The information geometry of large language models is shared, learned, and controllable
Dario Picozzi (University College London) studies the Fisher-Rao geometry of next-token probabilities and shows that it is shared across transformer, state-space and recurrent language models, that it tracks what the model learns, and that it gives a principled way to make local edits with minimal side effects.
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Shared structure: Output geometries agree across architectures more strongly than activation geometries, because behavior fixes the output geometry up to symmetries while activation geometry depends on the coordinate system. The shared geometry supports transfer of semantic categories.
Learning: Agreement with human word choices increases with predictive accuracy, scale and training. Pretraining corpus statistics predict held-out fact acquisition, and deeper evidence delays acquisition in every tested architecture.
Control: The geometry prescribes minimum-disturbance interventions and predicts their relative cost. Updates learned on donor prompts transfer to unseen prompts and preserve behavior on reference prompts better than Euclidean updates.
Applications: The same geometric correction improves steering, editing, attribution, dictionary learning and fine-tuning.
Abstract
Large language models learn similar behaviours, yet it remains unclear what structure they share or how to change one behaviour without disturbing others. The Fisher-Rao geometry of next-token probabilities connects these questions: behaviour determines this geometry up to output-preserving symmetries, whereas activation geometry depends on coordinates. Across transformer, state-space and recurrent models, output geometries agree more strongly than activation geometries, and shared geometry supports semantic-category transfer. Agreement with human word choices increases with predictive accuracy, scale and training, and improves further after model-only calibration. Token probabilities and read-out geometry jointly predict the spectrum and its effective dimension. Controlled language assignments show that geometry follows the language law across architectures. Pretraining corpus statistics predict held-out fact acquisition without recalibration, while randomised experiments show that deeper evidence substantially delays acquisition across every tested architecture and evidence construction. Finally, the geometry prescribes minimum-disturbance local interventions, predicts their relative cost, and supports reusable control: updates learned on donor prompts transfer to unseen prompts while better preserving behaviour on reference prompts than Euclidean control. The same geometric correction improves steering, editing, attribution, dictionary learning and fine-tuning.