By Fyfe C.
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Extra resources for Artificial Neural Networks and Information Theory
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.
Artificial Neural Networks and Information Theory by Fyfe C.