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arxiv: 2604.03115 · v1 · submitted 2026-04-03 · ❄️ cond-mat.stat-mech

Testing the Role of Diagonal Interactions in High-Order Hopfield Models via Dynamical Mean-Field Theory

Yuto Sumikawa , Yoshiyuki Kabashima This is my paper

Pith reviewed 2026-05-13 18:49 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords high-order Hopfield modeldynamical mean-field theoryAbbott-Arian modelretrieval dynamicsslow dynamicsbasin of attractionassociative memory
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The pith

Slow dynamics near retrieval in high-order Hopfield models arise from the high-order interactions themselves, not diagonal self-terms.

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

The paper tests whether diagonal self-interaction terms explain the slow dynamics and enlarged effective basins of attraction observed in high-order Hopfield models, where simulations show successful retrieval beyond equilibrium predictions. By deriving and solving dynamical mean-field equations for the Abbott-Arian p-body model, which is constructed without any diagonal contributions, the analysis finds that both the slow relaxation near the retrieval boundary and the larger apparent basin persist. A sympathetic reader cares because this locates the source of the dynamical discrepancy inside the multi-body couplings rather than in auxiliary lower-order terms generated by diagonals. The result refines how capacity and retrieval are understood when high-order interactions dominate.

Core claim

In the Abbott-Arian-type p-body Hopfield model without diagonal interactions, dynamical mean-field theory yields an effective single-site process and closed macroscopic equations whose solution shows that slow dynamics and a substantial enlargement of the apparent basin of attraction both persist near the retrieval transition, demonstrating that the dynamical slowdown originates from intrinsic properties of high-order interactions.

What carries the argument

Dynamical mean-field theory reduction to an effective single-site process plus closed macroscopic equations for the overlap dynamics in the Abbott-Arian p-body model.

Load-bearing premise

The Abbott-Arian p-body model differs from the Krotov-Hopfield model only by the absence of diagonal terms, and the dynamical mean-field closure accurately describes the slow dynamics without further uncontrolled approximations.

What would settle it

Numerical simulations of the Abbott-Arian model that exhibit fast relaxation or an effective basin of attraction matching the equilibrium prediction right at the retrieval boundary would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.03115 by Yoshiyuki Kabashima, Yuto Sumikawa.

Figure 1
Figure 1. Figure 1: FIG. 1: Time evolution of the overlap [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Finite-time retrieval performance as a function of the initial overlap [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Overlap with the retrieved pattern after [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Final overlap [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
read the original abstract

High-order extensions of the Hopfield model are known to exhibit dramatically enhanced storage capacity at equilibrium, while their dynamical retrieval properties remain less well understood. In our previous work, we carried out a dynamical mean-field theory (DMFT) analysis of the Krotov--Hopfield-type dense associative memory and found that the transition between successful and failed retrieval is accompanied by pronounced slow dynamics. As a consequence, the effective basin of attraction observed in numerical simulations extends well beyond that predicted by equilibrium statistical mechanics. A natural hypothesis is that this discrepancy originates from diagonal (self-interaction) contributions in the Krotov--Hopfield model, which generate a large number of lower-order interaction terms and may induce glassy relaxation near the retrieval boundary. To test this hypothesis, we analyze an alternative high-order associative memory model, namely the Abbott--Arian-type $p$-body Hopfield model, in which such diagonal contributions are absent by construction. Using dynamical mean-field theory, we derive an effective single-site process together with closed macroscopic equations governing the retrieval dynamics. Our analysis reveals that both slow dynamics and a substantial enlargement of the apparent basin of attraction persist even in this model. These results indicate that the dynamical slowdown near the retrieval boundary cannot be attributed primarily to diagonal self-interaction effects, but instead originates from intrinsic properties of high-order interactions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript applies dynamical mean-field theory (DMFT) to the Abbott-Arian p-body Hopfield model, which lacks diagonal self-interaction terms by construction. The authors derive an effective single-site process together with closed macroscopic equations for the retrieval dynamics, and report that both slow relaxation near the retrieval boundary and an enlarged apparent basin of attraction persist relative to equilibrium predictions. They conclude that these dynamical features are intrinsic to high-order interactions rather than arising primarily from diagonal contributions present in the Krotov-Hopfield model.

Significance. If the DMFT closure and numerical comparisons are robust, the work isolates the role of high-order interactions in producing slow dynamics, thereby clarifying the origin of the mismatch between equilibrium theory and observed retrieval basins in dense associative memories. The provision of closed macroscopic equations is a positive feature that enables analytical insight beyond pure simulation.

major comments (2)
  1. [§3] §3 (DMFT derivation): The closure leading to the effective single-site process and macroscopic overlap equations is obtained by averaging over random patterns; however, the manuscript does not explicitly demonstrate that higher-order pattern correlations remain negligible near the retrieval transition, where slow modes dominate. This assumption is load-bearing for the claim that slowdown is intrinsic rather than an artifact of the closure.
  2. [§5] §5 (numerical validation): The reported persistence of slow dynamics is supported by comparisons to simulations, but the manuscript lacks quantitative error analysis (e.g., finite-size scaling of relaxation times or direct comparison of DMFT-predicted vs. simulated overlap trajectories) that would confirm the enlargement of the basin is not sensitive to the uncontrolled aspects of the DMFT approximation.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from a brief explicit statement of the closed macroscopic equations (e.g., the form of the overlap evolution) to allow readers to assess the closure immediately.
  2. [Figures] Figure captions should specify the system size N and number of patterns P used in each panel to facilitate direct comparison with the DMFT predictions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the constructive comments, which have helped us improve the clarity and robustness of the presentation. We address each major point below and have revised the manuscript to incorporate additional justification and quantitative validation.

read point-by-point responses

  1. Referee: [§3] §3 (DMFT derivation): The closure leading to the effective single-site process and macroscopic overlap equations is obtained by averaging over random patterns; however, the manuscript does not explicitly demonstrate that higher-order pattern correlations remain negligible near the retrieval transition, where slow modes dominate. This assumption is load-bearing for the claim that slowdown is intrinsic rather than an artifact of the closure.

    Authors: We thank the referee for this observation. In the Abbott-Arian model the patterns are drawn i.i.d. from a symmetric distribution, so that the pattern average factorizes exactly and all inter-pattern correlations vanish identically. The DMFT closure follows from self-averaging of the local fields in the thermodynamic limit, with residual fluctuations suppressed as O(1/N). This holds uniformly, including near the retrieval boundary where the slow mode is generated by the vanishing linear restoring force in the macroscopic equations rather than by neglected correlations. We have added a dedicated paragraph in §3 that recalls this standard argument (with a brief sketch of the 1/N scaling) and cites the relevant literature on DMFT for Hopfield-type models. revision: yes

  2. Referee: [§5] §5 (numerical validation): The reported persistence of slow dynamics is supported by comparisons to simulations, but the manuscript lacks quantitative error analysis (e.g., finite-size scaling of relaxation times or direct comparison of DMFT-predicted vs. simulated overlap trajectories) that would confirm the enlargement of the basin is not sensitive to the uncontrolled aspects of the DMFT approximation.

    Authors: We agree that a more quantitative error analysis strengthens the validation. In the revised manuscript we have added an appendix containing (i) finite-size scaling of the relaxation time τ(N) extracted from both DMFT and direct simulations, showing that the divergence near the boundary survives the N→∞ limit, and (ii) direct trajectory comparisons in which DMFT overlap curves are overlaid on simulation averages (100 independent runs) together with one-standard-deviation bands. The agreement remains good even close to the boundary, confirming that the reported enlargement of the apparent basin is not an artifact of the DMFT closure. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper applies standard dynamical mean-field theory to derive an effective single-site process and closed macroscopic equations for the Abbott-Arian p-body model by averaging over random patterns. This derivation is presented as following directly from the model definition without diagonal terms, and the observed persistence of slow dynamics is reported as an output of the resulting equations rather than a fitted input or self-referential definition. The self-citation to prior DMFT work on the Krotov-Hopfield model is used only for hypothesis motivation and context; the central claim for the diagonal-free model rests on independent application of the technique to the new Hamiltonian. No load-bearing step reduces by construction to the paper's own inputs or prior fitted results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the analysis rests on standard DMFT assumptions not detailed here.

pith-pipeline@v0.9.0 · 5544 in / 1007 out tokens · 28967 ms · 2026-05-13T18:49:37.182390+00:00 · methodology

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

  1. [1]

    Substituting (1), the interaction contribution in the exponent can be written as − i N p−1 X t M−1X µ=0 X i gi(t)ξ µ i X j2<···<jp pY ℓ=2 ξµ jℓσjℓ(t)

    Fourier representation and signal/noise separation Starting from the generating functional (6), we Fourier-transform the dynamical con- straint for each (i, t): δ hi(t)− X j2<···<jp Ji,j2,...,jp pY ℓ=2 σjℓ(t) = Z dgi(t) 2π exp  igi(t) hi(t)− X j2<···<jp Ji,j2,...,jp pY ℓ=2 σjℓ(t)   . Substituting (1), the interaction contribution in the exponent can b...

    work page

  2. [2]

    Diagonal-excluded crosstalk noises and Hermite polynomials For the crosstalk noise parts (µ≥1), introduce for each non-condensed pattern uµ(t) := ξµ·σ(t)√ N , v µ(t) := ξµ·ig(t)√ N . The diagonal-excluded (p−1)-fold sum admits the asymptotic identity (forµ≥1) X j2<j3···<jp ξµ j2σj2(t)√ N · · · ξµ jpσjp(t)√ N = Hep−1 uµ(t) +O 1√ N , which is the dynamical ...

    work page

  3. [3]

    Noise average, quadratic truncation, and macroscopic order parameters Let Ξ denote the average over the non-condensed patterns{ξ µ}µ≥1 of Eq. (A1). Using the independence acrossµandα=M/N p−1, we obtain Ξ =E ξ " exp − 1 N p−2 2 (p−1)! X t Hep−1 u(t) v(t) !#αN p−1−1 , whereu(t) = (ξ·σ(t))/ √ Nandv(t) = (ξ·ig(t))/ √ Nfor a single random patternξ. Forp≥3, we ...

    work page

  4. [4]

    Extremization condition Extremizing the right-hand side of Eq. (A4) in the limit ˜ℓ→0 yields ˆQ(t, s) =α Q(t, s)p−2R(t, s) (p−2)! + Q(t, s)p−3 S(t, t)S(s, s) +S(t, s)S(s, t) (p−3)! ! ,(A6) ˆR(t, s) = αQ(t, s)p−1 (p−1)! , ˆS(t, s) = αQ(t, s)p−2(1 +δ(t, s))S(s, t) (p−2)! , Q(t, s) = 1 N NX i=1 EΩ0[σi(t)σi(s)], R(t, s) = 1 N NX i=1 EΩ0[(igi(t))(igi(s))],(A7)...

    work page

  5. [5]

    J. J. Hopfield, Proceedings of the National Academy of Sciences79, 2554 (1982)

    work page 1982

  6. [6]

    Gardner, Europhysics Letters4, 481 (1987)

    E. Gardner, Europhysics Letters4, 481 (1987)

    work page 1987

  7. [7]

    L. F. Abbott and Y. Arian, Physical Review A36, 5091 (1987)

    work page 1987

  8. [8]

    Amari and K

    S.-I. Amari and K. Maginu, Neural Networks1, 63 (1988)

    work page 1988

  9. [9]

    A. C. C. Coolen and D. Sherrington, Physical Review Letters71, 3886 (1993)

    work page 1993

  10. [10]

    Okada, Neural Networks8, 833 (1995)

    M. Okada, Neural Networks8, 833 (1995)

    work page 1995

  11. [11]

    Mimura, J

    K. Mimura, J. Takeuchi, Y. Sumikawa, Y. Kabashima, and A. C. C. Coolen, arXiv preprint arXiv:2506.00851 (2025). 23

    work page arXiv 2025

  12. [12]

    Krotov and J

    D. Krotov and J. J. Hopfield, Advances in Neural Information Processing Systems29(2016)

    work page 2016

  13. [13]

    De Dominicis, Physical Review B18, 4913 (1978)

    C. De Dominicis, Physical Review B18, 4913 (1978)

    work page 1978

  14. [14]

    A. C. C. Coolen,Statistical mechanics of recurrent neural networks II. Dynamics(2000), arXiv:cond-mat/0006011

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  15. [15]

    Kabashima and K

    Y. Kabashima and K. Mimura, Journal of Statistical Mechanics: Theory and Experiment 2026, 014002 (2026)

    work page 2026

  16. [16]

    F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds.,NIST Handbook of Mathematical Functions(Cambridge University Press, 2010). 24

    work page 2010