AI News, Artificial Neural Networks/Hopfield Networks

Artificial Neural Networks/Hopfield Networks

The activation function of a binary Hopfield network is given by the signum function of a biased weighted sum:

This means that mathematical minimization or optimization problems can be solved automatically by the Hopfield network if that problem can be formulated in terms of the network energy.

Each attractor represents a different data value that is stored in the network, and a range of associated patterns can be used to retrieve the data pattern.

Artificial Neural Networks/Hopfield Networks

The activation function of a binary Hopfield network is given by the signum function of a biased weighted sum:

This means that mathematical minimization or optimization problems can be solved automatically by the Hopfield network if that problem can be formulated in terms of the network energy.

Each attractor represents a different data value that is stored in the network, and a range of associated patterns can be used to retrieve the data pattern.

Pattern recall analysis of the Hopfield neural network with a genetic algorithm

This paper describes the implementation of a genetic algorithm to evolve the population of weight matrices for storing and recalling the patterns in a Hopfield type neural network model.

In the Hopfield type neural network of associative memory, the appropriate arrangement of synaptic weights provides an associative function in the network.

For this, we explore the population generation technique (mutation and elitism), crossover and the fitness evaluation function for generating the new population of the weight matrices.

Hopfield network

A Hopfield network is a form of recurrent artificial neural network popularized by John Hopfield in 1982, but described earlier by Little in 1974.[1][2]

They are guaranteed to converge to a local minimum, but will sometimes converge to a false pattern (wrong local minimum) rather than the stored pattern (expected local minimum).

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however, Hopfield found that this behavior is confined to relatively small parts of the phase space and does not impair the network's ability to act as a content-addressable associative memory system.

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This value is called the 'energy' because: the definition ensures that, when units are randomly chosen to update, the value of the energy, E, will either lower or stay the same.

Furthermore, under repeated updating the network will eventually converge to a state which is a local minimum in the energy function (which is considered to be a Lyapunov function).

Convergence is generally assured, as Hopfield proved that the attractors of this nonlinear dynamical system are stable, not periodic or chaotic as in some other systems.

This allows the net to serve as a content addressable memory system, that is to say, the network will converge to a 'remembered' state if it is given only part of the state.

For example, if we train a Hopfield net with five units so that the state (1, -1, 1, -1, 1) is an energy minimum, and we give the network the state (1, -1, -1, -1, 1) it will converge to (1, -1, 1, -1, 1).

The Hebbian Theory was introduced by Donald Hebb in 1949, in order to explain 'associative learning', in which simultaneous activation of neuron cells leads to pronounced increases in synaptic strength between those cells.[5]

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{\displaystyle w_{ij}={\frac {1}{n}}\sum _{\mu =1}^{n}\epsilon _{i}^{\mu }\epsilon _{j}^{\mu }}

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{\displaystyle w_{ij}^{\nu }=w_{ij}^{\nu -1}+{\frac {1}{n}}\epsilon _{i}^{\nu }\epsilon _{j}^{\nu }-{\frac {1}{n}}\epsilon _{i}^{\nu }h_{ji}^{\nu }-{\frac {1}{n}}\epsilon _{j}^{\nu }h_{ij}^{\nu }}

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{\displaystyle \epsilon _{i}^{\rm {mix}}=\pm \operatorname {sgn}(\pm \epsilon _{i}^{\mu _{1}}\pm \epsilon _{i}^{\mu _{2}}\pm \epsilon _{i}^{\mu _{3}})}

Furthermore, it was shown that the recall accuracy between vectors and nodes was 0.138 (approximately 138 vectors can be recalled from storage for every 1000 nodes) (Hertz et al., 1991).

When the Hopfield model does not recall the right pattern, it is possible that an intrusion has taken place, since semantically related items tend to confuse the individual, and recollection of the wrong pattern occurs.

Furthermore, both types of operations are possible to store within a single memory matrix, but only if that given representation matrix is not one or the other of the operations, but rather the combination (auto-associative and hetero-associative) of the two.

It is important to note that Hopfield’s network model utilizes the same learning rule as Hebb’s (1949) learning rule, which basically tried to show that learning occurs as a result of the strengthening of the weights by when activity is occurring.

Rizzuto and Kahana (2001) were able to show that the neural network model can account for repetition on recall accuracy by incorporating a probabilistic-learning algorithm.

As a result, the weights of the network remain fixed, showing that the model is able to switch from a learning stage to a recall stage.

McCulloch and Pitts' (1943) dynamical rule, which describes the behavior of neurons, does so in a way that shows how the activations of multiple neurons map onto the activation of a new neuron’s firing rate, and how the weights of the neurons strengthen the synaptic connections between the new activated neuron (and those that activated it).

This would therefore create the Hopfield dynamical rule and with this, Hopfield was able to show that with the nonlinear activation function, the dynamical rule will always modify the values of the state vector in the direction of one of the stored patterns.

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