By Kroese B., van der Smagt P.

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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 . When the quadratic portion is entered the new search direction is constructed from the previous direction and the gradient, resulting in a spiraling minimisation.

As a result, the crystal lattice will be highly ordered, without any impurities, such that the system is in a state of very low energy. 11) where T is a parameter comparable with the (synthetic) temperature of the system. This stochastic activation function is not to be confused with neurons having a sigmoid deterministic activation function. 12) where Pα is the probability of being in the α th global state, and Eα is the energy of that state. Note that at thermal equilibrium the units still change state, but the probability of finding the network in any global state remains constant.

6 Multi-layer perceptrons can do everything In the previous section we showed that by adding an extra hidden unit, the XOR problem can be solved. For binary units, one can prove that this architecture is able to perform any transformation given the correct connections and weights. The most primitive is the next one. 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 .

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

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