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Synchronization of mean-field models on the circle

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves a general third-derivative criterion for global synchronization of mean-field models on the circle and uses it to settle the synchronization threshold for self-attention dynamics at β ≥ -0.16.

desk verdict A genuinely useful general criterion and a convincing resolution of the β≥0 conjecture; the negative-β branch rests on an imported lemma, but this deserves a serious referee. read the letter →

arxiv 2507.22857 v1 pith:KESNBNN2 submitted 2025-07-30 math.DS cs.LGmath.APmath.OC

Yury Polyanskiy , Philippe Rigollet , Andrew Yao This is my paper
classification math.DScs.LGmath.APmath.OC MSC 34D0637C1037C7534C15
keywords synchronizationmean-fieldmodelonthecircleKuramotoself-attentiondynamicsgradientthird-derivativecriterionmeta-stableclusteringnormalizedattention
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

This paper asks when n particles moving on a circle, each feeling the same pairwise force from all others, are guaranteed to end up at one common point no matter how they start. The answer is a general criterion: if the positive part of the third derivative of the interaction force is small on a scale set by the force's slope at zero, then every stationary configuration is either locally unstable or fully synchronized. The paper applies this criterion to the self-attention dynamics $f_\beta(x)=\sin(x)e^{\beta\cos x}$, the stylized transformer model, and proves global synchronization whenever $\beta \ge -0.16$, which includes and extends the previously known range $0 \le \beta \le 1$. It also proves that global synchronization fails for $\beta < -2/3$, leaving only a narrow interval of $\beta$ unresolved.

What carries the argument

The engine is the synchronization ratio $4(1+\tau/M)f'(0)/(\tau\langle 1,|f'''|_+\rangle_{L^2})$: when it exceeds 1, stable non-synchronized states are impossible. The proof combines the cut-stability condition (C2), a linearized escape-direction test inherited from Kuramoto-type arguments that any locally stable stationary point must satisfy, with an integration-by-parts identity over inter-particle gaps. Summing the identity bounds $2\sum_i\chi'(x_i)$ from below by $2n f'(0)$ and from above by $(\tau/2)\int|\chi'''|_+$, producing the contradiction that forces synchronization. The remaining ingredients are the gradient structure of $f(x)=\sin(x)h(\cos x)$ dynamics, Lojasiewicz's theorem for convergence, the center-stable manifold theorem for generic avoidance of unstable points, and, for the negative-$\beta$ branch, the bound $\tau_{\max}(x)<\pi$ from Lemma C.4.

What would settle it

Solve the stationarity equations of (T1) at $\beta=-0.16$ with $n=3$ and test the Hessian of the energy at each non-synchronized solution: Corollary 2.4 says every such stationary point is locally unstable, so a stable two-cluster or three-cluster state would falsify the claim. Equivalently, a numerical ensemble from many random starts that leaves a positive-fraction basin around a clustered state at $\beta=-0.16$ contradicts the theorem.

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Extended reading notes

Core claim

The central discovery, Theorem 2.1, is a stability dichotomy for the mean-field flow $\dot{x}_i=-\sum_j f(x_i-x_j)$ on the circle. If a parameter $\tau\in(0,\pi]$ is chosen so that $f'(x)<0$ outside $[-\tau,\tau]$, and if $\tau\int_{-\pi}^{\pi}|f'''(x)|_+\,dx \le 4(1+\tau/2\pi)f'(0)$, then every stationary point is either locally unstable or synchronized, meaning all $x_i$ coincide. Because for $f(x)=\sin(x)h(\cos x)$ the flow is a gradient ascent on a compact manifold, Lojasiewicz's theorem makes convergence to a stationary point almost sure, and the center-stable manifold theorem makes the limiting point almost surely locally stable; the dichotomy then upgrades to global synchronization. For the self-attention interaction $f_\beta(x)=\sin(x)e^{\beta\cos x}$, the paper verifies a strengthened version of the inequality with the better constant $4(1+\tau/\pi)$ for all $\beta\ge -0.16$, yielding Corollaries 2.4 and 2.5 for the unnormalized and normalized dynamics, and it constructs a stable three-cluster stationary point for $\beta<-2/3$, yielding Corollary 2.6.

Load-bearing premise

For the negative-$\beta$ branch, the proof inherits the bound $\tau_{\max}(x)<\pi$ for every stable, stationary, non-synchronized configuration from [11, Lemma 10] (reproduced here as Lemma C.4) and does not re-prove it.

Editorial extensions

If this is right

  • For every $\beta \ge 0$, both the unnormalized (T1) and normalized (T2) self-attention dynamics globally synchronize, resolving the conjecture that was previously known only for $0\le \beta\le 1$ and for $\beta=\Omega(1/n)$.
  • For $\beta\in[-0.16,0)$, global synchronization still occurs, establishing a provable negative-temperature synchronization regime.
  • For $\beta<-2/3$, global synchronization fails when $n$ is divisible by 3 or $n$ is sufficiently large, so the synchronization phenomenon has a proven lower boundary.
  • The criterion extends to weighted dynamics (S3) and (S4), and to rank-1 weight matrices, so positive particle-dependent weights do not break the synchronization guarantee.
  • The synchronization ratio can be plotted for any candidate interaction function, giving a direct numerical certificate of where global synchronization is guaranteed.

Reading between the lines

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

  • The numerical plot in the paper suggests the criterion already holds near $\beta=-0.25$; if a sharper analytic version of Lemma C.5 can be proved, the theorem likely extends beyond $-0.16$.
  • The same third-derivative ratio can be evaluated for other interactions of the form $\sin(x)h(\cos x)$, offering a cheap way to map synchronization regions for many one-dimensional mean-field models.
  • The interval $(-2/3,-0.16)$ is left open; locating where the stable three-cluster state of Lemma C.6 first appears would decide whether the transition to synchronization is sharp.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. This paper studies the mean-field particle system ẋ_i = -Σ_j f(x_i - x_j) on the circle and its normalized variant. The main theoretical result, Theorem 2.1, gives a sufficient criterion for every stationary point of the unnormalized flow to be either locally unstable or synchronized, expressed through τ∫|f'''|_+ ≤ 4(1 + τ/2π)f'(0). Corollary 3.3 strengthens the criterion when a uniform bound τmax(x) < M is available. For the self-attention dynamics f_β(x) = sin(x)e^{β cos(x)}, the paper proves global synchronization for all β ≥ -0.16 (Corollaries 2.4 and 2.5) and constructs stable non-synchronized stationary points for β < -2/3 (Corollary 2.6). The proof combines the stationary-point criterion with the gradient structure, Lojasiewicz's theorem, and the center-stable manifold theorem.

Significance. Theorem 2.1 is an attractive and apparently new sufficient condition that is parameter-free and derived from the ODE rather than fitted to data; Lemma 3.2 is proved self-containedly, and the surrounding machinery is standard and clearly presented. The application resolves the open conjecture β ≥ 0 for self-attention on the circle and extends the synchronization regime to a range of negative β, while the β < -2/3 obstruction gives a qualitative boundary. The paper is also honest in identifying the remaining open interval (-2/3, -0.25). The main concerns are the external lemma supporting the negative-β branch and the heavily hand-verified appendix inequalities; neither undermines the β ≥ 0 branch, which is the core of the conjecture.

major comments (2)
  1. [§5 (Corollary 2.4; Lemma C.4)] For -0.16 ≤ β < 0, the proof sets M = π in Corollary 3.3 solely on the basis of Lemma C.4, which is imported from [11, Lemma 10] and not proved. This is load-bearing: Lemma C.5 is verified only in the M = π form, so if τmax(x) < π failed, no alternative inequality would establish the criterion. The β ≥ 0 branch and the β < -2/3 obstruction are unaffected, but the negative-β half of the main application is not self-contained as written. Please reproduce the proof of [11, Lemma 10] specialized to f_β, give a direct proof, or explicitly state that Corollaries 2.4 and 2.5 for negative β are conditional on that external result.
  2. [Appendix A (Lemmas A.4-A.7), Lemma B.4, Lemma C.5] The verifications for β > 1/3 and for β ∈ (0, 1/3] rest on many asserted numerical inequalities, for example g'_β(0.18) < 0 and g'_β(1.4) > 0 in Lemma A.4 and the interval bounds in Lemma B.4, which are not derived; Lemma A.2 also dismisses a final algebraic step as 'straightforward to verify'. These inequalities are exactly what makes the synchronization ratio exceed one, so the formal claims need a complete, checkable justification. I suggest either analytic proofs for the finite set of constants or a short supplementary file containing exact or rigorous interval-arithmetic verification.
minor comments (5)
  1. [§5, Proof of Corollary 2.4] In the -0.16 ≤ β < 0 case, the display reads τ f''(a) ≥ 2(1 + τ/π), but Lemma C.5 proves τ f''(a) ≥ -2(1 + τ/π); the sign is missing.
  2. [§6, Theorem 6.2] The statement says global synchronization occurs in (S2), but the theorem is formulated for (S4); the conclusion should refer to (S4).
  3. [§5, Proof of Corollary 2.5] The phrase 'same argument as the proof of Corollary 2.4 but with g_i...' is imprecise because Corollary 3.3 is for (S1), not (S2); since the M = π inequality implies the weaker M = 2π condition of Theorem 2.3, invoking Theorem 2.3 directly would make the step rigorous.
  4. [§C, Lemma C.5] The monotonicity statement 'both τ and (1 - 3β)e^{β^2 - β} - 2/π increase as β decreases from 0' is confusing; it should state explicitly that the product inequality is checked at the endpoint β = -0.16 and that monotonicity in the relevant direction makes it hold throughout [-0.16, 0).
  5. [Figure 1] The caption suggests the criterion is verified for β ≥ -0.25, while the paper only formally proves β ≥ -0.16; please state clearly that the plot is numerical and not part of the proof.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the synchronization criterion and the beta >= -0.16 threshold are derived by in-text inequalities, not fitted or self-referential inputs.

full rationale

The paper's central criterion, Theorem 2.1, is proved in Section 3 from the stationarity and cut-stability conditions (C1)-(C2). Lemma 3.1 and Lemma 3.2 are derived in the text, and Corollary 3.3 follows from the inequality in Lemma 3.2 together with the assumed bound on tau_max. The positive-beta branch of Corollary 2.4 is then a direct verification of the criterion: Appendices A and B prove explicit inequalities on f''' and f' for f(x)=sin(x)e^{beta(cos x -1)}, with no fitted parameters and no quantity defined in terms of the synchronization conclusion. The negative-beta branch uses Lemma C.4 to set M=pi in Corollary 3.3; that lemma is imported from the non-overlapping prior work [11, Lemma 10] and is not proved in this manuscript. This is an external dependency and a possible proof gap if the cited lemma does not apply, but it is not circular: the cited result is not an input used to define the criterion, it does not presuppose the theorem being proved, and it is not a self-citation of the present authors. Self-citations to [17] for the gradient representation and Lemma 2.2 invoke standard Lojasiewicz and center-stable manifold facts that are independently established, so the central derivation does not reduce to those citations. The threshold -0.16 is obtained by proving the inequality tau f''(a) >= 2(1+tau/pi) in Lemma C.5, not by tuning a constant to make the conclusion true. No fitted input is renamed as a prediction, and no uniqueness claim is imported from the authors' own prior work.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new entities and fits no data; the only hand-chosen numbers are internal proof constants. The argument rests on standard theorems (Lojasiewicz, center-stable manifold), the gradient representation from [17], and the imported τmax < π bound from [11]. The latter is the only borrowed result with substantial content.

free parameters (1)
  • Hand-chosen proof constants in Appendices A-C = 0.148, 0.228, 0.278, 0.321 (Lemma B.4); interval endpoints such as 0.75, 1, 2 (Lemmas A.4-A.7); endpoint -0.16 (Lemma…
    These are chosen by hand to make elementary inequalities work. They do not appear in the final theorems, but the formal verification of β ≥ -0.16 depends on the inequalities holding at these cutpoints.
assumptions (5)
  • standard math Lojasiewicz's theorem: real-analytic gradient ascent on a compact manifold converges to a critical point.
    Invoked in Section 2.1 to guarantee convergence for the energy E when h is real-analytic.
  • standard math Center-stable manifold theorem: the set of initial conditions converging to an unstable critical point has volume zero.
    Used through Lemma 2.2, attributed to [17, Lemma A.1] and Shub [31, Theorem III.7], to rule out convergence to locally unstable stationary points from almost every start.
  • domain assumption For f(x) = sin(x)h(cos(x)), the dynamics (S1) is gradient ascent on E = Σ φ(cos(xi-xj)), where φ' = h.
    This direct computation, from [17, (3.5)], is the bridge between Theorem 2.1 and global synchronization for self-attention dynamics.
  • domain assumption Imported lemma [11, Lemma 10]: for β < 0, every stable, stationary, non-synchronized point of (S1) has τmax(x) < π.
    Stated as Lemma C.4 here; it upgrades the criterion constant from 4(1+τ/2π) to 4(1+τ/π), which is needed to prove the -0.16 threshold. The present paper does not reproduce its proof.
  • domain assumption Real-analyticity of h on an open set containing [-1,1] and smooth positivity of the normalization factors gi in (S2).
    Required for Lojasiewicz's theorem and for the Riemannian gradient construction in Lemma 4.1. For fβ this holds for all real β.

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Cite this review

Pith. "Pith review of Synchronization of mean-field models on the circle." pith.science (2026). https://pith.science/paper/KESNBNN2

@misc{pith2026250722857,
  author       = {Pith},
  title        = {Pith review of: Synchronization of mean-field models on the circle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KESNBNN2}},
  note         = {Machine review of arXiv:2507.22857}
}
abstract

This paper considers a mean-field model of $n$ interacting particles whose state space is the unit circle, a generalization of the classical Kuramoto model. Global synchronization is said to occur if after starting from almost any initial state, all particles coalesce to a common point on the circle. We propose a general synchronization criterion in terms of $L_1$-norm of the third derivative of the particle interaction function. As an application we resolve a conjecture for the so-called self-attention dynamics (stylized model of transformers), by showing synchronization for all $\beta \ge -0.16$, which significantly extends the previous bound of $0\le \beta \le 1$ from Criscitiello, Rebjock, McRae, and Boumal (2024). We also show that global synchronization does not occur when $\beta < -2/3$.

Figures

Figures reproduced from arXiv: 2507.22857 by the authors.

Figure 1
Figure 1. The figure plots the synchronization ratio [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. The function fβ (red) and its derivative f ′ β (blue) for β = 2. The parameter τ = τ (β) is defined as the unique solution to the equation f ′ β (τ ) = 0 over [0, π]. For β = 2, τ ≃ 0.6749. In this case the particles tend to equi-disperse on the circle. Indeed, the unique global minimizer of (3) is the n-gon as shown in [10]. At the level of the evolution of measures, the unique global minimizer of the functions µ 7… view at source ↗

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Reference graph

Works this paper leans on

40 extracted references · 31 canonical work pages · cited by 5 Pith papers

  1. [17]

    A math- ematical perspective on transformers

    Borjan Geshkovski, Cyril Letrouit, Yury Polyanskiy, and Philippe Rigollet. A math- ematical perspective on transformers. Bull. Amer. Math. Soc. , 62:427–479, 2025

    work page 2025

  2. [1]

    Bandeira, Martin Kassabov, Victor Souza, Steven H

    Pedro Abdalla, Afonso S. Bandeira, Martin Kassabov, Victor Souza, Steven H. Stro- gatz, and Alex Townsend. Expander graphs are globally synchronizing.arXiv preprint arXiv:2210.12788, 2024

    arXiv 2024

  3. [2]

    The asymptotic behavior of attention in transformers

    ´Alvaro Rodr ´ ıguez Abella, Jo˜ ao Pedro Silvestre, and Paulo Tabuada. The asymptotic behavior of attention in transformers. arXiv preprint arXiv:2412.02682 , 2024

    arXiv 2024

  4. [3]

    The Kuramoto model: A simple paradigm for synchronization phenomena

    Juan A Acebr´ on, Luis L Bonilla, Conrad J P´ erez Vicente, F´ elix Ritort, and Renato Spigler. The Kuramoto model: A simple paradigm for synchronization phenomena. Reviews of Modern Physics , 77(1):137–185, 2005. 20

    work page 2005

  5. [4]

    Convergence in multiagent coordination, consensus, and flocking

    Vincent D Blondel, Julien M Hendrickx, Alex Olshevsky, and John N Tsitsiklis. Convergence in multiagent coordination, consensus, and flocking. In Proceedings of the 44th IEEE Conference on Decision and Control , pages 2996–3000. IEEE, 2005

    work page 2005

  6. [5]

    An introduction to optimization on smooth manifolds

    Nicolas Boumal. An introduction to optimization on smooth manifolds . Cambridge University Press, 2023

    work page 2023

  7. [6]

    Emergence of meta- stable clustering in mean-field transformer models

    Giuseppe Bruno, Federico Pasqualotto, and Andrea Agazzi. Emergence of meta- stable clustering in mean-field transformer models. In The Thirteenth International Conference on Learning Representations, 2025

    work page 2025

  8. [7]

    Anal- ysis of mean-field models arising from self-attention dynamics in transformer archi- tectures with layer normalization

    Martin Burger, Samira Kabri, Yury Korolev, Tim Roith, and Lukas Weigand. Anal- ysis of mean-field models arising from self-attention dynamics in transformer archi- tectures with layer normalization. arXiv preprint arXiv:2501.03096 , 2025

    arXiv 2025

Show all 40 references
  1. [8]

    A unified perspective on the dynamics of deep transformers

    Val´ erie Castin, Pierre Ablin, Jos´ e Antonio Carrillo, and Gabriel Peyr´ e. A unified perspective on the dynamics of deep transformers. arXiv preprint arXiv:2501.18322, 2025

    2025 arXiv

  2. [9]

    Quantitative clustering in mean-field transformer models

    Shi Chen, Zhengjiang Lin, Yury Polyanskiy, and Philippe Rigollet. Quantitative clustering in mean-field transformer models. arXiv preprint arXiv:2504.14697 , 2025

    2025 arXiv

  3. [10]

    Universally optimal distribution of points on spheres

    Henry Cohn and Abhinav Kumar. Universally optimal distribution of points on spheres. Journal of the American Mathematical Society , 20(1):99–148, 2007

    2007

  4. [11]

    McRae, and Nicolas Boumal

    Christopher Criscitiello, Quentin Rebjock, Andrew D. McRae, and Nicolas Boumal. Synchronization on circles and spheres with nonlinear interactions. arXiv preprint arXiv:2405.18273, 2024

    2024

  5. [12]

    Setting the record straight on transformer oversmoothing

    Gb` etondji JS Dovonon, Michael M Bronstein, and Matt J Kusner. Setting the record straight on transformer oversmoothing. arXiv preprint arXiv:2401.04301 , 2024

    2024 arXiv

  6. [13]

    Rank diminishing in deep neural networks

    Ruili Feng, Kecheng Zheng, Yukun Huang, Deli Zhao, Michael Jordan, and Zheng- Jun Zha. Rank diminishing in deep neural networks. Advances in Neural Information Processing Systems, 35:33054–33065, 2022

    2022

  7. [14]

    Long-time dynamics for a simple aggregation equation on the sphere

    Amic Frouvelle and Jian-Guo Liu. Long-time dynamics for a simple aggregation equation on the sphere. In International workshop on Stochastic Dynamics out of Equilibrium, pages 457–479. Springer, 2017

    2017

  8. [15]

    The emergence of clusters in self-attention dynamics

    Borjan Geshkovski, Cyril Letrouit, Yury Polyanskiy, and Philippe Rigollet. The emergence of clusters in self-attention dynamics. In Thirty-seventh Conference on Neural Information Processing Systems , 2023

    2023

  9. [16]

    Dynamic metastability in the self-attention model

    Borjan Geshkovski, Hugo Koubbi, Yury Polyanskiy, and Philippe Rigollet. Dynamic metastability in the self-attention model. arXiv preprint arXiv:2410.06833 , 2024. 21

    2024 arXiv

  10. [18]

    On the Relaxation Dynamics of Lohe Oscillators on Some Riemannian Manifolds

    Seung-Yeal Ha, Dongnam Ko, and Seung-Yeon Ryoo. On the Relaxation Dynamics of Lohe Oscillators on Some Riemannian Manifolds. Journal of Statistical Physics , 172:1427–1478, 06 2018

    2018

  11. [19]

    Coordination of groups of mobile autonomous agents using nearest neighbor rules

    Ali Jadbabaie, Jie Lin, and A Stephen Morse. Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 48(6):988–1001, 2003

    2003

  12. [20]

    The random graph process is globally synchronizing

    Vishesh Jain, Clayton Mizgerd, and Mehtaab Sawhney. The random graph process is globally synchronizing. arXiv preprint arXiv:2501.12205 , 2025

    2025 arXiv

  13. [21]

    Clustering in causal at- tention masking

    Nikita Karagodin, Yury Polyanskiy, and Philippe Rigollet. Clustering in causal at- tention masking. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024

    2024

  14. [22]

    Strogatz, and Alex Townsend

    Martin Kassabov, Steven H. Strogatz, and Alex Townsend. Sufficiently dense ku- ramoto networks are globally synchronizing. Chaos: An Interdisciplinary Journal of Nonlinear Science, 31(7):073135, Jul 2021

    2021

  15. [23]

    The impact of LoRA on the emergence of clusters in transformers

    Hugo Koubbi, Matthieu Boussard, and Louis Hernandez. The impact of LoRA on the emergence of clusters in transformers. arXiv preprint arXiv:2402.15415 , 2024

    2024 arXiv

  16. [24]

    Self-entrainment of a population of coupled non-linear oscillators

    Yoshiki Kuramoto. Self-entrainment of a population of coupled non-linear oscillators. In International Symposium on Mathematical Problems in Theoretical Physics, pages 420–422, Berlin, Heidelberg, 1975. Springer Berlin Heidelberg

    1975

  17. [25]

    Une propri´ et´ e topologique des sous-ensembles analytiques r´ eels.Les ´Equations aux D´ eriv´ ees Partielles, 117:87–89, 1963

    Stanislaw Lojasiewicz. Une propri´ et´ e topologique des sous-ensembles analytiques r´ eels.Les ´Equations aux D´ eriv´ ees Partielles, 117:87–89, 1963

    1963

  18. [26]

    Almost global consensus on the n -sphere

    Johan Markdahl, Johan Thunberg, and Jorge Gon¸ calves. Almost global consensus on the n -sphere. IEEE Transactions on Automatic Control , 63(6):1664–1675, 2018

    2018

  19. [27]

    Synchronization of pulse-coupled biological oscillators

    Renato E Mirollo and Steven H Strogatz. Synchronization of pulse-coupled biological oscillators. SIAM Journal on Applied Mathematics , 50(6):1645–1662, 1990

    1990

  20. [28]

    On the trend to global equilibrium for kuramoto oscillators

    Javier Morales and David Poyato. On the trend to global equilibrium for kuramoto oscillators. Annales de l’Institut Henri Poincar´ e C, 40(3):631–716, 2022

    2022

  21. [29]

    Sinkform- ers: Transformers with doubly stochastic attention

    Michael E Sander, Pierre Ablin, Mathieu Blondel, and Gabriel Peyr´ e. Sinkform- ers: Transformers with doubly stochastic attention. In International Conference on Artificial Intelligence and Statistics , pages 3515–3530. PMLR, 2022

    2022

  22. [30]

    Han Shi, Jiahui Gao, Hang Xu, Xiaodan Liang, Zhenguo Li, Lingpeng Kong, Stephen M. S. Lee, and James Kwok. Revisiting over-smoothing in BERT from the perspective of graph. In International Conference on Learning Representations , 2022. 22

    2022

  23. [31]

    Global Stability of Dynamical Systems

    Michael Shub. Global Stability of Dynamical Systems . Springer New York, NY, 1 edition, 2013

    2013

  24. [32]

    From Kuramoto to Crawford: exploring the onset of synchro- nization in populations of coupled oscillators

    Steven H Strogatz. From Kuramoto to Crawford: exploring the onset of synchro- nization in populations of coupled oscillators. Physica D: Nonlinear Phenomena, 143 (1-4):1–20, 2000

    2000

  25. [33]

    There is no non-zero stable fixed point for dense networks in the homogeneous Kuramoto model

    Richard Taylor. There is no non-zero stable fixed point for dense networks in the homogeneous Kuramoto model. Journal of Physics A: Mathematical and Theoretical, 45(5):055102, Jan 2012

    2012

  26. [34]

    Attention is all you need

    Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017

    2017

  27. [35]

    Novel type of phase transition in a system of self-driven particles

    Tam´ as Vicsek, Andr´ as Czir´ ok, Eshel Ben-Jacob, Inon Cohen, and Ofer Shochet. Novel type of phase transition in a system of self-driven particles. Physical Review Letters, 75(6):1226, 1995

    1995

  28. [36]

    Biological rhythms and the behavior of populations of coupled oscillators

    Arthur T Winfree. Biological rhythms and the behavior of populations of coupled oscillators. Journal of Theoretical Biology , 16(1):15–42, 1967

    1967

  29. [37]

    The lower bound of the network connectivity guaranteeing in-phase synchronization

    Ryosuke Yoneda, Tsuyoshi Tatsukawa, and Jun-nosuke Teramae. The lower bound of the network connectivity guaranteeing in-phase synchronization. Chaos: An In- terdisciplinary Journal of Nonlinear Science , 31(6):063124, Jun 2021. A Results for β >1 3 In this subsection, f (x) = ...

    2021

  30. [38]

    Suppose β ∈ [0.5, 0.75)

    ■ Lemma A.6. Suppose β ∈ [0.5, 0.75). Then, τ (f ′′(a) − f ′′(b)) < 2. Proof. Observe that g′ β(0.14) < 0 and g′ β(0.9) > 0 for β ∈ [0.5, 0.75). Thus, 0.14 < β(1 − cos(b)) < 0.9 < β(1 − cos(a)). Similarly, gβ(β(1 − cos(a))) > 0 > gβ(β(1 − cos(b))), because gβ(0) = 0, gβ(0.1) <...

  31. [39]

    There exists three elements p1,p2, and p3 of T such that ⌊ n 3 ⌋, ⌊ n 3 ⌋, and n − 2⌊ n 3 ⌋ points of x are placed at p1, p2, and p3, respectively

  32. [40]

    The vector x is a critical point of E and the Hessian of E over Tn at x is negative semidefinite, with only one eigenvalue whose eigenvectors are the scalar multiples of the vector [1, . . . ,1]⊤. Proof. Suppose n ≥ 3. Let p1 = 0, p2 = α, and p3 = 2π − α, where α is an element...

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