The DOI matches its Crossref record; 13 of the 13 references it cites were found in a registry or an index.
On a Model of Associative Memory with Huge Storage Capacity
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“Interaction functions of the form F ( x ) = exp( x ) lead to exponential storage capacity of 2 d/ 2 where all stored patterns are fixed points but the radius of attraction vanishes (Demircigil et al., 2017).”
From Hopfield Networks is All You Need (2020) · cites this paper as Demircigil et al., 2017
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“Modern binary Hopfield networks with energy functions based on interaction functions of the form F ( x ) = exp( x ) even lead to storage capacity of 2 d/ 2 , where all stored binary patterns are fixed points but the radius of attraction vanishes (Demircigil et al., 2017).”
From Hopfield Networks is All You Need (2020) · cites this paper as Demircigil et al., 2017
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“This energy leads to an exponential storage capacity of N = 2 d/ 2 for binary patterns.”
From Hopfield Networks is All You Need (2020) · cites this paper as Demircigil et al., 2017
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“This model was generalized by Demircigil et al. (Demircigil et al., 2017) to exponential interaction functions F ( x ) = exp( x ) , which gives the energy E = -exp(lse(1 , X T ξ )) .”
From Hopfield Networks is All You Need (2020) · cites this paper as Demircigil et al., 2017
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“Furthermore, with a single update the fixed point is recovered with high probability.”
From Hopfield Networks is All You Need (2020) · cites this paper as Demircigil et al., 2017
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“It has been shown that the network converges with high probability after one update (Demircigil et al., 2017).”
From Hopfield Networks is All You Need (2020) · cites this paper as Demircigil et al., 2017
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“Interaction functions of the form F ( x ) = x n lead to storage capacity of α n d n -1 , where α n depends on the allowed error probability (Krotov & Hopfield, 2016; 2018; Demircigil et al., 2017) (see (Krotov & Hopfield, 2018) for the non-binary case).”
From Hopfield Networks is All You Need (2020) · cites this paper as Demircigil et al., 2017
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“Theorem A10 (Storage Capacity for Binary Modern Hopfield Nets (Demircigil et al. 2017)) . Consider the generalized Hopfield model with the dynamics described in Eq. (545) and interaction function F given by F ( x ) = e x . For a fixed 0 < α < ln(2) / 2 let N = exp( αd ) + 1 and let x 1 , . . . , x N be N patterns chosen uniformly at random from {-1 , +1 } d .”
From Hopfield Networks is All You Need (2020) · cites this paper as Demircigil et al., 2017
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Citations
Cites
Neural networks and physical systems with emergent collective computational abilities.
Hopfield, J. J. (1982) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.79.8.2554 · Cited by 19,270
Spin-glass models of neural networks
Amit, D. J., Gutfreund, H., & Sompolinsky, H. (1985) · Physical Review A · doi:10.1103/physreva.32.1007 · Cited by 1,201
Storing Infinite Numbers of Patterns in a Spin-Glass Model of Neural Networks
Amit, D. J., Gutfreund, H., & Sompolinsky, H. (1985) · Physical Review Letters · doi:10.1103/physrevlett.55.1530 · Cited by 1,098
Large Deviations Techniques and Applications
Dembo, A., & Zeitouni, O. (2009) · Applications of mathematics · doi:10.1007/978-3-642-03311-7 · Cited by 1,056
The capacity of the Hopfield associative memory
McEliece, R., Posner, E., Rodemich, E., & Venkatesh, S. (1987) · IEEE Transactions on Information Theory · doi:10.1109/tit.1987.1057328 · Cited by 910
Memory capacity in neural network models: Rigorous lower bounds
Newman, C. M. (1988) · Neural Networks · doi:10.1016/0893-6080(88)90028-7 · Cited by 121
Rigorous results for the Hopfield model with many patterns
Talagrand, M. (1998) · Probability Theory and Related Fields · doi:10.1007/s004400050148 · Cited by 88
On the storage capacity of Hopfield models with correlated patterns
Löwe, M. (1998) · The Annals of Applied Probability · doi:10.1214/aoap/1028903378 · Cited by 51
Self-citationThe spin-glass phase-transition in the Hopfield model with $p$-spin interactions
Bovier, A., & Niederhauser, B. M. (2001) · Advances in Theoretical and Mathematical Physics · doi:10.4310/atmp.2001.v5.n6.a2 · Cited by 28
Lower bounds on the restitution error in the Hopfield Model
Daria, L. (1997) · Probability Theory and Related Fields · doi:10.1007/s004400050081 · Cited by 23
The capacity of q-state Potts neural networks with parallel retrieval dynamics
Löwe, M., & Vermet, F. (2007) · Statistics & Probability Letters · doi:10.1016/j.spl.2007.03.030 · Cited by 14
Self-citationThe storage capacity of the Hopfield model and moderate deviations
Löwe, M., & Vermet, F. (2005) · Statistics & Probability Letters · doi:10.1016/j.spl.2005.06.001 · Cited by 12
Self-citationSharp upper bounds on perfect retrieval in the Hopfield model
Bovier, A. (1999) · Journal of Applied Probability · doi:10.1017/s0021900200017708 · Cited by 2
Research works printed but not linked: 2
- Krotov, D., Hopfield, J.J.: Dense associative memory for pattern recognition. In: Lee, D.D., Sugiyama, M., Luxburg, U.V., Guyon, I., Garnett, R. (eds.) Advances in Neural Information Processing Systems, vol. 29, pp. 1172–1180. Curran Associates, Inc., Red Hook (2016)
- Löwe, M.: The storage capacity of generalized Hopfield models with semantically correlated patterns. Markov Process. Relat. Fields 5(1), 1–19 (1999)
Cited by
Sorted by citations
Hopfield Networks is All You Need
Ramsauer, H., Schäfl, B., Lehner, J., Seidl, P., Widrich, M., Adler, T., Gruber, L., Holzleitner, M., Pavlović, M., Sandve, G. K., et al. (2020) · arXiv · doi:10.48550/arxiv.2008.02217 · Cited by 236
“It has been shown that the network converges even after one update [23].”
BackgroundMethodInfluentialModern Hopfield Networks and Attention for Immune Repertoire ClassificationPreprint
Widrich, M., Schäfl, B., Pavlović, M., Ramsauer, H., Gruber, L., Holzleitner, M., Brandstetter, J., Sandve, G. K., Greiff, V., Hochreiter, S., et al. (2020) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2020.04.12.038158 · Cited by 111
A generative model of memory construction and consolidation
Spens, E., & Burgess, N. (2024) · Nature Human Behaviour · doi:10.1038/s41562-023-01799-z · Cited by 122
A new frontier for Hopfield networks
Krotov, D. (2023) · Nature Reviews Physics · doi:10.1038/s42254-023-00595-y · Cited by 97
CRS-Diff: Controllable Remote Sensing Image Generation With Diffusion Model
Tang, D., Cao, X., Hou, X., Jiang, Z., Liu, J., & Meng, D. (2024) · IEEE Transactions on Geoscience and Remote Sensing · doi:10.1109/tgrs.2024.3453414 · Cited by 103
Metadata
- Authors
- Demircigil, M., Heusel, J., Löwe, M., Upgang, S., & Vermet, F.
- Affiliations
- École Normale Supérieure - PSL (FR)
- University of Münster (DE)
- Centre National de la Recherche Scientifique (FR)
- Corresponding author
- Vermet, Franck
- Venue
- Journal of Statistical Physics
- Published
- 2017-05-19
- Volume
- 168
- Issue
- 2
- Pages
- 288-299
- ISSN
- 0022-4715 (ISSN-L)
- 1572-9613
- OpenAlex
- W2587333550
- Cited by
- 136
- References
- 13
- Citations by year
- 2026: 17 · 2025: 35 · 2024: 31 · 2023: 19 · 2022: 7 · 2021: 12 · 2020: 6 · 2019: 2 · 2018: 4 · 2017: 3
- FWCI
- 2.45
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Journal of Statistical Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Neural Networks and Applications
- Life Sciences › Neuroscience › Cognitive Neuroscience › Neural dynamics and brain function
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Theoretical and Computational Physics
- Keywords
- associative memory, Hopfield neural network, storage capacity, basins of attraction, artificial neural networks
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