Pith. sign in

REVIEW 2 major objections 4 minor 47 references

A binary Hopfield network with an exponential energy function stores exponentially many patterns in N and retrieves them from corrupted inputs in one step, provided the initial overlap exceeds about 0.337.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-04 22:51 UTC pith:MO4FZYI3

load-bearing objection Incremental but sound: the new cost function is equivalent to existing exponential Hopfield models, and the basin-threshold derivation is new and correct; the stress-test objection about skewed signals miscomputes the noise scale. the 2 major comments →

arxiv 2509.06905 v1 pith:MO4FZYI3 submitted 2025-09-08 cond-mat.dis-nn

Yet another exponential Hopfield model

Linda Albanese , Andrea Alessandrelli , Adriano Barra , Peter Sollich This is my paper
classification cond-mat.dis-nn MSC 82C3292B20
keywords exponential Hopfield modelassociative memorystorage capacitysignal-to-noise analysisbinary neuronsperfect recallbasins of attractiondense associative memories
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a binary-neuron Hopfield-style network whose energy is a sum of exponentials of quadratic losses, one per stored pattern, and argues that this model can store an exponentially large number of patterns in the number of neurons N. Using a signal-to-noise analysis of zero-temperature synchronous updates, the authors show that a stored pattern remains a fixed point as long as P is below about (1/(2 ln N))(2/(1+e^{-4}))^{N-1}, and that after one parallel update the pattern overlap is an error function of the storage load. They also analyze retrieval from corrupted inputs: as long as the initial overlap r exceeds about 0.337, the network recalls the pattern in one step, and although enlarging the basin of attraction lowers the exponent, the capacity stays exponential. A separate perfect-recall equilibrium argument reproduces the same bound up to logarithmic prefactors. The work matters because it shows that this non-mean-field, binary version of the exponential Hopfield family keeps the high storage scaling while remaining analytically tractable.

Core claim

The central claim is that the cost function H_N = -N \sum_\mu \exp(N(m_\mu - 1)), with m_\mu the Mattis magnetization of pattern \mu, makes perfect recall the natural low-energy state and yields exponential storage capacity for binary neurons. The paper shows by signal-to-noise that, starting from a stored pattern, the probability that any given spin does not flip tends to 1 provided P[(1+e^{-4})/2]^{N-1} \to 0, giving P \le (1/(2a \ln N))(2/(1+e^{-4}))^{N-1}. For corrupted inputs with overlap r, the same calculation gives a one-step retrieval condition and a capacity P = \gamma ([(1+r)+(1-r)e^{-2}]^2/[2(1+e^{-4})])^{N-1}, with perfect recall for r > 0.3374; the exponent shrinks with r but n

What carries the argument

The carrying object is the exponential cost function H_N = -N \sum_\mu \exp(-N L_\mu), where L_\mu = ||\xi^\mu - \sigma||^2/(2N) is the per-pattern quadratic loss; because -N L_\mu = N(m_\mu - 1), perfect recall m_\mu = 1 is exponentially favored. The argument then works through the post-synaptic field h_i(\sigma_{\backslash i}|\xi) obtained by separating neuron i from the sum; the zero-temperature update is \sigma_i^{\text{new}} = \sigma_i \operatorname{sign}(h_i \sigma_i). The field is approximated as Gaussian via the Central Limit Theorem, with mean e^{-1}\sinh(1) and variance proportional to P[(1+e^{-4})/2]^{N-1}, so stability, one-step magnetization, and basin size all reduce to error f

Load-bearing premise

The calculation approximates the field on each neuron as Gaussian with independent pattern contributions via the Central Limit Theorem, a step the paper itself calls a simplification, and correlations between noise variables are handled only in the appendix; the numerical verification reaches only N=22, so the asymptotic exponential scaling is not directly tested.

What would settle it

Simulate the proposed network at N=100 or larger with P set to the predicted bound P = (2/(1+e^{-4}))^{N-1}/(2 \ln N), start from a stored pattern, perform one parallel update, and measure the average Mattis magnetization: if it does not approach 1 as N grows, the exponential capacity claim fails. A complementary check is to compute the probability that all spins are stable, including correlated noise, and compare it with Eq. (5.14).

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

If this is right

  • A binary, non-mean-field network can store P ~ c^N / \ln N patterns, matching the exponential scaling of the original real-valued exponential Hopfield model.
  • Requiring recall from inputs corrupted up to Hamming distance d keeps capacity exponential as long as d < 0.331; the cost is only a smaller exponent.
  • The one-step update rule is identical to the one derived from the Ramsauer et al. cost function, so the two models share retrieval dynamics despite different energies.
  • Stability against all spin flips, rather than a single flip, changes only logarithmic prefactors, not the exponential scaling.
  • The perfect-recall equilibrium derivation gives the same bound independently, with prefactor 1/[2 \ln(N/\delta)], so the result does not depend on a single technical route.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A natural next target is finite temperatures: the exponential energy landscape likely exhibits a condensation-like transition where one pattern dominates, analogous to a random-energy-model phase, which the paper does not analyze.
  • The Gaussian approximation treats the noise variables z_i as independent; the appendix computes correlations only for all-spin-flip stability. A full treatment of correlated noise could shift the threshold r \approx 0.337, so numerical tests at larger N would discriminate.
  • Because the same update rule arises from the Ramsauer cost, this construction may be viewed as an energy-based dual of modern continuous Hopfield networks, suggesting that storage capacity is determined more by the exponential kernel than by the specific energy function.
  • The theory suggests a practical recipe: in a learned associative memory, choosing a loss whose exponential weights perfect recall should yield exponentially large capacity, a prediction testable in small-N simulations before scaling to hardware.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper introduces a binary-neuron exponential Hopfield model with cost function H_N = -N Σ_μ exp(N(m_μ-1)), derives the zero-temperature parallel update rule, and analyzes retrieval by a signal-to-noise/CLT approach. It claims that stored patterns are stable fixed points up to P ∼ (1/(2a ln N))(2/(1+e^{-4}))^{N-1} patterns, and that corrupted inputs with initial overlap r > 0.337 are retrieved after one parallel step while preserving exponential capacity. An independent equilibrium argument in the perfect-recall regime and an all-spin-flip stability appendix are presented as corroboration.

Significance. If the fixed-point claim stood alone, this would be a clean and useful contribution: the derivation is self-contained, the prefactor in the exponential capacity is explicit and falsifiable, and the MCMC data in Fig. 1 are consistent with the prediction at the tested system size. The paper contains no parameter fitting and the central scaling is derived rather than assumed. However, the paper's advertised extension beyond fixed points—the basin-of-attraction analysis of Sec. 2.2—is not sound. Since robustness to corrupted inputs is one of the two main results emphasized in the abstract and conclusions, the manuscript in its current form overclaims.

major comments (2)
  1. [Sec. 2.2, Eq. (2.37)] The second moment μ_2 is computed incorrectly. For the μ=1 contribution, S_i = e^{-1} sinh(1) e^{-2K_i} with K_i ~ Bin(N-1,(1-r)/2), so E[S_i^2] = e^{-2} sinh^2(1) [((1+r)+(1-r)e^{-4})/2]^{N-1}. Equation (2.37) instead uses (E[S_i])^2 = e^{-2} sinh^2(1) [((1+r)+(1-r)e^{-2})/2]^{2(N-1)}. This mistake enters the variance μ_2-μ_1^2 and propagates into (2.38)-(2.41), which are the basis of the claimed basin threshold and capacity.
  2. [Sec. 2.2, Eqs. (2.38)-(2.43)] Even with a corrected μ_2, the CLT treatment of the signal term is invalid for corrupted inputs. The term S_i is strictly positive and heavy-tailed across neurons; its typical value is e^{-1} sinh(1) e^{-(1-r)(N-1)}, while under the capacity choice (2.39) the noise standard deviation is e^{-1} sinh(1)√γ A_0^{N-1}, A_0 = ((1+r)+(1-r)e^{-2})/2. Since e^{-(1-r)}/A_0 < 1 for every r<1, the typical signal is exponentially smaller than the noise for any fixed γ. Therefore P(X_i<0) → 1/2 for typical neurons and m_1^{(1)} → 0, not erf(1/√(2γ)). The P=1 case for r<r_c exposes the failure: exact dynamics gives m_1=1, while (2.40) gives m_1→0. The basin threshold r≈0.337 is an artifact of using the mean instead of the typical signal and of the Gaussian approximation for a positive non-Gaussian term.
minor comments (4)
  1. [Sec. 2.3, after Eq. (2.50)] The text reads 'variances 2 = e^{-2}P[(1+e^{-4})/2]^{N-1}' with a missing symbol; it should be σ_z^2 or similar. Also 's_e' in Eq. (2.51) is not defined.
  2. [Sec. 2.1, Eq. (2.18)] The inequality P ≤ ... is used as an asymptotic capacity condition; it would be clearer to state that the choice P = 1/(2a ln N) (...) gives failure probability decaying as N^{-a}, since the inequality itself allows smaller P.
  3. [Fig. 2] The numerical evidence for the basin claim is limited to one or few small system sizes; the asymptotic separation between typical signal and noise that invalidates (2.40) is not probed. The caption should state the exact values of N and γ used.
  4. [Appendix, Eq. (5.14)] The constant c_3 = 1 - sinh^2(1)/cosh(2) appears without derivation; please show the intermediate algebra or justify the simplification.

Circularity Check

0 steps flagged

No significant circularity: the derivation is self-contained from the model definition with explicit CLT approximations and direct numerical checks.

full rationale

The paper's central results — the exponential storage bound (2.18), the one-update magnetization (2.26), the corrupted-input capacity (2.39)/(2.41), and the perfect-recall consistency condition (2.53) — are all obtained by algebraic manipulation from the Hamiltonian (2.3) and the zero-temperature update rule (2.11). No parameter is fitted to simulation data and then renamed as a prediction; the only inputs are the model definition and the explicitly stated Gaussian/CLT approximation. The Appendix refines the single-spin-flip stability criterion to all-spin-flips and shows that only logarithmic prefactors change, which is a robustness check rather than a circular step. Self-citations in the introduction and conclusion are contextual and not load-bearing for the main derivation. The 'perfect recall regime' argument of Sec. 2.3 is a consistency check using the same noise moments, not an independent external theorem, but the paper itself frames it as corroboration; it does not reduce the central claim to an assumption of the conclusion. The skeptical concern about heavy-tailed noise in Sec. 2.2 is a correctness/accuracy objection to the Gaussian approximation, not a circularity. Therefore no enumerated circular step is present, and the honest finding is a score of 0.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The model has no data-fitted parameters. The only free parameters are asymptotic control parameters (a, delta) and a remark variant C that set prefactors. The analysis relies on standard CLT and the Rademacher pattern assumption.

free parameters (3)
  • a = not fitted, arbitrary >0
    Controls the decay of the failure probability in the single-spin stability criterion (2.15)-(2.18); a larger a gives a smaller prefactor in the capacity bound.
  • delta = arbitrary small probability
    Failure probability in the perfect-recall derivation (2.49)-(2.53); appearance is only logarithmic, choice does not affect exponential scaling.
  • C = not fitted, arbitrary >0
    Scaling factor in the exponent of a generalized cost (Remark 1, Eq. (2.19)); not used in the main results, only in a remark.
axioms (4)
  • domain assumption Patterns are i.i.d. Rademacher random variables (each component +1 or -1 with equal probability)
    Used throughout; defines the quenched disorder. Section 2, Eq. (2.1).
  • standard math The Central Limit Theorem applies to the sum over patterns in the effective field, giving a Gaussian distribution with computed moments
    Used in Secs. 2.1 and 2.2 to replace the field by a Gaussian; no Berry-Esseen type bound is provided.
  • domain assumption Zero-temperature parallel Glauber dynamics always decreases the cost function, so the update rule (2.11) is valid
    Defines the dynamics in Sec. 2.1, Eq. (2.11).
  • ad hoc to paper The z_i noise variables in the equilibrium calculation are treated as independent and identically distributed (quenched)
    Explicitly called a simplification in Sec. 2.3; the appendix partially relaxes this for the all-flip criterion.

pith-pipeline@v1.3.0-alltime-deepseek · 13895 in / 15771 out tokens · 153504 ms · 2026-08-04T22:51:12.001669+00:00 · methodology

0 comments
Cite this review

Pith. "Pith review of Yet another exponential Hopfield model." pith.science (2026). https://pith.science/paper/MO4FZYI3

@misc{pith2026250906905,
  author       = {Pith},
  title        = {Pith review of: Yet another exponential Hopfield model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MO4FZYI3}},
  note         = {Machine review of arXiv:2509.06905}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

We propose and analyze a new variation of the so-called {\em exponential Hopfield model}, a recently introduced family of associative neural networks with unprecedented storage capacity. Our construction is based on a cost function defined through exponentials of standard quadratic loss functions, which naturally favors configurations corresponding to perfect recall. Despite not being a mean-field system, the model admits a tractable mathematical analysis of its dynamics and retrieval properties that agree with those for the original exponential model introduced by Ramsauer and coworkers. By means of a signal-to-noise approach, we demonstrate that stored patterns remain stable fixed points of the zero-temperature dynamics up to an exponentially large number of patterns in the system size. We further quantify the basins of attraction of the retrieved memories, showing that while enlarging their radius reduces the overall load, the storage capacity nonetheless retains its exponential scaling. An independent derivation within the perfect recall regime confirms these results and provides an estimate of the relevant prefactors. Our findings thus complement and extend previous studies on exponential Hopfield networks, establishing that even under robustness constraints these models preserve their exceptional storage capabilities. Beyond their theoretical interest, such networks point towards principled mechanisms for massively scalable associative memory, with potential implications for both neuroscience-inspired computation and high-capacity machine learning architectures.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

47 extracted references · 45 canonical work pages · 1 internal anchor

  1. [1]

    Albanese, et al.,Hebbian Learning from First Principles, J

    L. Albanese, et al.,Hebbian Learning from First Principles, J. Math. Phys.65, 113302 (2024)

    work page 2024

  2. [2]

    Agliari, et al.,Generalized Guerra’s interpolation schemes for dense associative neural networks, Neural Networks128:254-267, (2020)

    E. Agliari, et al.,Generalized Guerra’s interpolation schemes for dense associative neural networks, Neural Networks128:254-267, (2020)

    work page 2020

  3. [3]

    Barra, M

    A. Barra, M. Beccaria, A. Fachechi,A new mechanical approach to handle generalized Hopfield neural networks, Neural Networks106, 205-222, (2018)

    work page 2018

  4. [4]

    Albanese, F

    L. Albanese, F. Alemanno, A. Alessandrelli, A. Barra,Replica symmetry breaking in dense Hebbian neural networks, J. Stat. Phys.189(2), 24, (2022)

    work page 2022

  5. [5]

    Agliari, et al.,Dense Hebbian neural networks: a replica symmetric picture of supervised learning, Physica A626:129076, (2023)

    E. Agliari, et al.,Dense Hebbian neural networks: a replica symmetric picture of supervised learning, Physica A626:129076, (2023)

    work page 2023

  6. [6]

    Agliari, et al.,Dense Hebbian neural networks: A replica symmetric picture of unsupervised learning, Physica A627:129143, (2023)

    E. Agliari, et al.,Dense Hebbian neural networks: A replica symmetric picture of unsupervised learning, Physica A627:129143, (2023)

    work page 2023

  7. [7]

    Agliari, A

    E. Agliari, A. Barra, P. Sollich, L. Zdeborov´ a,Machine learning and statistical physics, J. Phys. A53, special issue, (2020)

    work page 2020

  8. [8]

    Agliari, et al.,Neural networks with a redundant representation: Detecting the undetectable, Phys

    E. Agliari, et al.,Neural networks with a redundant representation: Detecting the undetectable, Phys. Rev. Lett.124.2:028301, (2020)

    work page 2020

  9. [9]

    Agliari, et al.,Notes on the p-spin glass studied via Hamilton-Jacobi and smooth-cavity techniques, J

    E. Agliari, et al.,Notes on the p-spin glass studied via Hamilton-Jacobi and smooth-cavity techniques, J. Math. Phys.53.6, (2012)

    work page 2012

  10. [10]

    Gross, M

    D.J. Gross, M. Mezard,The simplest spin glass, Nucl. Phys. B240.4:431-452, (1984)

    work page 1984

  11. [11]

    D.J. Amit, H. Gutfreund, H. Sompolinsky,Spin-glass models of neural networks, Phys. Rev. A. 32(2),1007–1018, (1985a)

    work page

  12. [12]

    D.J. Amit, H. Gutfreund, H. Sompolinsky,Storing infinite numbers of patterns in a spin-glass model of neural networks, Phys. Rev. Lett.55, 1530–1533, (1985b)

    work page

  13. [13]

    Baldi, S.S

    P. Baldi, S.S. Venkatesh,Number of stable points for spin-glasses and neural networks of higher orders, Phys. Rev. Lett.58, (1987)

    work page 1987

  14. [14]

    H. Bao, R. Zhang, Y. Mao,The capacity of the dense associative memory networks, Neurocomp. 469:198-208, (2022)

    work page 2022

  15. [15]

    Dense Hopfield Networks in the Teacher-Student Setting

    R. Theriault, D. Tantari,Dense Hopfield Networks in the Teacher-Student Setting, arXiv preprint arXiv:2401.04191, (2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  16. [16]

    Theriault, D

    R. Theriault, D. Tantari,Saddle Hierarchy in Dense Associative Memory, arXiv preprint arXiv:2508.19151, (2025)

    work page arXiv 2025

  17. [17]

    Baldassi, et al.,Unveiling the Structure of Wide Flat Minima in Neural Networks, Phys

    C. Baldassi, et al.,Unveiling the Structure of Wide Flat Minima in Neural Networks, Phys. Rev. Lett. 127, 278301, (2021). – 15 –

    work page 2021

  18. [18]

    Coolen, R

    A.C.C. Coolen, R. K¨ uhn, P. Sollich,Theory of neural information processing systems, Oxford University Press (2005)

    work page 2005

  19. [19]

    Demircigil, et al.,On a model of associative memory with huge storage capacity, J

    M. Demircigil, et al.,On a model of associative memory with huge storage capacity, J. Stat. Phys. 168:288-299, (2017)

    work page 2017

  20. [20]

    Derrida,Random-energy model: An exactly solvable model of disordered systems, Physical Review B 24.5:2613, (1981)

    B. Derrida,Random-energy model: An exactly solvable model of disordered systems, Physical Review B 24.5:2613, (1981)

    work page 1981

  21. [21]

    Ramsauer, et al,.Hopfield networks is all you need, Int

    H. Ramsauer, et al,.Hopfield networks is all you need, Int. Conf. Learn. Repr. (2021)

    work page 2021

  22. [22]

    Gardner,Multiconnected neural network models, J

    E. Gardner,Multiconnected neural network models, J. Phys. A20.11, 3453, (1987)

    work page 1987

  23. [23]

    Gardner,Spin glasses with p-spin interactions, Nucl

    E. Gardner,Spin glasses with p-spin interactions, Nucl. Phys. B257, 747, (1985)

    work page 1985

  24. [24]

    Hillar, N.M

    C.J. Hillar, N.M. Tran,Robust exponential memory in Hopfield networks. J. Math. Neurosci.8.1:1-20, (2018)

    work page 2018

  25. [25]

    Hopfield,Neural networks and physical systems with emergent collective computational abilities, Proc

    J.J. Hopfield,Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. U.S.A.79(8), 2554–2558, (1982)

    work page 1982

  26. [26]

    Krotov, J.J

    D. Krotov, J.J. Hopfield,Dense associative memory for pattern recognition, Adv. Neural Inf. Process. Syst. 1180-1188, (2016)

    work page 2016

  27. [27]

    Krotov, J.J

    D. Krotov, J.J. Hopfield,Dense associative memory is robust to adversarial inputs, Neural Comput. 30:3151-3167, (2018)

    work page 2018

  28. [28]

    Lucibello, M

    C. Lucibello, M. M´ ezard,The Exponential Capacity of Dense Associative Memories, arXiv preprint arXiv:2304.14964 (2023)

    Pith/arXiv arXiv 2023

  29. [29]

    Periola, A

    A. Periola, A. Alone, K.A. Agudo,Ocean warming events resilience capability in underwater computing platforms, Sci. Rep.14, 1, 3781, (2024)

    work page 2024

  30. [30]

    Cutler, et al.,Dunking the data center, IEEE Spectrum54, 3, 26–31, (2017)

    B. Cutler, et al.,Dunking the data center, IEEE Spectrum54, 3, 26–31, (2017)

    work page 2017

  31. [31]

    Kanter, H

    I. Kanter, H. Sompolinsky,Associative recall of memory without errors, Phys. Rev. A35.1, 380, (1987)

    work page 1987

  32. [32]

    Bovier, V

    A. Bovier, V. Gayrard, P. Picco,Gibbs states of the Hopfield model with extensively many patterns, J. Stat. Phys.79, 395–414, (1995)

    work page 1995

  33. [33]

    Krotov,A new frontier for Hopfield networks, Nature Rev

    D. Krotov,A new frontier for Hopfield networks, Nature Rev. Phys.5.7:366-367, (2023)

    work page 2023

  34. [34]

    Krotov, J

    D. Krotov, J. Hopfield,Large associative memory problem in neurobiology and machine learning, arXiv:2008.06996, (2020)

    Pith/arXiv arXiv 2008

  35. [35]

    Krotov, J

    D. Krotov, J. Hopfield,Unsupervised learning by competing hidden units, Proceedings of the National Academy of Sciences USA,116, 16, 7723–7731, (2019)

    work page 2019

  36. [36]

    Chaudhry, et al.,Long sequence hopfield memory, Adv

    H. Chaudhry, et al.,Long sequence hopfield memory, Adv. Neur. Inf. Proc. Sys.36, (2024)

    work page 2024

  37. [37]

    M´ ezard, G

    M. M´ ezard, G. Parisi, M.A. Virasoro,Spin glass theory and beyond, World Sci. Publ. (1987)

    work page 1987

  38. [38]

    Kirkpatrick, D

    S. Kirkpatrick, D. Sherrington,Infinite-ranged models of spin-glasses, Phys. Rev. B17.11, 4384, (1978)

    work page 1978

  39. [39]

    Carreira-Perpinan, G

    M. Carreira-Perpinan, G. Hinton,On contrastive divergence learning, PMLR, Int. Workshop AI & statistics627, 33-40, (2005)

    work page 2005

  40. [40]

    MacKay,Failures of the one-step learning algorithm, Inference Phys

    D. MacKay,Failures of the one-step learning algorithm, Inference Phys. Cam.9, 1-9, (2001)

    work page 2001

  41. [41]

    Hoover, et al.Energy transformers, Adv

    B. Hoover, et al.Energy transformers, Adv. Neur. Inf. Proc. Sys.36, (2024)

    work page 2024

  42. [42]

    Belanger, A

    D.P. Belanger, A. P. Young,The random field Ising model, J. Magn. & Magn. Mat.100.1-3:272-291, (1991)

    work page 1991

  43. [43]

    Nattermann,Theory of the random field Ising model, spin glasses and random fields 277-298, (1998)

    T. Nattermann,Theory of the random field Ising model, spin glasses and random fields 277-298, (1998). – 16 –

    work page 1998

  44. [44]

    Derrida,Random-energy model: An exactly solvable model of disordered systems, Phys

    B. Derrida,Random-energy model: An exactly solvable model of disordered systems, Phys. Rev. B 24.5:2613, (1981)

    work page 1981

  45. [45]

    Derrida,Random-energy model: Limit of a family of disordered models, Phys

    B. Derrida,Random-energy model: Limit of a family of disordered models, Phys. Rev. Letts45.2:79, (1980)

    work page 1980

  46. [46]

    Fachechi, et al.,Dreaming neural networks: forgetting spurious memories and reinforcing pure ones, Neural Networks112:24-40, (2019)

    A. Fachechi, et al.,Dreaming neural networks: forgetting spurious memories and reinforcing pure ones, Neural Networks112:24-40, (2019)

    work page 2019

  47. [47]

    Agliari, et al.,Dreaming neural networks: rigorous results, JSTAT8, 083503, (2019)

    E. Agliari, et al.,Dreaming neural networks: rigorous results, JSTAT8, 083503, (2019). – 17 –

    work page 2019