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Now, the initial weights have been considered as w, ~ 0 (near to zero) for all i’s and /’s. Pattern recalling analysis of English alphabets using Hopfield model From the synaptic dynamics, we have:

Figure 1 Now, the initial weights have been considered as w, ~ 0 (near to zero) for all i’s and /’s. Pattern recalling analysis of English alphabets using Hopfield model From the synaptic dynamics, we have:

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Table 1 Genetic operator used in experiments The parameters used in different runs of the experiments are described in Table 2 and Table 3. Four runs of the experiments were taken on same Hopfield network architecture (i.e., 35 neuron’s network). Each run is based on one of the two experiments — recalling English alphabets with Hebbian rule and recalling the same alphabet with GA. The inputs for four different runs are 0-bit, 1-bit, 2-bits and 3-bits errors induced randomly in the pattern already stored in the network. In each experiment, the Hebbian learning rule is used to store patterns in the Hopfield neural network (using equation 9). The genetic operators used in each experiment are summarised in Table 1.
Table 1 Genetic operator used in experiments The parameters used in different runs of the experiments are described in Table 2 and Table 3. Four runs of the experiments were taken on same Hopfield network architecture (i.e., 35 neuron’s network). Each run is based on one of the two experiments — recalling English alphabets with Hebbian rule and recalling the same alphabet with GA. The inputs for four different runs are 0-bit, 1-bit, 2-bits and 3-bits errors induced randomly in the pattern already stored in the network. In each experiment, the Hebbian learning rule is used to store patterns in the Hopfield neural network (using equation 9). The genetic operators used in each experiment are summarised in Table 1.
The task associated to the Hopfield NNs in performed experiments is to store the English alphabets as patterns with the appropriate recalling of the same patterns with inducec noise. Table 2 Parameters used for Hebbian learning rule
The task associated to the Hopfield NNs in performed experiments is to store the English alphabets as patterns with the appropriate recalling of the same patterns with inducec noise. Table 2 Parameters used for Hebbian learning rule
This square matrix is known as parent weight matrix for storing the given input patterns. Hopfield suggested that the maximum limit for the storage is 0.15 N in a network with N neurons, if a small error in recalling is allowed. Later, this was theoretically calculated as p=0.14N by using replica method (Amit et. al., 1985).
This square matrix is known as parent weight matrix for storing the given input patterns. Hopfield suggested that the maximum limit for the storage is 0.15 N in a network with N neurons, if a small error in recalling is allowed. Later, this was theoretically calculated as p=0.14N by using replica method (Amit et. al., 1985).
3.4 The GA implementation In this simulation, we are not starting the GA with random solutions, as it generally starts; instead, we start with a sub-optimal solution to achieve the optimal solution. In each iteration, this population of sub-optimal solution is modified through uniform random mutations and discrete crossovers and their fitness values are evaluated. According to the fitness values, individuals of the next generation are selected, using a (u + A) — strategy in ES terminology.
3.4 The GA implementation In this simulation, we are not starting the GA with random solutions, as it generally starts; instead, we start with a sub-optimal solution to achieve the optimal solution. In each iteration, this population of sub-optimal solution is modified through uniform random mutations and discrete crossovers and their fitness values are evaluated. According to the fitness values, individuals of the next generation are selected, using a (u + A) — strategy in ES terminology.
Figure 3 Flowchart of GA implementation
Figure 3 Flowchart of GA implementation
Table 4 The results for recalling letters of English alphabet with no error Table 5, 6 and 7 represent the results for recalling the corresponding stored patterns while these are presented with induced noise. In these cases, noise was created by reverting 1-bit, 2-bits and 3-bits in the presented prototype input patterns of the already stored patterns. These positions of the bit(s) to be reverted to create noise are taken randomly.
Table 4 The results for recalling letters of English alphabet with no error Table 5, 6 and 7 represent the results for recalling the corresponding stored patterns while these are presented with induced noise. In these cases, noise was created by reverting 1-bit, 2-bits and 3-bits in the presented prototype input patterns of the already stored patterns. These positions of the bit(s) to be reverted to create noise are taken randomly.
Table 7 The results for recalling letters of English alphabet with 3-bit error It is pa has been claimed by Amit et al. (1985) that the capacity of deterministic Hopfield model with Hebbian rule is about 0.15 N for the noisy prototype input patterns, where N he number o f nodes in the network. If such a network is overloaded wit h a number of terns exceeding its capacity, its performance rapidly deteriorates toward zero. Here, we are storing t suggests that on inducing 5-bit error in presented prototype input pattern e to recall the stored patterns. It implies that the network capacity has increased up to 0.75 N. Thus, during the recal suitable optima number of attractions. Hence, the Hebbian rule, which has been used pattern information is not the optimal weight matrix for finding the global minima of the problem due to the limited capacity of the Hopfield model. Thus, capacity has been increased with GA by exploring the optimal weight matrices for the encoded patterns. ab he numbers of attractions are existing here and success ing process. It is quit obvious to understand that the GA ha weight matrices, which are responsible to generate suf: he 26 alphabets in a network of 35 nodes and the performance of the GA he network is fully explored s searched the ficiently large o encode the
Table 7 The results for recalling letters of English alphabet with 3-bit error It is pa has been claimed by Amit et al. (1985) that the capacity of deterministic Hopfield model with Hebbian rule is about 0.15 N for the noisy prototype input patterns, where N he number o f nodes in the network. If such a network is overloaded wit h a number of terns exceeding its capacity, its performance rapidly deteriorates toward zero. Here, we are storing t suggests that on inducing 5-bit error in presented prototype input pattern e to recall the stored patterns. It implies that the network capacity has increased up to 0.75 N. Thus, during the recal suitable optima number of attractions. Hence, the Hebbian rule, which has been used pattern information is not the optimal weight matrix for finding the global minima of the problem due to the limited capacity of the Hopfield model. Thus, capacity has been increased with GA by exploring the optimal weight matrices for the encoded patterns. ab he numbers of attractions are existing here and success ing process. It is quit obvious to understand that the GA ha weight matrices, which are responsible to generate suf: he 26 alphabets in a network of 35 nodes and the performance of the GA he network is fully explored s searched the ficiently large o encode the
The simulation program, which was developed in MATLAB 7.0, to test the two algorithms for the problem of recalling English alphabets (capital only), stores the patterns in feedback Hopfield neural network of 35 neurons. Due to limitation of resources, we could execute only 4-runs (i.e., for no error, 1-bit error, 2-bit error and 3-bit error) and could not go beyond that. It is to note that during GA implementation, the success is considered only if the recalling is done within 20 iterations (i.e., mutation, elitism, first fitness evaluation function, crossover and second fitness evaluation function).
The simulation program, which was developed in MATLAB 7.0, to test the two algorithms for the problem of recalling English alphabets (capital only), stores the patterns in feedback Hopfield neural network of 35 neurons. Due to limitation of resources, we could execute only 4-runs (i.e., for no error, 1-bit error, 2-bit error and 3-bit error) and could not go beyond that. It is to note that during GA implementation, the success is considered only if the recalling is done within 20 iterations (i.e., mutation, elitism, first fitness evaluation function, crossover and second fitness evaluation function).
Figure 5 Comparison chart for Table 4 (see online version for colours)
Figure 5 Comparison chart for Table 4 (see online version for colours)
The simulation resu Hebbian rule for recalling the letters of English alp ts (i.e., Tables 4-7) indicate tha’ GA has more success rate than the habet, which are containing 0, 1, 2, and 3 bit errors from stored patterns in Hopfield neural network. Sometimes it has also been observed that t he performance of GA was less t han what was expected to be. One of the reasons for this deviation may be the position(s) of bits reverted to induce noise in the recalling pattern. Another interesting observation is t some of the letters t hat there is confusion while recalling hrough both, Hebbian and GA. For example, on inducing 2-bit error in letter D at positions (1, 5), (1, 2) and (31, 35), the recalling was confused with letter O.
The simulation resu Hebbian rule for recalling the letters of English alp ts (i.e., Tables 4-7) indicate tha’ GA has more success rate than the habet, which are containing 0, 1, 2, and 3 bit errors from stored patterns in Hopfield neural network. Sometimes it has also been observed that t he performance of GA was less t han what was expected to be. One of the reasons for this deviation may be the position(s) of bits reverted to induce noise in the recalling pattern. Another interesting observation is t some of the letters t hat there is confusion while recalling hrough both, Hebbian and GA. For example, on inducing 2-bit error in letter D at positions (1, 5), (1, 2) and (31, 35), the recalling was confused with letter O.
Figure 7 Comparison chart for Table 6 (see online version for colours)
Figure 7 Comparison chart for Table 6 (see online version for colours)
6 Conclusions
6 Conclusions
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