Cohen's kappa screening, and residual held-out gain selection, matching baseline accuracy while achieving higher human agreement and lower label leakage on ten tasks.
ASRU combines activation redirection and reward-optimized fine-tuning to unlearn cross-modal sensitive knowledge in MLLMs, reporting +24.6% better unlearning effectiveness and 5.8x higher generation quality on Qwen3-VL while preserving utility with limited retained data.
A latent mediation framework with sparse autoencoders enables non-additive token-level influence attribution in LLMs by learning orthogonal features and back-propagating attributions.
LLMs perform in-context learning as trajectories through a structured low-dimensional conceptual belief space, with the structure visible in both behavior and internal representations and causally manipulable via interventions.
Temporal knowledge drift is encoded as a geometrically orthogonal direction in LLM residual streams, independent of correctness and uncertainty.
LLMs exhibit three geometric phases in next-token prediction—seeding multiplexing, hoisting overriding, and focal convergence—where predictive subspaces rise, stabilize, and converge across layers.
Tool identity is linearly readable and steerable in LLMs via mean activation differences, with 77-100% switch accuracy and error prediction from activation gaps.
Tree SAE learns hierarchical feature structures by combining activation coverage with a new reconstruction condition, outperforming prior SAEs on hierarchical pair detection while matching state-of-the-art benchmark performance.
Health foundation model embeddings contain an interpretable symbolic organization shared across modalities that supports cross-domain transfer without joint training.
citing papers explorer
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Is Dimensionality a Barrier for Retrieval Models?
Dimension d = O(m^{-2} log n) nearly achieves the optimal margin m^rd(+∞, A) for retrieval embeddings, with matching lower bounds showing d = O(k log(n/k)) suffices and is necessary for m = Θ(k^{-1/2}) on k-sparse query matrices.
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SLAM: Structural Linguistic Activation Marking for Language Models
SLAM achieves 100% detection on Gemma-2 models with only 1-2 point quality cost by causally steering SAE-identified residual-stream directions for linguistic structure.
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Steerable but Not Decodable: Function Vectors Operate Beyond the Logit Lens
Function vectors steer LLMs successfully where the logit lens fails to decode the target answer, showing the two properties come apart.
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Subliminal Learning is a LoRA Artifact
Subliminal learning is a LoRA artifact that disappears with full finetuning, depends on context tokens like system prompts, and localizes to overlapping finetuning-evaluation tokens.
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The Hidden Signal of Verifier Strictness: Controlling and Improving Step-Wise Verification via Selective Latent Steering
VerifySteer selectively steers hidden states at paragraph boundaries using latent correctness signals to control verifier strictness and outperform baselines on ProcessBench and Hard2Verify with lower compute.
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Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behavior
Manifold steering along activation geometry induces behavioral trajectories matching the natural manifold of outputs, while linear steering produces off-manifold unnatural behaviors.
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The Right Answer, the Wrong Direction: Why Transformers Fail at Counting and How to Fix It
Transformers store count information internally but cannot read it out as digits due to near-orthogonal alignment with output-head rows; updating digit rows or applying LoRA to attention layers improves constrained and unconstrained counting respectively.
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Cell-Based Representation of Relational Binding in Language Models
Large language models encode relational bindings via a cell-based representation: a low-dimensional linear subspace in which each cell corresponds to an entity-relation index pair and attributes are retrieved from the matching cell.
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Local Linearity of LLMs Enables Activation Steering via Model-Based Linear Optimal Control
Local linearity of LLM layers enables LQR-based closed-loop activation steering with theoretical tracking guarantees.
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Refusal in Language Models Is Mediated by a Single Direction
Refusal in language models is mediated by a single direction in residual stream activations that can be erased to disable safety or added to elicit refusal.
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Steering Language Models With Activation Engineering
Activation Addition steers language models by adding contrastive activation vectors from prompt pairs to control high-level properties like sentiment and toxicity at inference time without training.
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Abstract representational geometry supports inference in large language models
LLMs show hippocampal-like abstract representational geometry in higher layers during inference, with layer-wise hierarchy and interventions linking geometry to reasoning success.
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LLM Self-Recognition: Steering and Retrieving Activation Signatures
Steering LLM residual streams with random sparse vectors creates detectable self-recognition fingerprints that enable over 98% accurate attribution of generated text to specific models without degrading output quality.
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Shared Doubt: Zero-Shot Cross-Lingual Confidence Estimation for Language Models
A monolingually trained linear probe on intermediate LLM representations predicts answer correctness zero-shot across typologically diverse languages, with confidence signals concentrated in middle layers.
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Manifold-Guided Attention Steering
MAGS learns low-dimensional subspaces from correct versus incorrect reasoning traces and applies targeted projection corrections to attention heads when they deviate from the correctness manifold during inference.
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Interpretable Discriminative Text Representations via Agreement and Label Disentanglement
LFD discovers predictive text features via LLM contrastive proposals, cross-LLM Cohen's kappa screening, and residual held-out gain selection, matching baseline accuracy while achieving higher human agreement and lower label leakage on ten tasks.
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ASRU: Activation Steering Meets Reinforcement Unlearning for Multimodal Large Language Models
ASRU combines activation redirection and reward-optimized fine-tuning to unlearn cross-modal sensitive knowledge in MLLMs, reporting +24.6% better unlearning effectiveness and 5.8x higher generation quality on Qwen3-VL while preserving utility with limited retained data.
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Correcting Influence: Unboxing LLM Outputs with Orthogonal Latent Spaces
A latent mediation framework with sparse autoencoders enables non-additive token-level influence attribution in LLMs by learning orthogonal features and back-propagating attributions.
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Stories in Space: In-Context Learning Trajectories in Conceptual Belief Space
LLMs perform in-context learning as trajectories through a structured low-dimensional conceptual belief space, with the structure visible in both behavior and internal representations and causally manipulable via interventions.
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The Geometry of Forgetting: Temporal Knowledge Drift as an Independent Axis in LLM Representations
Temporal knowledge drift is encoded as a geometrically orthogonal direction in LLM residual streams, independent of correctness and uncertainty.
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A Geometric Perspective on Next-Token Prediction in Large Language Models: Three Emerging Phases
LLMs exhibit three geometric phases in next-token prediction—seeding multiplexing, hoisting overriding, and focal convergence—where predictive subspaces rise, stabilize, and converge across layers.
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Tool Calling is Linearly Readable and Steerable in Language Models
Tool identity is linearly readable and steerable in LLMs via mean activation differences, with 77-100% switch accuracy and error prediction from activation gaps.
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Tree SAE: Learning Hierarchical Feature Structures in Sparse Autoencoders
Tree SAE learns hierarchical feature structures by combining activation coverage with a new reconstruction condition, outperforming prior SAEs on hierarchical pair detection while matching state-of-the-art benchmark performance.
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Emergent Symbolic Structure in Health Foundation Models: Extraction, Alignment, and Cross-Modal Transfer
Health foundation model embeddings contain an interpretable symbolic organization shared across modalities that supports cross-domain transfer without joint training.
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Uncovering and Shaping the Latent Representation of 3D Scene Topology in Vision-Language Models
VLMs possess a latent 3D scene topology subspace corresponding to Laplacian eigenmaps that can be causally shaped via Dirichlet energy regularization to improve spatial task performance by up to 12.1%.
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Pairwise matrices for sparse autoencoders: single-feature inspection mislabels causal axes
Pairwise matrices for SAEs demonstrate that single-feature inspection mislabels causal axes, with joint suppression and matched-geometry controls revealing distinct output regimes not captured by single-feature or random perturbations.
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Harmful Intent as a Geometrically Recoverable Feature of LLM Residual Streams
Harmful intent is linearly separable in LLM residual streams across 12 models and multiple architectures, reaching mean AUROC 0.982 while showing protocol-dependent directions and strong generalization to held-out harm benchmarks.
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LLM Safety From Within: Detecting Harmful Content with Internal Representations
SIREN identifies safety neurons via linear probing on internal LLM layers and combines them with adaptive weighting to detect harm, outperforming prior guard models with 250x fewer parameters.
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Characterizing Model-Native Skills
Recovering an orthogonal basis from model activations yields a model-native skill characterization that improves reasoning Pass@1 by up to 41% via targeted data selection and supports inference steering, outperforming human-characterized alternatives.
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Rhetorical Questions in LLM Representations: A Linear Probing Study
Linear probes show rhetorical questions are encoded via multiple dataset-specific directions in LLM representations, with low cross-probe agreement on the same data.
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Shared Emotion Geometry Across Small Language Models: A Cross-Architecture Study of Representation, Behavior, and Methodological Confounds
Mature small language models share nearly identical 21-emotion geometries across architectures with Spearman correlations 0.74-0.92 despite opposite behavioral profiles, while immature models restructure under RLHF and prior comprehension-generation differences decompose into four distinct layers.
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Dictionary-Aligned Concept Control for Safeguarding Multimodal LLMs
DACO curates a 15,000-concept dictionary from 400K image-caption pairs and uses it to initialize an SAE that enables granular, concept-specific steering of MLLM activations, raising safety scores on MM-SafetyBench and JailBreakV while preserving general capabilities.
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When Safety Geometry Collapses: Fine-Tuning Vulnerabilities in Agentic Guard Models
Benign fine-tuning collapses safety geometry in guard models like Granite Guardian, dropping refusal to 0%, but Fisher-Weighted Safety Subspace Regularization restores it to 75% while improving robustness.
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The Master Key Hypothesis: Unlocking Cross-Model Capability Transfer via Linear Subspace Alignment
The Master Key Hypothesis states that capabilities are low-dimensional directions transferable across models through linear subspace alignment, with UNLOCK demonstrating gains such as 12.1% accuracy improvement on MATH when transferring CoT from 14B to 7B models.
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Under Pressure: Emotional Framing Induces Measurable Behavioral Shifts and Structured Internal Geometry in Small Language Models
Emotional framings induce distinct behavioral shifts and form a structured geometry in the final-layer activations of small language models, with pressure linked to shortcuts and calm to honesty.
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ActivationReasoning: Logical Reasoning in Latent Activation Spaces
ActivationReasoning grounds logical reasoning in LLM latent activations via SAEs to enable structured inference, concept composition, and behavior steering on multi-hop, abstraction, and safety tasks.
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Steering Llama 2 via Contrastive Activation Addition
Contrastive Activation Addition steers Llama 2 Chat by adding averaged residual-stream activation differences from contrastive example pairs to control targeted behaviors at inference time.
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Sparse probes and murky physics: a case study of interpretability challenges in a foundation model for continuum dynamics
Case study applies SAE probing with enstrophy triage to a continuum-dynamics foundation model and reports intermittent feature consistency that does not align with standard physics while linking some output discrepancies to specific feature changes.
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The Latin Substrate: How Language Models Represent and Mediate Script Choice
LLMs route script choice through shared latent representations and a small set of late-layer attention heads, with non-Latin scripts gated compactly and Latin emerging diffusely.
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Steered Generation via Gradient-Based Optimization on Sparse Query Features
Prototype-Based Sparse Steering decomposes query activations with SAEs and optimizes sparse features via gradients to steer LLM outputs toward specific behaviors.
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Decodable but Not Corrected by Fixed Residual-Stream Linear Steering: Evidence from Medical LLM Failure Regimes
Overthinking in medical QA is linearly decodable at 71.6% accuracy yet fixed residual-stream steering yields no correction across 29 configurations, while enabling selective abstention with AUROC 0.610.
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Negative Before Positive: Asymmetric Valence Processing in Large Language Models
Negative valence localizes to early layers and positive valence to mid-to-late layers in LLMs, with the directions being causally steerable.
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Semantic Structure of Feature Space in Large Language Models
LLM hidden states encode semantic features whose geometric relations, including axis projections, cosine similarities, low-dimensional subspaces, and steering spillovers, closely mirror human psychological associations.
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H-Probes: Extracting Hierarchical Structures From Latent Representations of Language Models
H-probes locate low-dimensional subspaces encoding hierarchy in LLM activations for synthetic tree tasks, show causal importance and generalization, and detect weaker signals in mathematical reasoning traces.
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At the Edge of Understanding: Sparse Autoencoders Trace The Limits of Transformer Generalization
Sparse autoencoders show OOD prompts increase fallacious concept activation in transformers, offering a mechanistic measure of shift and a path to robust fine-tuning.
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From Weights to Activations: Is Steering the Next Frontier of Adaptation?
Steering is positioned as a distinct adaptation paradigm that uses targeted activation interventions for local, reversible behavioral changes without parameter updates.
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There Will Be a Scientific Theory of Deep Learning
A mechanics of the learning process is emerging in deep learning theory, characterized by dynamics, coarse statistics, and falsifiable predictions across idealized settings, limits, laws, hyperparameters, and universal behaviors.
- Relational Linear Properties in Language Models: An Empirical Investigation
- REALISTA: Realistic Latent Adversarial Attacks that Elicit LLM Hallucinations