The Geometry of Updates: Fisher Alignment at Vocabulary Scale

Activation steering methods modify intermediate representations of language models to control output behavior, but universally assume the activation space is Euclidean. We show this assumption fails drastically: the local geometry induced by the model's own output behavior -- the Fisher information metric of the softmax layer, pulled back through the Jacobian of subsequent layers -- deviates from the Euclidean metric by over 97% in relative spectral norm on GPT-2, with an effective dimensionality of only 2--17% of the ambient space. From this pullback Fisher metric, we derive a closed-form steering equation that identifies the minimum-distortion direction for any target concept, yielding a closed-form optimal direction at each point that can be applied iteratively without manifold fitting or data-driven geometry estimation. We call the resulting framework FishBack. The metric admits a layer-wise recursive decomposition, which reveals that existing methods -- CAA, ActAdd, ITI, and others -- each implicitly adopt a particular approximate metric, and that their performance gaps are quantitatively predicted by a single spectral diagnostic: the ratio of their implicit metric's cost to the Fisher-optimal cost. On GPT-2, iterative pullback steering consistently outperforms all Euclidean baselines across three verb-morphology concepts and four layers, with off-target KL reductions of $1.3\times$--$2.5\times$ relative to Euclidean gradient ascent and $1.5\times$ relative to CAA at matched concept probability.

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.

The human ability to translate diverse perceptual and linguistic inputs into structured behavior has been thought to rest on learning robust representations of concepts. The rapid advancement of transformer-based large language models (LLMs) has surfaced a diversity of computational ingredients relevant for model building - architectures, fine-tuning methods, and training datasets among others - yet it remains unclear which are most crucial for developing human-like conceptual representations. Further, most current benchmarks are ill-suited to measuring representational alignment, making LLMs' scores on them unreliable for assessing whether they are progressing as cognitive models. We address these limitations by evaluating over 75 models on a triplet similarity task, a method well established in cognitive science for measuring conceptual representations, using concepts from the THINGS database. We find that instruction fine-tuning and larger attention head dimensionality are among the strongest predictors of human alignment, while activation function choice, multimodal pretraining, and parameter size have limited influence on alignment. Correlations between alignment scores and existing benchmark scores reveal that while some benchmarks (e.g., BigBenchHard) better capture representational alignment than others (e.g., MUSR), none fully accounts for the variance in human-model alignment, demonstrating their insufficiency. Taken together, our findings highlight key computational ingredients for advancing LLMs as models of human conceptual representation and address a key gap in LLM evaluation.