PDE/Statistical Mechanics Duality: Relation Between Guerra’s Interpolated p-Spin Ferromagnets and the Burgers Hierarchy

Fachechi, Alberto

doi:10.1007/s10955-021-02747-9

Cited by 6 publications

(16 citation statements)

References 82 publications

(101 reference statements)

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“…Remark 5. If we do not take into account the disordered part of the system, namely J = 0, in (1.28), we recover for this P-spin ferromagnetic model the same result in [19]. Remark 6.…”

Section: Rs Solutionsupporting

confidence: 63%

Rigorous approaches for spin glass and Gaussian spin glass with P-wise interactions

Albanese¹,

Alessandrelli²

2021

Preprint

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Purpose of this paper is to face up to P-spin glass and Gaussian P-spin model, i.e. spin glasses with polynomial interactions of degree P > 2. We consider the replica symmetry and first step of replica simmetry breaking assumptions and we solve the models via transport equation and Guerra's interpolating technique, showing that we reach the same results.Thus, using rigorous approaches, we recover the same expression for quenched statistical pressure and self-consistency equation in both assumption found with other techniques, including the wellknown replica trick technique.At the end, we show that for P = 2 the Gaussian P-spin glass model is intrinsecally RS.

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“…Remark 5. If we do not take into account the disordered part of the system, namely J = 0, in (1.28), we recover for this P-spin ferromagnetic model the same result in [19]. Remark 6.…”

Section: Rs Solutionsupporting

confidence: 63%

Rigorous approaches for spin glass and Gaussian spin glass with P-wise interactions

Albanese¹,

Alessandrelli²

2021

Preprint

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“…The present section is a compendium of the results reported in [36], so we refer to that work for a detailed derivation. In p-spin ferromagnets, the interaction between spins is fixed by the requirement J i1,i2,...,ip = J for each i 1 , .…”

Section: Generalities and Notationmentioning

confidence: 99%

“…In this case, we can take advantage of rigorous mathematical methods by applying sum rules [26] or by mapping the relevant quantities (the free energy or the model order parameters) of the statistical setting to the solutions of PDE systems. Indeed, differential equations involving the partition functions (or related quantities) of thermodynamic models have been extensively investigated in the literature, see for example [27][28][29][30][31][32][33][34][35][36][37]. In particular, they allow us to express the equation of state (or the self-consistency equations) governing the equilibrium dynamics of the system in terms of solutions of non-linear differential equations, and to describe phase transition phenomena as the development of shock waves, thus linking critical behaviours to gradient catastrophe theory [38][39][40][41].…”

Section: Introductionmentioning

confidence: 99%

“…In particular, they allow us to express the equation of state (or the self-consistency equations) governing the equilibrium dynamics of the system in terms of solutions of non-linear differential equations, and to describe phase transition phenomena as the development of shock waves, thus linking critical behaviours to gradient catastrophe theory [38][39][40][41]. In a recent work [36], a direct connection between the thermodynamics of ferromagnetic models with interactions of order p and the equations of the Burgers hierarchy was established by linking the solution of the latter as the equilibrium solution of the order parameter of the former (i.e. the global magnetization m).…”

Section: Introductionmentioning

confidence: 99%

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Non-linear PDEs approach to statistical mechanics of Dense Associative Memories

Agliari¹,

Fachechi²,

Marullo³

2022

Preprint

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Dense associative memories (DAM), are widespread models in artificial intelligence used for pattern recognition tasks; computationally, they have been proven to be robust against adversarial input and theoretically, leveraging their analogy with spin-glass systems, they are usually treated by means of statistical-mechanics tools. Here we develop analytical methods, based on nonlinear PDEs, to investigate their functioning. In particular, we prove differential identities involving DAM's partition function and macroscopic observables useful for a qualitative and quantitative analysis of the system. These results allow for a deeper comprehension of the mechanisms underlying DAMs and provide interdisciplinary tools for their study.

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“…Moreover, it is easy to check that the only no trivial case is a = P − 1, otherwise our model turns into a ferromagnet [39] with polynomial interaction of degree P . So, if we consider P ≥ 4 (since we have rescaled N P/2−1 p in p), the previous expression reads as (A.8) in the thermodynamics limit (N → ∞) the correction terms will vanish.…”

Section: Conclusion and Outlooksmentioning

confidence: 99%

Replica symmetry breaking in dense neural networks

Albanese,

Alemanno,

Alessandrelli

et al. 2021

Preprint

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Understanding the glassy nature of neural networks is pivotal both for theoretical and computational advances in Machine Learning and Theoretical Artificial Intelligence. Keeping the focus on dense associative Hebbian neural networks (i.e. Hopfield networks with polynomial interactions of even degree P > 2), the purpose of this paper is two-fold: at first we develop rigorous mathematical approaches to address properly a statistical mechanical picture of the phenomenon of replica symmetry breaking (RSB) in these networks, then -deepening results stemmed via these routes-we aim to inspect the glassiness that they hide. In particular, regarding the methodology, we provide two techniques: the former (closer to mathematical physics in spirit) is an adaptation of the transport PDE to the case, while the latter (more probabilistic in its nature) is an extension of Guerra's interpolation breakthrough. Beyond coherence among the results, either in replica symmetric and in the one-step replica symmetry breaking level of description, we prove the Gardner's picture (heuristically achieved through the replica trick) and we identify the maximal storage capacity by a ground-state analysis in the Baldi-Venkatesh high-storage regime.In the second part of the paper we investigate the glassy structure of these networks: at difference with the replica symmetric scenario (RS), RSB actually stabilizes the spin-glass phase. We report huge differences w.r.t. the standard pairwise Hopfield limit: in particular, it is known that it is possible to express the free energy of the Hopfield neural network (and, in a cascade fashion, all its properties) as a linear combination of the free energies of an hard spin glass (i.e. the Sherrington-Kirkpatrick model) and a soft spin glass (the Gaussian or "spherical" model). While this continues to hold also in the first step of RSB for the Hopfield model, this is no longer true when interactions are more than pairwise (whatever the level of description, RS or RSB). For dense networks solely the free energy of the hard spin glass survives. As the Sherrington-Kirkpatrick spin glass is full-RSB (i.e. Parisi theory holds for that model), while the Gaussian spin-glass is replica symmetric, these different representation theorems prove a huge diversity in the underlying glassiness of associative neural networks.

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PDE/Statistical Mechanics Duality: Relation Between Guerra’s Interpolated p-Spin Ferromagnets and the Burgers Hierarchy

Cited by 6 publications

References 82 publications

Rigorous approaches for spin glass and Gaussian spin glass with P-wise interactions

Rigorous approaches for spin glass and Gaussian spin glass with P-wise interactions

Non-linear PDEs approach to statistical mechanics of Dense Associative Memories

Replica symmetry breaking in dense neural networks

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