Representation Geometry in Language Models

Does structure in representations imply structure in computation? We study how Llama-3.1-8B reasons over cyclic concepts (e.g., "what month is six months after August?"). Even though Llama-3.1-8B's representations for these concepts are circularly structured, we find that instead of directly computing modular addition in the period of the cyclic concept (e.g., 12 for months), the model re-uses a generic addition mechanism across tasks that operates independently of concept-specific geometry. First, it computes the sum of its two inputs using base-10 addition (six + August=14). Then, it maps this sum back to cyclic concept space (14->February). We show that Llama-3.1-8B uses task-agnostic Fourier features to compute these sums--in fact, these features have periods that respect standard base-10 addition, e.g., 2, 5, and 10, rather than the cyclic concept period (e.g., 12 for months). Furthermore, we identify a sparse set of 28 MLP neurons re-used across all tasks (approximately 0.2% of the MLP at layer 18) that can be partitioned into disjoint clusters, each computing the sum for a Fourier feature with a different period. Our work highlights how an interplay between causal abstraction and feature geometry can deepen our mechanistic understanding of LMs.

Mode collapse is a persistent challenge in generative modeling and appears in autoregressive text generation as behaviors ranging from explicit looping to gradual loss of diversity and premature trajectory convergence. We take a dynamical-systems view and reinterpret mode collapse as reduced state-space accessibility caused by geometric collapse: during generation, the model's internal trajectory becomes confined to a low-dimensional region of its representation space. This implies mode collapse is not purely a token-level phenomenon and cannot be reliably solved by symbolic constraints or probability-only decoding heuristics. Guided by this perspective, we propose Reinforced Mode Regulation (RMR), a lightweight, online state-space intervention that regulates dominant self-reinforcing directions in the Transformer value cache (implemented as low-rank damping). Across multiple large language models, RMR substantially reduces mode collapse and enables stable generation at extremely low entropy rates (down to 0.8 nats/step), whereas standard decoding typically collapses near 2.0 nats/step.

In autoregressive large language models (LLMs), temporal straightening offers an account of how the next-token prediction objective shapes representations. Models learn to progressively straighten the representational trajectory of input sequences across layers, potentially facilitating next-token prediction via linear extrapolation. However, a direct link between this trajectory and token-level behavior has been missing. We provide such a link by relating contextual curvature-a geometric measure of how sharply the representational trajectory bends over recent context-to next-token entropy. Across two models (GPT-2 XL and Pythia-2.8B), contextual curvature is correlated with entropy, and this relationship emerges during training. Perturbation experiments reveal selective dependence: manipulating curvature through trajectory-aligned interventions reliably modulates entropy, while geometrically misaligned perturbations have no effect. Finally, regularizing representations to be straighter during training modestly reduces token-level entropy without degrading validation loss. These results identify trajectory curvature as a task-aligned representational feature that influences behavioral uncertainty in LLMs.

Language models trained on natural text learn to represent numbers using periodic features with dominant periods at $T=2, 5, 10$. In this paper, we identify a two-tiered hierarchy of these features: while Transformers, Linear RNNs, LSTMs, and classical word embeddings trained in different ways all learn features that have period-$T$ spikes in the Fourier domain, only some learn geometrically separable features that can be used to linearly classify a number mod-$T$. To explain this incongruity, we prove that Fourier domain sparsity is necessary but not sufficient for mod-$T$ geometric separability. Empirically, we investigate when model training yields geometrically separable features, finding that the data, architecture, optimizer, and tokenizer all play key roles. In particular, we identify two different routes through which models can acquire geometrically separable features: they can learn them from complementary co-occurrence signals in general language data, including text-number co-occurrence and cross-number interaction, or from multi-token (but not single-token) addition problems. Overall, our results highlight the phenomenon of convergent evolution in feature learning: A diverse range of models learn similar features from different training signals.

Understanding a discourse requires tracking entities and the relations that hold between them. While Large Language Models (LLMs) perform well on relational reasoning, the mechanism by which they bind entities, relations, and attributes remains unclear. We study discourse-level relational binding and show that LLMs encode it via a Cell-based Binding Representation (CBR): a low-dimensional linear subspace in which each ``cell'' corresponds to an entity--relation index pair, and bound attributes are retrieved from the corresponding cell during inference. Using controlled multi-sentence data annotated with entity and relation indices, we identify the CBR subspace by decoding these indices from attribute-token activations with Partial Least Squares regression. Across domains and two model families, the indices are linearly decodable and form a grid-like geometry in the projected space. We further find that context-specific CBR representations are related by translation vectors in activation space, enabling cross-context transfer. Finally, activation patching shows that manipulating this subspace systematically changes relational predictions and that perturbing it disrupts performance, providing causal evidence that LLMs rely on CBR for relational binding.

Static concreteness ratings are widely used in NLP, yet a word's concreteness can shift with context, especially in figurative language such as metaphor, where common concrete nouns can take abstract interpretations. While such shifts are evident from context, it remains unclear how LLMs understand concreteness internally. We conduct a layer-wise and geometric analysis of LLM hidden representations across four model families, examining how models distinguish literal vs figurative uses of the same noun and how concreteness is organized in representation space. We find that LLMs separate literal and figurative usage in early layers, and that mid-to-late layers compress concreteness into a one-dimensional direction that is consistent across models. Finally, we show that this geometric structure is practically useful: a single concreteness direction supports efficient figurative-language classification and enables training-free steering of generation toward more literal or more figurative rewrites.

Masked language modelling (MLM) has been the dominant pre-training objective for text encoders since BERT, yet it encourages representations that are strongly anchored to surface-form token identity rather than deeper semantic structure. Inspired by the success of Joint Embedding Predictive Architectures (JEPA) (LeCun, 2022) in vision and audio, we propose a hybrid pre-training objective that combines a JEPA-style latent-space prediction loss with a standard MLM objective over a single shared encoder. A learnable scalar parameter continuously balances the two objectives during training. We pre-train both a hybrid model and a pure-MLM baseline on English Wikipedia using identical architectures and compute budgets (NVIDIA H100). Extensive representation analysis across five GLUE benchmarks (SST-2, MRPC, MNLI, CoLA, STS-B) using four pooling strategies reveals that the hybrid encoder produces significantly more uniform embeddings (uniformity less than -0.16 vs -0.05 for MLM), exhibits richer spectral geometry under max pooling, encodes less surface-level lexical information, and achieves a better semantic-to-lexical balance. Despite similar linear-probe downstream accuracy, the geometric differences are consistent and significant, suggesting that the JEPA predictive objective reshapes the latent space in ways that standard accuracy metrics alone cannot capture.

When a decoder-only transformer is forced to process matched correct and incorrect single-token continuations of a factual query, the two pathways through hidden-state space diverge: displacement vectors from the query-only representation keep near-equal magnitude but rotate apart, with angular separation growing through mid-depth before late layers resolve an asymmetric outcome. A logit-lens preference in the incorrect run falls far below the equal-probability prior (roughly 11.5x more mass on the incorrect token than the correct one). We read this pattern, rotational divergence then late-layer asymmetric commitment, as the geometric signature of the model externally appearing to reject a wrong continuation, while staying explicit that it is observational, not causal: the incorrect run could equally reflect the model conforming to the token it is forced to carry, which only a random-token control can settle. It holds across six decoder-only transformers spanning four architecture families (Llama, Mistral, Gemma, StableLM) from 1B to 13B parameters; a seventh (Qwen2 1.5B) is flat under our protocol, plausibly a tokenizer artefact, leaving an emergence threshold open. Linear probes recover the distinction at intermediate depth, and cross-domain transfer is structurally asymmetric, a financial-medical corridor transferring far better than transport pairs. Where single-layer activation patching is cleanly interpretable (LLaMA-2 13B, Mistral 7B) it yields no layer band of consistent recovery; a third model (StableLM-2 1.6B) recovers uniformly above the ceiling, which we diagnose as a code-path artefact and exclude. Under this scoped null the late-layer asymmetry is not localized to a single component, fitting a distributed-by-trajectory account rather than single-layer localized recall. We document this with forced-completion probing across seven models, three domains, and 300 queries.

The internal representations learned by language models consistently exhibit striking geometric structure: calendar months organize into a circle, historical years form a smooth one-dimensional manifold, and cities' latitudes and longitudes can be decoded using a linear probe. To explain this neural code, we first show that language statistics exhibit translation symmetry (for example, the frequency with which any two months co-occur in text depends only on the time interval between them). We prove that this symmetry governs these geometric structures in high-dimensional word embedding models, and we analytically derive the manifold geometry of word representations. These predictions empirically match large text embedding models and large language models. Moreover, the representational geometry persists at moderate embedding dimension even when the relevant statistics are perturbed (e.g., by removing all sentences in which two months co-occur). We prove that this robustness emerges naturally when the co-occurrence statistics are controlled by an underlying latent variable. Our results indicate that these representational manifolds originate in the statistical symmetries of natural language.

Activation decomposition methods in language models are tightly coupled to geometric assumptions on how concepts are realized in activation space. Existing approaches search for individual global directions, implicitly assuming linear separability, which overlooks concepts with nonlinear or multi-dimensional structure. In this work, we leverage Mixture of Factor Analyzers (MFA) as a scalable, unsupervised alternative that models the activation space as a collection of Gaussian regions with their local covariance structure. MFA decomposes activations into two compositional geometric objects: the region's centroid in activation space, and the local variation from the centroid. We train large-scale MFAs for Llama-3.1-8B and Gemma-2-2B, and show they capture complex, nonlinear structures in activation space. Moreover, evaluations on localization and steering benchmarks show that MFA outperforms unsupervised baselines, is competitive with supervised localization methods, and often achieves stronger steering performance than sparse autoencoders. Together, our findings position local geometry, expressed through subspaces, as a promising unit of analysis for scalable concept discovery and model control, accounting for complex structures that isolated directions fail to capture.

Large language models (LLMs) were invented for natural language tasks such as translation, but they have proved that they can perform highly complex functions across domains. Additionally, they have been thought to develop new skills without being trained on them. These learning capabilities lead to LLMs adoption in a wide range of domains. Thus, it is imperative that we understand their operating mechanisms and limitations for proper diagnostics and repair. The earlier studies proposed that high level concepts are encoded as linear directions in LLMs activation space and that the geometry of embeddings have semantic meanings. Inspired by these studies, we hypothesize that LLMs may use subspaces and vector algebra in subspaces to perform tasks. To address this hypothesis, we analyze LLMs' functional modules and residual streams collected from LLMs engaging in in-context learning (ICL), one of the emergent abilities. Our analyses suggest that 1) LLMs can create subspaces, where evidence can be accumulated and 2) ICL tasks can be solved via simple algebraic operations in subspaces.