The dimensionality of the Hopfield model

We use the Binary Intrinsic Dimension (BID), a geometrical measure designed for binary data, to analyze the Hopfield model, a paradigmatic spin system from statistical mechanics, machine learning and neuroscience. The BID allows us to characterize the phases and transitions of this system, and moreover it is robust against finite-size effects that interfere with the correct numerical estimation of the spin-glass order parameter ($q$). We observe that the BID scales linearly with system size in the retrieval and paramagnetic phases, where the correlations between spins are small, and exhibits sublinear scaling in the whole spin-glass phase, highlighting its correlated structure. Furthermore, we establish a direct relationship between the BID and the overlap distribution, unveiling a novel connection between the geometry of the state-space and standard spin order parameters.