By Kroese B., van der Smagt P.

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I = gTi+1 gi+1 gTi gi gk = −∇f |pk with for all k ≥ 0. 33) Next, calculate pi+2 = pi+1 + λi+1 ui+1 where λi+1 is chosen so as to minimise f (p i+2 )3 . , see (Stoer & Bulirsch, 1980)). The process described above is known as the Fletcher-Reeves method, but there are many variants which work more or less the same (Hestenes & Stiefel, 1952; Polak, 1971; Powell, 1977). 6). Powell introduced some improvements to correct for behaviour in non-quadratic systems. The resulting cost is O(n) which is significantly better than the linear convergence 4 of steepest descent.

2). 4), the weight update must be changed to implement a shift towards the input: wk (t + 1) = wk (t) + γ(x (t) − wk (t)). 6) Again only the weights of the winner are updated. A point of attention in these recursive clustering techniques is the initialisation. Especially if the input vectors are drawn from a large or high-dimensional input space, it is not beyond imagination that a randomly initialised weight vector w o will never be chosen as the winner and will thus never be moved and never be used.

For a given transformation y = d(x ), we can divide the set of all possible input vectors into two classes: X + = { x | d(x ) = 1 } and X − = { x | d(x ) = −1 }. 19) Since there are N input units, the total number of possible input vectors x is 2 N . 7. CONCLUSIONS 31 is equal to 1 for x p = wh only. Similarly, the weights to the output neuron can be chosen such that the output is one as soon as one of the M predicate neurons is one: M yop = sgn h=1 yh + M − 1 2 . 21) This perceptron will give y o = 1 only if x ∈ X + : it performs the desired mapping.