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.
“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
The discussion before Theorem 1.3 (p. 3) explicitly rejects the claim that exponential capacity requires a vanishing radius of attraction: it says the dynamics can repair a number of random errors of order N. Theorem 1.3 (p. 4) considers errors at a positive Hamming radius ρN. It gives capacity M = exp(αN) + 1 for α < log(2)/2, rather than asserting capacity exactly 2^{d/2}. (LLM, conf=0.90)
Checked by a reasoning model — may be wrong.
Experimental — accuracy not yet measuredModern binary Hopfield networks with energy functions based on interaction functions of the form F(x)=exp(x) lead to storage capacity of 2^{d/2}, where all stored binary patterns are fixed points but the radius of attraction vanishes.
“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
Theorem 1.3 (p. 4) establishes exponential storage for F(x)=exp(x) with M=exp(αd)+1 for α<log(2)/2, and Remark 1.4 says all stored patterns are fixed points with probability tending to one. The claimed 2^{d/2} is the limiting exponent, not a capacity established at equality. The assertion that the radius of attraction vanishes is not supported as stated: the discussion preceding the theorem (p. 3) says the model can repair a number of random errors proportional to d, and the theorem allows a positive attraction radius at lower storage exponents. (LLM, conf=0.86)
Experimental — accuracy not yet measuredThe exponential interaction function energy leads to an exponential storage capacity of N = 2^{d/2} for binary patterns.
“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
Theorem 1.3 (p. 4) establishes exponential storage for binary patterns, but specifies M = exp(αN) + 1 only for α < log(2)/2. Thus the paper supports rates approaching 2^(d/2), not the stated capacity exactly at 2^(d/2). (LLM, conf=0.89)
Experimental — accuracy not yet measuredDemircigil et al. generalized the model to exponential interaction functions F(x)=exp(x), giving energy E = -exp(lse(1, X^T ξ)).
“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
Theorem 1.3 (p. 4) explicitly studies the generalized Hopfield model with interaction function F(x)=e^x. The available text does not state the claimed energy formula E=-exp(lse(1, Xᵀξ)); the relevant formulas are not decoded in our copy, so that part cannot be verified. (LLM, conf=0.83)
Experimental — accuracy not yet measuredWith a single update, the fixed point is recovered with high probability in the Demircigil et al. model.
“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
The Proof of Theorem 1.3 (p. 6) says the model corrects random errors by updating each neuron once, and the proof (p. 7) says the probability of an incorrect neuron update vanishes as the number of neurons grows. This supports high-probability recovery after one pass through the neurons, not necessarily after a single neuron update. (LLM, conf=0.87)
Experimental — accuracy not yet measured
It has been shown that the network converges with high probability after one update.
“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
The proof of Theorem 1.3 (p. 6) says the model can correct random errors by updating each neuron once, and the theorem (p. 4) concerns randomly corrupted stored patterns under specified conditions. That supports high-probability retrieval after a full pass through the neurons, not convergence after a single neuron update or from an arbitrary network state. (LLM, conf=0.87)
Experimental — accuracy not yet measuredInteraction 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.
“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
Theorem 1.1 (p. 3) states that F(x) = x^n permits storage of up to α_n N^(n−1) patterns when small retrieval errors are tolerated. It does not state that α_n depends on the allowed error probability; that dependence is not established by the passage. (LLM, conf=0.83)
Experimental — accuracy not yet measuredFor the generalized Hopfield model with interaction function F(x)=e^x, with N=exp(αd)+1 patterns chosen uniformly at random from {-1,+1}^d, and for any pattern perturbed within Hamming radius ρd, the update recovers the pattern with high probability if α is chosen appropriately.
“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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Wei, H., Li, F., & Miao, W. (2024) · PLOS complex systems. · doi:10.1371/journal.pcsy.0000019 · Cited by 1 · see citing works
Marullo, C., Buonaiuto, G., Ciani, D., Gargiulo, F., & Esposito, M. (2024) · Reviews in Mathematical Physics · doi:10.1142/s0129055x24300115 · Cited by 1 · see citing works
On the capacity of continuous-time Hopfield models
Betteti, S., Baggio, G., & Zampieri, S. (2024) · IEEE Conference on Decision and Control (CDC) · doi:10.1109/cdc56724.2024.10886497 · Cited by 1 · see citing works
Content-Addressable Memory with a Content-Free Energy Function
Benoist, F., Peliti, L., & Sartori, P. (2025) · Physical Review Letters · doi:10.1103/tcqx-586g · Cited by 1 · see citing works
Statistical mechanics of vector Hopfield network near and above saturation
Nicoletti, F., D’Amico, F. E., & Negri, M. (2025) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8121/ae2bd0 · Cited by 1 · see citing works
Dense associative memory in a nonlinear optical Hopfield neural network
Musa, K., Kumar, S., Katidis, M., & Huang, Y.-P. (2026) · Computational Optical Imaging and Artificial Intelligence in Biomedical Sciences · doi:10.1117/12.3080047 · Cited by 1 · see citing works
Hopfield Networks as Models of Emergent Function in Biology
Yampolskaya, M., & Mehta, P. (2026) · Annual Review of Biophysics · doi:10.1146/annurev-biophys-101425-023356 · Cited by 1 · see citing works
He, C., Jiang, M., Shan, K., Yang, S., Li, Z., Wang, S., Pedretti, G., Ignowski, J., & Li, C. (2026) · Nature Communications · doi:10.1038/s41467-026-69958-0 · Cited by 1 · see citing works
Associative Memories in the Feature Space
Salvatori, T., Millidge, B., Song, Y., Bogcaz, R., & Lukasiewicz, T. (2023) · Frontiers in artificial intelligence and applications · doi:10.3233/faia230500 · Cited by 1 · see citing works
Topological properties of basins of attraction of width bounded autoencoders
Beise, H.-P., & Cruz, S. (2024) · Analysis and Applications · doi:10.1142/s0219530524500076 · Cited by 1 · see citing works
Krasilenko, V. G., Kychak, V. M., Nikolsky, A. I., Lazarev, A. A., & Nikitovich, D. V. (2024) · Herald of Khmelnytskyi National University Technical sciences · doi:10.31891/2307-5732-2024-331-30 · Cited by 1 · see citing works
Exact Replica Symmetric Solution for Transverse Field Hopfield Model under Finite Trotter Size
Okajima, K., & Kabashima, Y. (2025) · Journal of the Physical Society of Japan · doi:10.7566/jpsj.94.044005 · Cited by 1 · see citing works
Dat, D. H., Mao, P.-Y., Nguyen, T. T., Buntine, W., & Bennamoun, M. (2025) · IEEE Winter Conference on Applications of Computer Vision · doi:10.1109/wacv61041.2025.00115 · Cited by 1 · see citing works
Strong Anti-Hebbian Plasticity Alters the Convexity of Network Attractor Landscapes
Gong, L., Chen, X., & Ching, S. (2025) · IEEE Transactions on Neural Networks and Learning Systems · doi:10.1109/tnnls.2025.3561217 · Cited by 1 · see citing works
Sequential Learning in the Dense Associative Memory
McAlister, H., Robins, A. V., & Szymanski, L. (2025) · Neural Computation · doi:10.1162/neco.a.20 · Cited by 1 · see citing works
An Application of the Associate Hopfield Network for Pattern Matching in Chart Analysis
Mai, W., & Lee, R. (2021) · Applied Sciences · doi:10.3390/app11093876 · Cited by 1 · see citing works
Abubakar, H. (2021) · Kuwait Journal of Science · doi:10.48129/kjs.11861 · Cited by 1 · see citing works
Using Hopfield Networks to Correct Instruction Faults
Köylü, T. Ç., Fieback, M., Hamdioui, S., & Taouil, M. (2022) · Proceedings - Asian Test Symposium/Proceedings · doi:10.1109/ats56056.2022.00030 · Cited by 1 · see citing works
ATHENA for Memory-Based InferencePreprint
Ralston, R. (2023) · doi:10.31234/osf.io/b6j9n · Cited by 1 · see citing works
Associative memories using complex-valued Hopfield networks based on spin-torque oscillator arrays
Prasad, N., Mukim, P., Madhavan, A., & Stiles, M. D. (2022) · Neuromorphic Computing and Engineering · doi:10.1088/2634-4386/ac7d05 · Cited by 12 · see citing works
One Shoot In-Door Surveillance Module Based On MCA Associative Memory
Mubarak, G. A., & Kareem, E. I. A. (2019) · International Conference of Computer and Applied Sciences (CAS) · doi:10.1109/cas47993.2019.9075724
Capsule Networks as Generative Models
Kiefer, A. B., Millidge, B., Tschantz, A., & Buckley, C. (2022) · International Workshop on Affective Interactions · doi:10.1007/978-3-031-28719-0_14
Inference Acceleration of Deep Learning Classifiers Based on RNN
Keddous, F. E., Shvai, N., Llanza, A., & Nakib, A. (2023) · Proceedings - International Conference on Image Processing · doi:10.1109/icip49359.2023.10222316
Comparison of Structural Associative Memory with Modern Hopfield NetworksPreprint
Raif, P., Starzyk, J. A., Stokłosa, P., Horzyk, A., & Bulanda, D. (2023) · SSRN Electronic Journal · doi:10.2139/ssrn.4630831
Krasilenko, V. G., Chikov, I. A., & Nikitovich, D. V. (2023) · Modeling Control and Information Technologies · doi:10.31713/mcit.2023.037
A Multi-layer Hopfield Neural Network for Semi-Supervised Anomaly Detection
Guo, P. W., & Kim, D. K. (2023) · Journal of Digital Contents Society · doi:10.9728/dcs.2023.24.11.2893
Ororbia, A. G., & Kelly, M. A. (2024) · Proceedings of the AAAI Symposium Series · doi:10.1609/aaaiss.v2i1.27702
Towards a Model of Associative Memory with Learned Distributed Representations
Fandl, M., & Takáč, M. (2024) · Lecture notes in computer science · doi:10.1007/978-3-031-72332-2_16
- 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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