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9) dw It is clear that the weights will be stable (when dtij = 0) at the points where wij = αE(yi xj ). Using a similar type of argument to that employed for simple Hebbian learning, we can show that at convergence we must have αCw = w. Thus w would have to be an eigenvector of the correlation matrix of the input data with corresponding eigenvalue α1 . We shall be interested in a somewhat more general result. g. 11) where the decay term is gated by the output term yi . These, while still falling some way short of the decay in which we will be interested, show that researchers of even 15 years ago were beginning to think of both differentially weighted decay terms and allowing the rate of decay to depend on the statistics of the data presented to the network.

Thus PCA provides a means of compressing the data whilst retaining as much information within the data as possible. 6: here we show the points of a two dimensional distribution on the plane; we therefore require two coordinates to describe each point exactly but if we are only allowed a single coordinate, our best bet is to choose to use the coordinate axis labelled “first principal component”. This axis will allow us to represent each point as accurately as possible with a single coordinate. It is the best possible linear compression of the information in the data.

3 and has anti-Hebbian connections between the output neurons. e. 6) 56 CHAPTER 4. ANTI-HEBBIAN LEARNING where ρ is the correlation coefficient. (T x)T ) = T Cxx T T . 7) The anti-Hebb rule reaches equilibrium when the the units are decorrelated and so the terms w12 = w21 = 0. Notice that this gives us a quadratic equation in w (which naturally we can solve). Let us consider the special case that the elements of x have the same variance so that σ1 = σ2 = σ. 10) −1 + ρ F¨oldi´ak further shows that this is a stable point in the weight space.