Renormalization flow of the hierarchical Anderson model at weak disorder

Metz, Fernando L.; Leuzzi, Luca; Parisi, G.

doi:10.1103/physrevb.89.064201

Cited by 6 publications

(8 citation statements)

References 28 publications

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“…In this section we show how to compute the trace over the spins using a simple change of integration variables combined with the model hierarchical structure. Such approach has been employed to study hierarchical models in the context of interacting spin systems (see [36] and references therein) and Anderson localisation [37,38,39].…”

Section: The Recursive Methodsmentioning

confidence: 99%

Statistical mechanics of the spherical hierarchical model with random fields

Metz,

Rocchi,

Urbani

2014

Preprint

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We study analytically the equilibrium properties of the spherical hierarchical model in the presence of random fields. The expression for the critical line separating a paramagnetic from a ferromagnetic phase is derived. The critical exponents characterising this phase transition are computed analytically and compared with those of the corresponding D-dimensional short-range model, leading to conclude that the usual mapping between one dimensional long-range models and D-dimensional short-range models holds exactly for this system, in contrast to models with Ising spins. Moreover, the critical exponents of the pure model and those of the random field model satisfy a relationship that mimics the dimensional reduction rule. The absence of a spin-glass phase is strongly supported by the local stability analysis of the replica symmetric saddle-point as well as by an independent computation of the free-energy using a renormalization-like approach. This latter result enlarges the class of random field models for which the spin-glass phase has been recently ruled out.

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Section: The Recursive Methodsmentioning

confidence: 99%

Statistical mechanics of the spherical hierarchical model with random fields

Metz,

Rocchi,

Urbani

2014

Preprint

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“…If we want the system to handle up to p patterns, we need p different blocks of spins and then M = log(p). 28) obtained for different values of σ (as explained by the legend) and plotted versus a rescaled noise. Note that by rescaling the noise the dependence on σ is lost and all curves are collapsed.…”

Section: Serial Versus Parallel Retrieval In Hopfield Hierarchical Modelmentioning

confidence: 99%

“…1: each pair of nearest-neighbor spins form a "dimer" connected with the strongest coupling, then spins belonging to nearest "dimers" interact each other with a weaker coupling and so on recursively [32]. In particular, the Sherrington-Kirkpatrick model for spin-glasses defined on the hierarchical topology has been investigated in [18]: despite a full analytic formulation of its solution still lacks, renormalization techniques, [19,30], rigorous bounds on its free-energies [17] and extensive numerics [27,28] can be achieved nowadays and they give extremely sharps hints on the thermodynamic behavior of systems defined on these peculiar topologies.…”

Section: Introductionmentioning

confidence: 99%

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Hierarchical neural networks perform both serial and parallel processing

et al. 2015

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In this work we study a Hebbian neural network, where neurons are arranged according to a hierarchical architecture such that their couplings scale with their reciprocal distance. As a full statistical mechanics solution is not yet available, after a streamlined introduction to the state of the art via that route, the problem is consistently approached through signal-to-noise technique and extensive numerical simulations. Focusing on the low-storage regime, where the amount of stored patterns grows at most logarithmical with the system size, we prove that these non-mean-field Hopfield-like networks display a richer phase diagram than their classical counterparts. In particular, these networks are able to perform serial processing (i.e. retrieve one pattern at a time through a complete rearrangement of the whole ensemble of neurons) as well as parallel processing (i.e. retrieve several patterns simultaneously, delegating the management of different patterns to diverse communities that build network). The tune between the two regimes is given by the rate of the coupling decay and by the level of noise affecting the system. The price to pay for those remarkable capabilities lies in a network's capacity smaller than the mean field counterpart, thus yielding a new budget principle: the wider the multitasking capabilities, the lower the network load and vice versa. This may have important implications in our understanding of biological complexity.

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“…Starting from the pioneering Dyson work [30], where the hierarchical ferromagnet was introduced and its phase transition (splitting an ergodic region from a ferromagnetic one) rigorously proven, recently its extensions to spin-glasses have also been investigated [31]. Although an analytical solution is still not available, giant step forward toward a deep comprehension of the hierarchical statistical mechanics have been obtained [32,33,34,35,36,37,38,39]. In this paper we aim to analyze in details the hierarchical neural networks, and to this task we start by considering the statistical mechanics of the Dyson model from a novel perspective: we investigate its metastabilities.In Section One we deal with Dyson's model: once fundamental definitions have been introduced, in the first subsection we prove the existence of the thermodynamic limit of its related free energy within the spirit of the classical Guerra-Toninelli scheme [41] to the case.…”

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confidence: 99%

Metastable states in the hierarchical Dyson model drive parallel processing in the hierarchical Hopfield network

Agliari

Barra

Galluzzi

et al. 2014

J. Phys. A: Math. Theor.

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In this paper we introduce and investigate the statistical mechanics of hierarchical neural networks: First, we approach these systems à la Mattis, by thinking at the Dyson model as a single-pattern hierarchical neural network and we discuss the stability of different retrievable states as predicted by the related self-consistencies obtained from a mean-field bound and from a bound that bypasses the mean-field limitation. The latter is worked out by properly reabsorbing fluctuations of the magnetization related to higher levels of the hierarchy into effective fields for the lower levels. Remarkably, mixing Amit's ansatz technique (to select candidate retrievable states) with the interpolation procedure (to solve for the free energy of these states) we prove that (due to gauge symmetry) the Dyson model accomplishes both serial and parallel processing. One step forward, we extend this scenario toward multiple stored patterns by implementing the Hebb prescription for learning within the couplings. This results in an Hopfield-like networks constrained on a hierarchical topology, for which, restricting to the low storage regime (where the number of patterns grows at most logarithmical with the amount of neurons), we prove the existence of the thermodynamic limit for the free energy and we give an explicit expression of its mean field bound and of the related improved bound. The resulting self-consistencies for the Mattis magnetizations (that act as order parameters) are studied and the stability of solutions is analyzed to get a picture of the overall retrieval capabilities of the system according to the mean field and to the non-mean-field scenarios. Our main finding is that embedding the Hebbian rule on a hierarchical topology allows the network to accomplish both serial and parallel processing. By tuning the level of fast noise affecting it, or triggering the decay of the interactions with the distance among neurons, the system may switch from sequential retrieval to multitasking features and vice versa. However, as these multitasking capabilities are basically due to the vanishing "dialogue" between spins at long distance, such an effective penury of links strongly penalizes the network's capacity, which results bounded by the low storage.

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Renormalization flow of the hierarchical Anderson model at weak disorder

Cited by 6 publications

References 28 publications

Statistical mechanics of the spherical hierarchical model with random fields

Statistical mechanics of the spherical hierarchical model with random fields

Hierarchical neural networks perform both serial and parallel processing

Metastable states in the hierarchical Dyson model drive parallel processing in the hierarchical Hopfield network

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