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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.17427 |
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| _version_ | 1866914277608128512 |
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| author | Erazo, Cristopher Acevedo, Santiago Ingrosso, Alessandro |
| author_facet | Erazo, Cristopher Acevedo, Santiago Ingrosso, Alessandro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17427 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The dimensionality of the Hopfield model Erazo, Cristopher Acevedo, Santiago Ingrosso, Alessandro Disordered Systems and Neural Networks Computational Physics 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. |
| title | The dimensionality of the Hopfield model |
| topic | Disordered Systems and Neural Networks Computational Physics |
| url | https://arxiv.org/abs/2601.17427 |