Geometry of Cognitive Maps in Minimal Representations

An effective cognitive map organizes states into a compact coordinate system whose geometry—the relative arrangement of states—preserves task-relevant relations and makes them explicit for inference and planning. Yet determining how task structure constrains this geometry is difficult in neural population recordings and artificial neural networks, where representations are obscured by noise, finite sampling, learning dynamics, and architectural choices. Here, we model cognitive maps as regularized hidden-layer activations in linear neural networks trained to predict the next state. Building on previously derived theoretical representations, we prove that their dimensionality can be analytically computed as the number of connected components plus the number of globally compatible independent actions, minus one. Applying this theory across spatial and non-spatial tasks, our results reveal an organizing principle of cognitive maps not captured by standard state-transition models: how transitions are grouped into relational operations could shape cognitive-map geometry. The theory also provides several insights over neural representations, including how shared action encoding supports structural transfer and how increasing similarity among action representation (i.e. action merging) may serve as a mechanism for acquiring a low-dimensional cognitive map. Cognitive-map representations therefore reflect an agent's internal definitions of states and relational operations rather than the external environment alone. By translating candidate task models into quantitative predictions of neural population geometry, our theory may also open a principled route to inferring how an agent internally represents a cognitive map.