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The dimensionality of the Hopfield model

Author

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  • Erazo, Cristopher
  • Acevedo, Santiago
  • Ingrosso, Alessandro

Abstract

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.

Suggested Citation

  • Erazo, Cristopher & Acevedo, Santiago & Ingrosso, Alessandro, 2026. "The dimensionality of the Hopfield model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 695(C).
    • Handle: RePEc:eee:phsmap:v:695:y:2026:i:c:s0378437126003456
    DOI: 10.1016/j.physa.2026.131609
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