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There is no explicit weight decay, normalisation or clipping of weights in the model. The subtraction of the weighted sum of the output neuron values acts like anti-Hebbian learning. We will consider the network as a transformation from inputs x to outputs z; by considering the effects of these rules on individual neurons, we can show that the resultant network is equivalent to Oja’s Subspace Algorithm. 15) k where we have dropped the time (t) notation for simplicity. 14). The comparative results given in the various tables in Chapter 3 were from a negative feedback network.

5: Two outputs attempting to jointly convey as much information as possible about the inputs. 5. 21) j We will use the same assumptions as previously. 22) Now since the output neurons are both dependent on the same input vector x there will be a correlation between their outputs. Let the correlation matrix be R. 23) where rij = E(yi yj ). e. if the outputs are zero mean, the main diagonal contains the variances of each output while the off-diagonal terms contain the covariances. r12 r11 = r21 = = σ12 + σν2 σ1 σ2 ρ12 r22 = σ22 + σν2 where σi2 , i = 1, 2 is the variance of each output neuron in the absence of noise while ρ12 is the correlation coefficient of the output signals also in the absence of noise.

3 47 Sanger’s Generalized Hebbian Algorithm Sanger has developed a different algorithm (which he calls the “Generalized Hebbian Algorithm”) which also finds the actual Principal Components. 15) k=1 Note that the crucial difference between this rule and Oja’s Subspace Algorithm is that the decay term for the weights into the ith neuron is a weighted sum of the first i neurons’ activations. 16) We see that the central term comprises the residuals after the first j-1 Principal Components have been found, and therefore the rule is performing the equivalent of One Neuron learning on subsequent residual spaces.

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Artificial Neural Networks and Information Theory by Fyfe C.


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