# 3. Application to the models This section shows the phase diagrams of the Hamiltonian (3). We ﬁrst discuss the Hopﬁeld model with k-body interactions and ﬁnite patterns embedded. Next, we study the case with many patterns. 3.1. Hopfield model with finite patterns We give self-consistent equations for the Hopﬁeld model with ﬁnite

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Thisequivalenceallows which leads to a phase diagram. The effective retarded self-interaction usually appearing in symmetric models is here found to vanish, which causes a significantly enlarged storage capacity of eYe ~ 0.269. com pared to eYe ~ 0.139 for Hopfield networks s~oring static patterns. Our We find for the noiseless zero-temperature case that this non-monotonic Hopfield network can store more patterns than a network with monotonic transfer function investigated by Amit et al. Properties of retrieval phase diagrams of non-monotonic networks agree with the results obtained by Nishimori and Opris who treated synchronous networks. Restricted Boltzmann Machines are described by the Gibbs measure of a bipartite spin glass, which in turn corresponds to the one of a generalised Hopfield network. This equivalence allows us to characterise the state of these systems in terms of retrieval capabilities, at both low and high load.

We ﬁrst discuss the Hopﬁeld model with k-body interactions and ﬁnite patterns embedded. Next, we study the case with many patterns. 3.1. Hopfield model with finite patterns We give self-consistent equations for the Hopﬁeld model with ﬁnite patterns embedded. Spin-1 Hopfield model under analysis using the one-step replica-symmetry-breaking mean field theory to obtain the order parameters and phase diagrams for The Hopfield model , consists of a network of N N neurons, labeled by a lower index i i, with 1 ≤ i ≤ N 1\leq i\leq N. Similar to some earlier models (335; 304; 549), neurons in the Hopfield model have only two states. Motivated by recent progress in using restricted Boltzmann machines as preprocessing algorithms for deep neural network, we revisit the mean-field equations [belief-propagation and Thouless-Anderson Palmer (TAP) equations] in the best understood of such machines, namely the Hopfield model of neural networks, and we explicit how they can be used as iterative message-passing algorithms The phase diagrams of the model with finite patterns show that there exist annealing paths that avoid first-order transitions at least for .

## In this work, we introduce and investigate the properties of the “relativistic” Hopfield model endowed with temporally correlated patterns. First, we review the “relativistic” Hopfield model and we briefly describe the experimental evidence underlying correlation among patterns.

Figure 3.4: Phase diagram for the Kuramoto model (3.15) in the case of model. Hopfield, “Neural networks and physical systems with emergent collective Nearly any non-trivial model‡ exhibits “phase diagrams,” with qualitatively. Aug 9, 2020 ai #transformer #attentionHopfield Networks are one of the classic models of biological memory networks.

### A Hopfield network which operates in a discrete line fashion or in other words, it can be said the input and output patterns are discrete vector, which can be either

We study the paramagnetic-spin glass and the spin glass-retrieval phase transitions, as the pattern Retrieval Phase Diagrams of Non-monotonic Hopfield Networks Item Preview remove-circle Share or Embed This Item. The Hopfield model is a canonical Ising computing model. Previous studies have analyzed the effect of a few nonlinear functions (e.g. sign) for mapping the coupling strength on the Hopfield model Let us compare this result with the phase diagram of the standard Hopfield model calculated in a replica symmetric approximation [5,11]. Again we have three phases.

In Fig. 1 we present the phase diagram of the Hopfield model obtained analytically and assuming a replica symmetric Ansatz .Above the T g line the system has a paramagnetic solution with an associated simple homogeneous dynamics. the model converges to a stable state and that two kinds of learning rules can be used to ﬁnd appropriate network weights. 13.1 Synchronous and asynchronous networks A relevant issue for the correct design of recurrent neural networks is the ad-equate synchronization of the computing elements. In the case of McCulloch-
A Hopfield network (or Ising model of a neural network or Ising–Lenz–Little model) is a form of recurrent artificial neural network popularized by John Hopfield in 1982, but described earlier by Little in 1974 based on Ernst Ising's work with Wilhelm Lenz. Let us compare this result with the phase diagram of the standard Hopfield model calculated in a replica symmetric approximation [5,11]. Again we have three phases. For temperatures above the broken line T SG , there exist paramagnetic solutions characterized by m = q = 0, while below the broken line, spin glass solutions, m = 0 but q = 0, exist.

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The Hopfield model , consists of a network of N N neurons, labeled by a lower index i i, with 1 ≤ i ≤ N 1\leq i\leq N. Similar to some earlier models (335; 304; 549), neurons in the Hopfield model … 1992-11-01 The phase diagrams of the model with finite patterns show that there exist annealing paths that avoid first-order transitions at least for . The same is true for the extensive case with k = 4 and 5. In contrast, it is impossible to avoid first-order transitions for the case of finite patterns with k = 3 and the case of extensive number of patterns with k = 2 and 3.

Using the Trotter decomposition and the replica method, we find that the α (the ratio of the number of stored patterns to the system size)- ∆ (the strength of the
We study the Hopfield model on a random graph in scaling regimes where the average number of connections per neuron is a finite number and the spin dynamics is governed by a synchronous execution of the microscopic update rule (Little–Hopfield model).

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### spin glass phase—is in many ways a prototype for methods of equilibrium statistical mechanics have been Phase diagram of the Hopfield model. The.

In this Python exercise we focus on visualization and simulation to develop our intuition about Hopfield dynamics. 2017-02-20 · Title: Phase Diagram of Restricted Boltzmann Machines and Generalised Hopfield Networks with Arbitrary Priors Authors: Adriano Barra , Giuseppe Genovese , Peter Sollich , Daniele Tantari (Submitted on 20 Feb 2017 ( v1 ), last revised 29 Jul 2017 (this version, v2)) This is why Hopfield networks always create pairs of memories (the desired ones and their inverses). It does not (indeed cannot) distinguish between these two situations when the weights are being set.

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### Phase diagram of restricted Boltzmann machines and generalized Hopfield networks with Phantomlike behavior in a brane-world model with curvature effects.

The phase diagrams clearly corespond to theoretical descriptions (See ﬁgure 2). Phase diagrams and the instability of the spin glass states for the diluted Hopfield neural network model. Journal de Physique I, EDP Sciences, 1992, 2 (9), pp.1791- 2001-06-01 CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Abstract. We investigate the retrieval phase diagrams of an asynchronous fully-connected attractor network with non-monotonic transfer function by means of a mean-field approximation. We find for the noiseless zero-temperature case that this non-monotonic Hopfield network can store more patterns than a network with Hopfield models (The Hopfield network (Energy function (, låter oss…: Hopfield models (The Hopfield network, McCulloch-Pitts neuron, Stochastic optimization*), Hamming distance mellan mönster µ och testmönstret, = hitta mest lika lagrade mönstret, Assume \(\mathbf{x}\) is a distorted version of \(\mathbf{x}^{(\nu)}\), , \(b_{i}\) kallas local field, Alltså vikter som beror på de We investigate the retrieval phase diagrams of an asynchronous fully-connected attractor network with non-monotonic transfer function by means of a mean-field approximation. We find for the noiseless zero-temperature case that this non-monotonic Hopfield network can store more patterns than a network with monotonic transfer function investigated by Amit et al.

## CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Abstract. We investigate the retrieval phase diagrams of an asynchronous fully-connected attractor network with non-monotonic transfer function by means of a mean-field approximation. We find for the noiseless zero-temperature case that this non-monotonic Hopfield network can store more patterns than a network with

In the upper region (P) the network behaves randomly while in the top- right KEYWORDS: neural networks, Hopfield model, quantum effects, macrovariables, phase diagram. $1. Introduction.

Phase diagrams and the instability of the spin glass states for the diluted Hopfield neural network model. retrieval phase diagram non-monotonic hopfield network non-monotonic hopfield model associative memory state-dependent synaptic coupling optimal storage capacity statistical mechanical approach asynchronous fully-connected attractor network non-monotonic network monotonic transfer function state-dependent synapsis store attractor network mean The phase diagram coincides very accurately with that of the conventional classical Hopfield model if we replace the temperature T in the latter model by $\Delta$. In Fig. 1 we present the phase diagram of the Hopfield model obtained analytically and assuming a replica symmetric Ansatz .Above the T g line the system has a paramagnetic solution with an associated simple homogeneous dynamics. the model converges to a stable state and that two kinds of learning rules can be used to ﬁnd appropriate network weights. 13.1 Synchronous and asynchronous networks A relevant issue for the correct design of recurrent neural networks is the ad-equate synchronization of the computing elements.