Training oscillator Ising machines to assign the dynamic stability of their equilibrium points

Yi Cheng; Zongli Lin

REVIEW 3 major objections 6 minor 23 references

Training oscillator Ising machines to assign the dynamic stability of their equilibrium points

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read Training oscillator Ising machines can make them remember only the stored patterns

desk verdict A plausible but unproven training method for oscillator Ising machines; the key sampling proxy needs verification. read the letter →

arxiv 2507.14386 v1 pith:TW7MCCLE submitted 2025-07-18 cs.NE cs.IR

Yi Cheng , Zongli Lin This is my paper · ORCID

classification cs.NEcs.IR MSC 34D2037N2568T07

keywords oscillatorIsingmachineassociativememoryequilibriumpointstabilityHamiltonianenergyeigenvaluecontrastivemethodBoltzmannlearningGibbssamplingHopfieldnetwork

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

An oscillator Ising machine (OIM) has the useful property that every $0/\pi$ binary equilibrium point is structurally stable, so training cannot create or destroy these equilibria, only decide which ones are dynamically stable. This paper proposes a training method, the Hamiltonian-Regularized Eigenvalue Contrastive Method (HRECM), that aims to make exactly the desired binary patterns asymptotically stable and every other $0/\pi$ pattern unstable. The method rests on a connection between stability and energy: the largest eigenvalue $\lambda_N(D(s))$ of the stability matrix $D(s)$ is approximately positively correlated with the Ising Hamiltonian $H(s)$, and Theorem 1 makes this precise with two-sided bounds. Training updates the coupling weights by contrasting a cheaply sampled low-Hamiltonian pattern against the desired pattern with the largest $\lambda_N(D(s))$, plus a Hamiltonian regularization term. In numerical experiments the desired patterns' eigenvalues become the smallest among all $0/\pi$ patterns, so a single ratio $K_s/K$ can be chosen that makes the desired equilibria the only stable ones.

What carries the argument

The load-bearing object is the symmetric matrix $D(s)$, built from the coupling weights and the binary pattern $s$, whose largest eigenvalue $\lambda_N(D(s))$ decides asymptotic stability: a $0/\pi$ equilibrium point is stable exactly when $\lambda_N(D(s)) < K_s/(2K)$ (from condition (3)). The second load-bearing piece is Theorem 1's two-sided bound linking $\lambda_N(D(s))$ to the Ising Hamiltonian $H(s) = -\sum_{i<j} J_{ij} s_i s_j$, which lets the algorithm find low-eigenvalue negative examples by cheap Hamiltonian Gibbs sampling (update Eq. (5)) instead of by an expensive eigenvalue minimization. The training loop then pushes down the largest eigenvalue of the sampled negative pattern and pushes up the largest eigenvalue of the worst desired pattern, using the eigenvector-weighted update (12); a Hamiltonian regularization term biases the Gibbs distribution toward the desired patterns.

What would settle it

Take a small trained network (say $N \le 12$), enumerate all $2^N$ binary patterns, and compute $\lambda_N(D(s))$ for each one. If the pattern chosen by Hamiltonian Gibbs sampling as $\hat{s}_{\min}$ frequently differs from the exact minimizer of $\lambda_N(D(s))$ among non-desired patterns, or if inequality (6) holds on the sampled subset but is violated on the full enumeration, then the proxy that carries the training argument is false.

Watch

Extended reading notes

Core claim

The central claim is that OIMs can realize Hopfield-like associative memory purely by assigning dynamic stability to equilibrium points, without the usual existence-versus-stability trade-off. The paper states the perfect memory condition as inequality (6): the largest eigenvalue $\lambda_N(D(s^\star_d))$ of every desired pattern must be smaller than the largest eigenvalue $\lambda_N(D(s))$ of every non-desired binary pattern; then any $K_s/K$ in the gap makes the desired $0/\pi$ equilibrium points asymptotically stable and all others unstable. Theorem 1 shows $\lambda_N(D(s))$ is at least $\max\{0, 2H(s)/N\}$ and at most an affine increasing function of $H(s)$ plus a constant, giving the paper its justification for using Hamiltonian Gibbs sampling as a cheap proxy for the minimal-eigenvalue pattern. HRECM (Algorithm 1) updates $J_{ij}$ along the approximate Clarke gradient in Eq. (12), using the sample with smallest $\lambda_N(D(\hat{s}_{\min}))$ from Hamiltonian Gibbs sampling and the desired pattern with largest $\lambda_N(D(s_{\max}))$. The reported experiments on networks up to $N=70$ show the desired patterns' eigenvalues separating below all others, with the spurious rate reduced further by the Hamiltonian regularization term.

Load-bearing premise

The method assumes that the binary pattern with the smallest Ising Hamiltonian is essentially the same pattern as the one whose stability matrix has the smallest largest eigenvalue, so the cheaply sampled low-Hamiltonian pattern can stand in for the true hardest negative example in every gradient step.

Editorial extensions

If this is right

For OIMs, storing patterns no longer requires balancing existence against stability, because every $0/\pi$ equilibrium point already exists for any coupling weights; training only assigns which ones are stable.
If inequality (6) is achieved, a single scalar $K_s/K$ suffices to make all desired patterns asymptotically stable and all other $0/\pi$ patterns unstable, so retrieval is decided by eigenvalue ordering alone.
Hamiltonian Gibbs sampling replaces the per-step eigenvalue computation of Boltzmann-machine learning with a Hamiltonian update, reducing the per-sample cost from $O(N^3)$ to $O(N)$ and making larger networks trainable.
Adding the Hamiltonian regularization term systematically lowers the spurious rate, with the largest gains when the number of desired patterns exceeds the network size.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper validates the eigenvalue-Hamiltonian correlation only through the training's success; a direct comparison of the Gibbs-sampled minimizer with the exact minimizer on a small network would isolate whether the proxy or the contrastive update is doing the work.
Because the $0/\pi$ equilibrium points are structurally stable, the same stability-assignment idea could extend to other oscillator networks or to continuous-valued patterns, as long as a cheap energy proxy for the stability eigenvalue can be found.
The reported experiments stop at $N=70$ and use sampled spurious-rate estimates for large networks; a test that enumerates all patterns for moderate $N$ would show how close the training comes to the exact inequality (6) rather than to a sampled approximation.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Worth a look if you work on Ising machines or physically-inspired associative memories. The new thing here is that they train the couplings of an oscillator Ising machine (OIM) to assign dynamic stability to the 0/π equilibria directly, rather than trying to embed patterns as equilibria (which OIMs don't need since those EPs are structurally stable). The HRECM objective—eigenvalue contrastive term plus a Hamiltonian regularizer—is clean, and Theorem 1, modulo a typo, is a genuine attempt to relate the largest eigenvalue of D(s) to the Hamiltonian.

What the paper does well: it states the training loop precisely (Algorithm 1), uses Clarke gradients for the nonsmooth eigenvalue, and Figure 2 shows a real eigenvalue separation in one 20-oscillator run. The comparison with/without regularization (Fig. 3) suggests the regularizer helps when m>N. The idea of decoupling existence from stability is structurally appealing.

Where it gets soft: the load-bearing step is the replacement of smin = argmin_{s∈S\Sd} λN(D(s)) by ŝmin drawn from Hamiltonian Gibbs sampling. Nothing restricts the samples to S\Sd, and the regularizer actually lowers the Hamiltonian of desired patterns, making them more likely to be sampled. So ŝmin can easily be a desired pattern, in which case the gradient update (12) is not pushing up the smallest spurious eigenvalue. Theorem 1 does not fix this: it only gives a loose two-sided bound with no control on the distance between the H-minimizer and the λ-minimizer, and the printed upper bound has a misprint (the proof gives 1 + ..., not 2 + ...). The paper also claims 'extensive experiments' for the correlation but shows only one run without error bars, baselines, or code. None of this is fatal—the empirical eigenvalue separation is visible—but it means the central mechanism isn't yet established.

Bottom line: this is a plausible idea from a group that knows OIM dynamics, but the missing verification of the sampling proxy is the thing a referee should push on. I'd send it to peer review rather than desk reject; with a revision that either restricts sampling to non-desired states or reports how often ŝmin is spurious (plus a corrected Theorem 1 and baseline comparisons), it could become a solid contribution.

Referee Report

3 major / 6 minor

Summary. The paper proposes a training method, Hamiltonian-Regularized Eigenvalue Contrastive Method (HRECM), for oscillator Ising machines (OIMs) so that a chosen set of 0/pi binary patterns become asymptotically stable equilibrium points while all other 0/pi patterns are unstable. The key idea is to exploit the known criterion (3) that a 0/pi pattern s is stable exactly when lambda_N(D(s)) < K_s/K, and to train the coupling weights J_ij so that inequality (6) holds: all desired patterns have smaller largest eigenvalues of D(s) than all spurious patterns. Since direct minimization over all spurious patterns is expensive, the paper proposes to approximate the spurious hard negative by a state drawn from Hamiltonian Gibbs sampling, justified by a claimed qualitative relation (Theorem 1) between lambda_N(D(s)) and the Hamiltonian H(s). The gradient update is given in Eq. (12) and the training loop in Algorithm 1. Numerical experiments on networks of sizes N = 10 to 70 report decreasing spurious rates, with regularization improving results when the number of desired patterns exceeds N.

Significance. If the method works as claimed, it offers a useful separation between the existence and the dynamic stability of stored patterns, exploiting the structural stability of OIM binary equilibria established in prior work. The formulation is clean: the perfect-memory condition (6) is a direct and falsifiable eigenvalue separation criterion, and the Hamiltonian-Gibbs approximation is a concrete algorithmic proposal that avoids O(N^3) eigenvalue computations at every Gibbs step. The paper also ships precise equilibrium and gradient formulas, and the experiments cover a range of network sizes. However, the central theoretical bridge between the Hamiltonian and the largest eigenvalue is not established by the stated theorem, and the numerical experiments do not directly validate the proxy that the training algorithm relies on. The contribution is therefore promising but needs additional analysis or verification before it can be considered sound.

major comments (3)

[Section III.C, Eq. (12), Algorithm 1 Step 2.2] The replacement of the exact spurious hard negative s_min = argmin_{s in S\S_d} lambda_N(D(s)) by the Hamiltonian-Gibbs sample s_hat_min is load-bearing but unsupported. The Gibbs sampler of Eq. (5) draws from all of S, not from S\S_d, and nothing in Algorithm 1 prevents s_hat_min from being one of the desired patterns. After training has made desired patterns low in H(s), those patterns can become likely under the Hamiltonian Gibbs distribution, so s_hat_min may frequently lie in S_d; in that case the first term of Eq. (12) updates the eigenvalues of a desired pattern rather than raising the smallest spurious eigenvalue, and the mechanism claimed for satisfying (6) is not in operation. The experiments do not report how often s_hat_min belongs to S_d, nor do they compare s_hat_min with the exact s_min for small N. This gap needs either a theoretical argument that low-H samples are typically spurious hard negatives, a modification of the sampler to exclude S_d, or a direct empirical check of the proxy.
[Theorem 1, Eq. (7) and Appendix A] The printed upper bound in Eq. (7) is not consistent with the proof. Starting from lambda_N(D(s)) = lambda_N(hat D(s)) - c and the displayed bound lambda_N(hat D(s)) <= 1 + r tr(hat D(s))/N with r = sqrt((kappa N - 1)(N - 1)), the proof obtains lambda_N(D(s)) <= 1 + r tr(D(s))/N + r c - c. Using tr(D(s)) = 2H(s), this gives 1 + 2r H(s)/N + r c - c, not 2 + 2r H(s)/N + d unless an additional identity involving c is supplied. The theorem as stated therefore cannot serve as the stated justification that lambda_N(D(s)) 'tends to be smaller' when H(s) is smaller; the bounds are also too loose to control the gap between the H-minimizer and the lambda_N-minimizer. The statement of Theorem 1 and its proof should be corrected and restated with precise constants and explicit conditions on c.
[Section IV, Experiments 1-3] The numerical validation does not directly test the assumption on which Algorithm 1 rests. Experiment 1 shows eigenvalue separation in a single 20-oscillator run, but with no statistics over random initializations or over different random choices of desired patterns. Experiment 3 replaces the full spurious set by Hamiltonian-Gibbs samples, using the same sampling distribution as the training procedure; this can bias the reported spurious rate, because the sampler preferentially visits low-H states and may not represent all spurious patterns. The paper should report, at least for small N where exact enumeration is feasible, how often s_hat_min is spurious, how well s_hat_min approximates s_min, and the variance of the eigenvalue gap in Eq. (6) over multiple runs. Without such evidence, the central numerical claim that the method 'assigns' stability through the Hamiltonian proxy remains unverified.

minor comments (6)

[Introduction] The text says 'Section V presents numerical experiments' and 'Section V concludes the paper'; the experiments are actually in Section IV. This should be corrected.
[Abstract and Section II.B] Minor grammatical issues: 'an ideal candidates' should be 'an ideal candidate'. The phrase 'the design of coupling weights needs only to focus' could be made more concise. These do not affect the technical content.
[Section III.C] The text contains small terminology errors: 'Lipchitz' should be 'Lipschitz', 'Clark gradient' should be 'Clarke gradient', and 'convex hall' should be 'convex hull'.
[Section III.C, Eq. (11)] The update rule (12) is written for all J_ij, but the OIM requires J_ij = J_ji and J_ii = 0. The paper should state explicitly that the update is applied only to i < j and that symmetry is preserved; otherwise the resulting matrix may not satisfy the model assumptions.
[Section III.C, Eq. (12)] The final term in Eq. (12) is written as alpha( <s_i s_j>_{S_d} - <s_i s_j>_{S_-} ), but the objective (8) contains alpha P(s_d^*), not alpha log P(s_d^*). The derivative of P(s_d^*) carries a factor P(s_d^*), so the expression in (12) appears to be a different surrogate. This should be clarified, for example by stating that the regularizer is optimized in logarithmic form or that the missing factor is absorbed into alpha.
[Figure 2] The caption is incomplete because it does not define the red and blue markers explicitly; the text explains them in the body, but the caption alone should be self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: HRECM directly optimizes the eigenvalue-separation condition, and the self-cited stability theorem is an external, non-fitted result.

full rationale

The core training objective in Eq. (8), max over J of min_{S\S_d} λ_N(D(s)) − max_{S_d} λ_N(D(s)) + αP, is by construction the same quantity whose sign determines the perfect-associative-memory condition (6). Reporting that trained weights separate desired from spurious eigenvalues is therefore the outcome of direct optimization, not a circular derivation. The stability criterion (3) is imported from Ref. [19], which has overlapping authors, but it is an externally published dynamical-systems theorem with stated assumptions and is not fitted to the data in this paper; it is independent support, so the self-citation is not load-bearing in a circular sense. The Hamiltonian-Gibbs-sampling proxy in Eq. (12), which replaces the exact hard-negative minimizer s_min by ŝ_min over samples, is an unverified approximation and Theorem 1 gives only loose two-sided bounds, but this is a correctness or convergence risk rather than a circular equivalence: ŝ_min is not defined in terms of the target eigenvalue separation, and the experiments in Fig. 2 enumerate all 0/π EPs for N=20 rather than only sampled states. No step reduces its claimed output to its own inputs by definition.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the prior stability theorem from the authors' own [19], a standard eigenvalue bound, and an unproven heuristic that Hamiltonian sampling finds near-minimizers of the largest eigenvalue. The training hyperparameters are free but do not enter the theory. No new physical entities are introduced.

free parameters (5)

regularization factor alpha = 0.8 (Exp. 1), 1 (Exps. 2-3), 0 (ablation)
Hand-chosen hyperparameter controlling the Hamiltonian regularization term in the objective (8). Performance and spurious rate depend on it.
learning rate epsilon = 0.01
Fixed step size in Algorithm 1; no schedule or tuning reported.
Gibbs sample count n = 500
Number of Hamiltonian Gibbs samples used to approximate the non-desired state with minimal largest eigenvalue.
training iterations T = 100/300/2000 (Exp. 1), 5000 (Exps. 2-3)
Stopping criterion; fixed per experiment.
constant c in Theorem 1 = unspecified, must satisfy c > -2H(s)/N for all s
Proof device; the bound's tightness depends on c, and kappa depends on c through D_hat.

assumptions (5)

domain assumption Stability criterion for 0/pi EPs of OIMs: asymptotically stable iff Ks/K > lambda_N(D(theta*))/2
Taken from the authors' prior work [19]; the entire training target (6) relies on this equivalence. Not re-derived here.
domain assumption All 0/pi and +/-pi/2 binary EPs are structurally stable with respect to coupling weights
Also from [19]; supports the claim that existence of EPs does not need to be enforced during training.
standard math Wolkowicz-Styan eigenvalue bound
Used in the proof of Theorem 1 (Appendix A, reference [23]).
domain assumption Hamiltonian Gibbs sampling converges to the Boltzmann distribution p(s) proportional to exp(-H(s))
Standard result in Boltzmann machine learning [20,21]; used to generate candidate low-energy states.
ad hoc to paper Low-Hamiltonian states approximate low-lambda_N(D) states sufficiently well for training
The paper infers this from the weak bound in Theorem 1; no proof that the minimizer of H is close to the minimizer of lambda_N(D), nor that Gibbs samples cover the relevant low-lambda region. This is the load-bearing approximation behind replacing s_min by s_hat_min in Eq. (12).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Training oscillator Ising machines to assign the dynamic stability of their equilibrium points." pith.science (2026). https://pith.science/paper/TW7MCCLE

@misc{pith2026250714386,
  author       = {Pith},
  title        = {Pith review of: Training oscillator Ising machines to assign the dynamic stability of their equilibrium points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TW7MCCLE}},
  note         = {Machine review of arXiv:2507.14386}
}

abstract

We propose a neural network model, which, with appropriate assignment of the stability of its equilibrium points (EPs), achieves Hopfield-like associative memory. The oscillator Ising machine (OIM) is an ideal candidates for such a model, as all its $0/\pi$ binary EPs are structurally stable with their dynamic stability tunable by the coupling weights. Traditional Hopfield-based models store the desired patterns by designing the coupling weights between neurons. The design of coupling weights should simultaneously take into account both the existence and the dynamic stability of the EPs for the storage of the desired patterns. For OIMs, since all $0/\pi$ binary EPs are structurally stable, the design of the coupling weights needs only to focus on assigning appropriate stability for the $0/\pi$ binary EPs according to the desired patterns. In this paper, we establish a connection between the stability and the Hamiltonian energy of EPs for OIMs, and, based on this connection, provide a Hamiltonian-Regularized Eigenvalue Contrastive Method (HRECM) to train the coupling weights of OIMs for assigning appropriate stability to their EPs. Finally, numerical experiments are performed to validate the effectiveness of the proposed method.

Figures

Figures reproduced from arXiv: 2507.14386 by the authors.

**Figure 2.** FIG. 2. Evolution of the training process [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. The comparison of the algorithm with and without [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. The evolution of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 21 canonical work pages

[1]

Little, The existence of persistent states in the brain, Mathematical Biosciences 19, 101 (1974)

W. Little, The existence of persistent states in the brain, Mathematical Biosciences 19, 101 (1974)

work page 1974
[2]

J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities., Proceedings of the national academy of sciences 79, 2554 (1982)

work page 1982
[3]

D. J. Amit, H. Gutfreund, and H. Sompolinsky, Storing infinite numbers of patterns in a spin-glass model of neu- ral networks, Physical Review Letters 55, 1530 (1985)

work page 1985
[4]

Kanter and H

I. Kanter and H. Sompolinsky, Associative recall of mem- ory without errors, Physical Review A 35, 380 (1987)

work page 1987
[5]

A. J. Storkey and R. Valabregue, The basins of attraction of a new Hopfield learning rule, Neural Networks 12, 869 (1999)

work page 1999
[6]

Demircigil, J

M. Demircigil, J. Heusel, M. L¨ owe, S. Upgang, and F. Vermet, On a model of associative memory with huge storage capacity, Journal of Statistical Physics 168, 288 (2017)

work page 2017
[7]

Ramsauer, B

H. Ramsauer, B. Sch¨ afl, J. Lehner, P. Seidl, M. Widrich, T. Adler, L. Gruber, M. Holzleitner, M. Pavlovi´ c, G. K. Sandve, et al., Hopfield networks is all you need, arXiv preprint arXiv:2008.02217 (2020)

arXiv 2020
[8]

D. Wu, J. Y.-C. Hu, T.-Y. Hsiao, and H. Liu, Uniform memory retrieval with larger capacity for modern hop- field models, arXiv preprint arXiv:2404.03827 (2024)

arXiv 2024

Show all 23 references

[9]

Millidge, T

B. Millidge, T. Salvatori, Y. Song, T. Lukasiewicz, and R. Bogacz, Universal hopfield networks: A general frame- work for single-shot associative memory models, in Inter- national Conference on Machine Learning(PMLR, 2022) pp. 15561–15583

2022
[10]

Brain-State-in-a-Box

R. M. Golden, The “Brain-State-in-a-Box” neural model is a gradient descent algorithm, Journal of Mathematical Psychology 30, 73 (1986). 8

1986
[11]

Perfetti, A synthesis procedure for brain-state-in-a- box neural networks, IEEE Transactions on Neural Net- works 6, 1071 (1995)

R. Perfetti, A synthesis procedure for brain-state-in-a- box neural networks, IEEE Transactions on Neural Net- works 6, 1071 (1995)

1995
[12]

Schultz, Collective recall via the Brain-State-in-a-Box network, IEEE Transactions on Neural Networks 4, 580 (1993)

A. Schultz, Collective recall via the Brain-State-in-a-Box network, IEEE Transactions on Neural Networks 4, 580 (1993)

1993
[13]

Hui and S

S. Hui and S. H. Zak, Dynamical analysis of the brain- state-in-a-box (BSB) neural models, IEEE Transactions on Neural Networks 3, 86 (1992)

1992
[14]

Park, Optimal and robust design of brain-state-in-a- box neural associative memories, Neural Networks 23, 210 (2010)

Y. Park, Optimal and robust design of brain-state-in-a- box neural associative memories, Neural Networks 23, 210 (2010)

2010
[15]

D. J. Amit, H. Gutfreund, and H. Sompolinsky, Statisti- cal mechanics of neural networks near saturation, Annals of physics 173, 30 (1987)

1987
[16]

G. A. Carpenter and S. Grossberg, Pattern recognition by self-organizing neural networks(MIT Press, 1991)

1991
[17]

Folli, M

V. Folli, M. Leonetti, and G. Ruocco, On the maximum storage capacity of the Hopfield model, Frontiers in com- putational neuroscience 10, 144 (2017)

2017
[18]

Fusi, Memory capacity of neural network models, arXiv preprint arXiv:2108.07839 (2021)

S. Fusi, Memory capacity of neural network models, arXiv preprint arXiv:2108.07839 (2021)

2021 arXiv
[19]

Cheng, M

Y. Cheng, M. Khairul Bashar, N. Shukla, and Z. Lin, A control theoretic analysis of oscillator Ising machines, Chaos: An Interdisciplinary Journal of Nonlinear Science 34 (2024)

2024
[20]

G. E. Hinton, Training products of experts by minimiz- ing contrastive divergence, Neural computation 14, 1771 (2002)

2002
[21]

D. H. Ackley, G. E. Hinton, and T. J. Sejnowski, A learn- ing algorithm for Boltzmann machines, Cognitive science 9, 147 (1985)

1985
[22]

M. L. Overton, Large-scale optimization of eigenvalues, SIAM Journal on Optimization 2, 88 (1992)

1992
[23]

Wolkowicz and G

H. Wolkowicz and G. P. Styan, Bounds for eigenvalues using traces, Linear algebra and its applications 29, 471 (1980)

1980

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Little, The existence of persistent states in the brain, Mathematical Biosciences 19, 101 (1974)

W. Little, The existence of persistent states in the brain, Mathematical Biosciences 19, 101 (1974)

work page 1974

[2] [2]

J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities., Proceedings of the national academy of sciences 79, 2554 (1982)

work page 1982

[3] [3]

D. J. Amit, H. Gutfreund, and H. Sompolinsky, Storing infinite numbers of patterns in a spin-glass model of neu- ral networks, Physical Review Letters 55, 1530 (1985)

work page 1985

[4] [4]

Kanter and H

I. Kanter and H. Sompolinsky, Associative recall of mem- ory without errors, Physical Review A 35, 380 (1987)

work page 1987

[5] [5]

A. J. Storkey and R. Valabregue, The basins of attraction of a new Hopfield learning rule, Neural Networks 12, 869 (1999)

work page 1999

[6] [6]

Demircigil, J

M. Demircigil, J. Heusel, M. L¨ owe, S. Upgang, and F. Vermet, On a model of associative memory with huge storage capacity, Journal of Statistical Physics 168, 288 (2017)

work page 2017

[7] [7]

Ramsauer, B

H. Ramsauer, B. Sch¨ afl, J. Lehner, P. Seidl, M. Widrich, T. Adler, L. Gruber, M. Holzleitner, M. Pavlovi´ c, G. K. Sandve, et al., Hopfield networks is all you need, arXiv preprint arXiv:2008.02217 (2020)

arXiv 2020

[8] [8]

D. Wu, J. Y.-C. Hu, T.-Y. Hsiao, and H. Liu, Uniform memory retrieval with larger capacity for modern hop- field models, arXiv preprint arXiv:2404.03827 (2024)

arXiv 2024