# Originally, the Hopfield NN was introduced as a toy model of associative Phase diagram of the OQS generalization of the Hopfield model in the (T,Ω) plane.

A. Barra, G. Genovese, P. Sollich, D. Tantari, Phase diagram of restricted Boltzmann machines and generalized Hopfield networks with arbitrary priors , Physical Review E 97 (2), 022310, 2018 Restricted Boltzmann machines are described by the Gibbs measure of a bipartite spin glass, which in turn can be seen as a generalized Hopfield network.

Previous studies have analyzed the effect of a few nonlinear functions (e.g. sign) for mapping the coupling strength on the Hopfield model 13.7 Hopfield Model (1/7) • The Hopfield network (model) consists of a set of neurons and a corresponding set of unit-time delays, forming a multiple-loop feedback system. • To study the dynamics of the Hopfield network, we use the neurodynamic model which is based on the additive model of a neuron. Figure 13.9 Architectural graph of a Hopfield 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 A symmetrically dilute Hopfield model with a Hebbian learning rule is used to study the effects of gradual dilution and of synaptic noise on the categorization ability of an attractor neural network with hierarchically correlated patterns in a two-level structure of ancestors and descendants. The purpose of a Hopfield net is to store 1 or more patterns and to recall the full patterns based on partial input. For example, consider the problem of optical character recognition.

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But at any > 0, m eventually peels off from this asymptote to reach m = 1 for T 0. Lower panels show the behaviour of : it tends to zero linearly at low temperature, T/ , while for T > , = . - "Phase Diagram of Restricted Boltzmann Machines and Generalised Hopfield Networks with Arbitrary Priors" 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.

It does not (indeed cannot) distinguish between these two situations when the weights are being set. Summary of Hopfield Network Equations Weight setting (training) for n memories in an N node Hopfield Network - Ground-state phase diagram as a function of U ∞, μ, and t (in units of U 0) obtained by numerically determining the phases using the mean-field model of Eq. .

## CSE 5526: Hopfield Nets 5 Hopfield (1982) describes the problem • “Any physical system whose dynamics in phase space is dominated by a substantial number of locally stable states to which it is attracted can therefore be regarded as a general content-addressable memory. The physical system will be a potentially useful memory if, in addition

Properties of retrieval phase A Hopfield network (or Ising model of a neural network or Ising–Lenz–Little model) is a form of recurrent artificial neural network and a type of spin glass system popularised by John Hopfield in 1982 as described earlier by Little in 1974 based on Ernst Ising's work with Wilhelm Lenz on Ising Model. Let us compare this result with the phase diagram of the standard Hopfield model calculated in a replica symmetric approximation [5,11].

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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.

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). We solve this model within replica symmetry, and by using bifurcation analysis we prove that the spin-glass/paramagnetic and the retrieval
7. Hopfield Network model of associative memory¶.

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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.

In this video I present the graphs used in visualizing the Ramsey Cass Koopmans model.

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### As 0, m approaches the value (3.5) at low T . But at any > 0, m eventually peels off from this asymptote to reach m = 1 for T 0. Lower panels show the behaviour of : it tends to zero linearly at low temperature, T/ , while for T > , = . - "Phase Diagram of Restricted Boltzmann Machines and Generalised Hopfield Networks with Arbitrary Priors"

local minima of the energy function- But these are not the only attractors a

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PHASE DIAGRAM OF RESTRICTED BOLTZMANN MACHINES AND GENERALISED HOPFIELD NETWORKS WITH ARBITRARY PRIORS ADRIANOBARRA,GIUSEPPEGENOVESE,PETERSOLLICH,ANDDANIELETANTARI Abstract. Restricted Boltzmann Machines are described by the Gibbs measure of a bipartite spin glass,whichinturncorrespondstotheoneofageneralisedHopﬁeldnetwork. 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.

model of McCulloch and Pitts [38], the Rosenblatt perceptron [42], … 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 study the paramagnetic-spin glass and the spin glass-retrieval phase transitions, as the pattern 2018-02-20 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.