An Introduction to Neural Networks by Kroese B., van der Smagt P.

By Kroese B., van der Smagt P.

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By Kroese B., van der Smagt P.

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However, the pattern and weight vectors are not normalised, and in this case w2 should be considered the ‘winner’ when x is applied. However, the dot product xT w1 is still larger than xT w2 . Winner selection: Euclidean distance Previously it was assumed that both inputs x and weight vectors w were normalised. 1) gives a ‘biological plausible’ solution. 3 it is shown how the algorithm would fail if unnormalised vectors were to be used. Naturally one would like to accommodate the algorithm for unnormalised input data.

The resulting cost is O(n) which is significantly better than the linear convergence 4 of steepest descent. 2 A matrix A is called positive definite if ∀y = 0, yT Ay > 0. ) However, line minimisation methods exist with super-linear convergence (see footnote 4). 4 A method is said to converge linearly if E i+1 = cE i with c < 1. , E i+1 = c(E i )m with m > 1 are called super-linear. 42 CHAPTER 4. 6: Slow decrease with conjugate gradient in non-quadratic systems. The hills on the left are very steep, resulting in a large search vector ui .

B. , but with different lengths. , vectors x and w1 are nearest to each other, and their dot product xT w1 = |x||w1 | cos α is larger than the dot product of x and w2 . , however, the pattern and weight vectors are not normalised, and in this case w2 should be considered the ‘winner’ when x is applied. However, the dot product xT w1 is still larger than xT w2 . Winner selection: Euclidean distance Previously it was assumed that both inputs x and weight vectors w were normalised. 1) gives a ‘biological plausible’ solution.

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