It is found that 1D manifolds with place-cell feature tiling emerge for tasks where the ordinal variable is locally computable from token identity, while tasks requiring cross-position integration or semantic extraction produce higher-dimensional or incoherent representations.
Abstract
Recent work showed that language models represent character counts on curved 1D manifolds, with attention heads performing geometric transformations to enable computation. We test whether this generalizes across four ordinal tasks (bracket depth, indentation, table position, numeric magnitude) in Gemma-2-2B, Gemma-2-9B, and Qwen3-4B. We find that 1D manifolds with place-cell feature tiling emerge for tasks where the ordinal variable is locally computable from token identity, while tasks requiring cross-position integration or semantic extraction produce higher-dimensional or incoherent representations. Geometric computation is architecture-dependent: Qwen3-4B shows substantially stronger twisting than Gemma models for indentation, and its twisters preserve ordinal order, unlike its numeric twisters. Activation patching confirms that the identified manifold subspaces concentrate task-relevant information, with manifold-direction ablation causing dramatically larger probe accuracy drops than random-direction controls.
A scale-dependent transition between two ID regimes is found: at low lexical diversity, conditions with fewer unique final words produce higher ID, while at high lexical diversity, this ordering reverses, and conditions with more unique words produce higher ID.
Arwa Osman, Marco Baroni, Iuri Macocco· 0 citations
Transformer representations describe trajectories through high-dimensional vector spaces, which are shaped dynamically as tokens incorporate relational context across layers. Such data tend to concentrate on lower-dimensional sub-manifolds, a form of compression quantified by the Intrinsic Dimensionality (ID), the minimum number of independent variables needed to represent them without significant information loss. In this work, we ask whether the grammatical role of tokens, as marked by their part-of-speech (PoS) tag, shapes the local geometry of this manifold. To this end: (1) We investigate the layer-wise evolution of ID, finding that closed-class items expand earlier and collapse sooner than open-class ones; (2) We show its expansion and contraction to be explained by changes in the neighborhood structure, and hence in the relations between words within a sentence; (3) We compare encoders (ModernBERT, bigbird-roberta-large) and decoders (gemma-2-2B, Llama-3.2-3B), finding that the two families evolve differently across layers, consistently with how each integrates context;(4) We show that geometric features alone recover a token's grammatical role, and use them to interpret how the semantic content of each PoS evolves across layers in a downstream classification task.
S. Vallisa, Federico Ravenda, C. Palominos et al.· 0 citations
Large language models place structured concepts on geometrically faithful manifolds: weekdays lie on a circle, months on another, usually taken to be a fixed world-model the network stores and looks up. We show that context is king: the structure a model actually uses is set by the in-context specification. A declarative rule fixes not only which relations the geometry encodes but its topology type: the same tokens form a cycle or a branching tree on command, built even on arbitrary, meaning-free tokens with no prior to inherit, which a relabeled stored shape cannot do. When the specification conflicts with a strong pretrained prior, the context-set geometry dominates it in capable models, read from the same activations (representational similarity 0.6--0.9 to the imposed structure versus near-zero to the prior), across the priors we test and both families we study (Gemma, Qwen). Activation patching shows the map is causally used, not a probe correlate: swapping one entity's activation for another's makes the model answer with the other entity's successor under the imposed order. A rough map forms readily, present even in small and base models; what scale gates is using it cleanly: clean dominance and the causal crossover emerge only in the larger models (up to Gemma-31B and Qwen-27B) and weaken or reverse below, so a mechanism present in a large model can be absent in a smaller one of the same family. Whether the model builds this geometry anew or reconfigures a stored one we leave open; operationally, the geometry it uses is the one the context specifies.
Behavior manifold analysis is introduced, which isolates behavior-specific geometry by selecting sparse behavior-associated coordinates and lifting them into low-dimensional local charts, and provides a unified framework for understanding the mechanistic distinction between the two objectives.
Juntong Wang, Shengkun Yang, Xiyuan Wang et al.· 0 citations
Existing hypotheses represent a concept in an LLM as a single point, a linear direction, or a Gaussian cluster, yet it remains unclear how and why such structures emerge. Here, we show that concept geometry can be precisely characterized via Laguerre Geometry, in which a concept is defined as a region--a Laguerre-Voronoi cell or a union of cells--allowing us to strictly define, measure, and separate concepts. Building on this formulation, we show that finer-grained concept structures, such as inclusion and hierarchy, are naturally revealed by the Laguerre weights. We then push this geometry inside the transformer. Decomposing each layer into piecewise-linear operators, we show that a token's hidden trajectory is governed by two coupled mechanisms: a static tree of self-contained piecewise-linear flow, and a dynamic transport that hops the trajectory across trees when cross-token attention fires. This decomposition yields Geometric Lens, a training-free, hyperparameter-free method for reading out the exact concept a hidden vector encodes at any layer. We also develop Laguerre Autoencoder, a 2D visualizer that renders both the decision geometry and a model's full reasoning trajectory in one view. Finally, we move beyond explanatory geometry toward actionable interpretability, showing that Geometric Lens recovers the correct factual token when a model is prompted with in-context interference. The code is available on GitHub.
Hyperbolic vision-language models are designed to encode abstraction geometrically: general concepts near the origin, specific ones farther out, and entailment cones representing directed order. We ask whether trained MERU, HyCoCLIP, and PHyCLIP models actually use these mechanisms. We audit seven released checkpoints and matched from-scratch interventions, using diagnostics that distinguish active hyperbolic geometry from angular structure and supervision effects. All audited converged checkpoints remain near-Euclidean in the dimensionless radius $u=\sqrt{c}\rho$, which measures how strongly embeddings experience hyperbolic geometry: the largest observed image-side value is $0.37$ -- well below $u\approx0.84$, where local metric distortion reaches $10\%$. Releasing the curvature floor changes curvature and norms but not this regime, with mixed, generally modest downstream shifts. Trained entailment cones are saturated or nearly saturated, so low violation rates can arise from trivially wide cones rather than learned order. Preregistered semantic traversal detects weak within-branch order but no operative full-hierarchy readout. Shuffle-controlled tests detect no pair-specific radial ordering in released checkpoints, and no positive result is consistent across all three matched ViT-B seeds. We trace this to a low-curvature shortcut: lowering curvature widens entailment cones and suppresses violations without learning order. In the probed trajectories, gradient decomposition identifies entailment as the dominant curvature-lowering pressure during collapse. Yet curvature contracts even when entailment is removed, so the shortcut is not the sole cause. Under our diagnostics, the audited formulations do not demonstrate an operative radial or cone-based hierarchy. We distill the audit into a five-number geometry report for evaluating future hierarchy claims.
Jaeyoung Kim, Eunseok Kim, Dongsuk Jang· 0 citations