By virtue of linguistic compositionality, few syntactic rules and a finite lexicon can generate an unbounded number of sentences. That is, language, though seemingly high-dimensional, can be explained using relatively few degrees of freedom. An open question is whether contemporary language models (LMs) reflect the intrinsic simplicity of language that is enabled by compositionality. We take a geometric view of this problem by relating the degree of compositionality in a dataset to the intrinsic dimension (ID) of its representations under an LM, a measure of feature complexity. We find not only that the degree of dataset compositionality is reflected in representations’ ID, but that the relationship between compositionality and geometric complexity arises due to learned linguistic features over training. Finally, our analyses reveal a striking contrast between nonlinear and linear dimensionality, showing they respectively encode semantic and superficial aspects of linguistic composition.

Why do biological and artificial neurons sometimes modularise, each encoding a single meaningful variable, and sometimes entangle their representation of many variables? In this work, we develop a theory of when biologically inspired networks---those that are nonnegative and energy efficient---modularise their representation of source variables (sources). We derive necessary and sufficient conditions on a sample of sources that determine whether the neurons in an optimal biologically-inspired linear autoencoder modularise. Our theory applies to any dataset, extending far beyond the case of statistical independence studied in previous work. Rather we show that sources modularise if their support is ``sufficiently spread''. From this theory, we extract and validate predictions in a variety of empirical studies on how data distribution affects modularisation in nonlinear feedforward and recurrent neural networks trained on supervised and unsupervised tasks. Furthermore, we apply these ideas to neuroscience data, showing that range independence can be used to understand the mixing or modularising of spatial and reward information in entorhinal recordings in seemingly conflicting experiments. Further, we use these results to suggest alternate origins of mixed-selectivity, beyond the predominant theory of flexible nonlinear classification. In sum, our theory prescribes precise conditions on when neural activities modularise, providing tools for inducing and elucidating modular representations in brains and machines.

We investigate the robustness of Large Language Models (LLMs) to structural interventions by deleting and swapping adjacent layers during inference. Surprisingly, models retain 72–95\% of their original top-1 prediction accuracy without any fine-tuning. We find that performance degradation is not uniform across layers: interventions to the early and final layers cause the most degradation, while the model is remarkably robust to dropping middle layers. This pattern of localized sensitivity motivates our hypothesis of four stages of inference, observed across diverse model families and sizes: (1) detokenization, where local context is integrated to lift raw token embeddings into higher-level representations; (2) feature engineering, where task- and entity-specific features are iteratively refined; (3) prediction ensembling, where hidden states are aggregated into plausible next-token predictions; and (4) residual calibration, where irrelevant features are suppressed to finalize the output distribution. Synthesizing behavioral and mechanistic evidence, we provide a hypothesis for interpreting depth-dependent computations in LLMs.

Diverse studies in systems neuroscience begin with extended periods of curriculum training known as ‘shaping’ procedures. These involve progressively studying component parts of more complex tasks, and can make the difference between learning a task quickly, slowly or not at all. Despite the importance of shaping to the acquisition of complex tasks, there is as yet no theory that can help guide the design of shaping procedures, or more fundamentally, provide insight into its key role in learning. Modern deep reinforcement learning systems might implicitly learn compositional primitives within their multilayer policy networks. Inspired by these models, we propose and analyse a model of deep policy gradient learning of simple compositional reinforcement learning tasks. Using the tools of statistical physics, we solve for exact learning dynamics and characterise different learning strategies including primitives pre-training, in which task primitives are studied individually before learning compositional tasks. We find a complex interplay between task complexity and the efficacy of shaping strategies. Overall, our theory provides an analytical understanding of the benefits of shaping in a class of compositional tasks and a quantitative account of how training protocols can disclose useful task primitives, ultimately yielding faster and more robust learning.