Cisco - Introduction to xDSL Technology 203

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BACK-PROPAGATION the characteristics. The input of the system is given by the two-dimensional vector x and the output is given by the one-dimensional vector d . 3 (top left). 3: Example of function approximation with a feedforward network. Top left: The original learning samples Top right: The approximation with the network Bottom left: The function which generated the learning samples Bottom right: The error in the approximation. programmed with two inputs, 10 hidden units with sigmoid activation function and an output unit with a linear activation function.

Stable states which do not correspond with stored patterns). The rst of these two problems can be solved by an algorithm proposed by Bruce et al. 7) k= 1 otherwise. Now modify wjk by wjk = yj yk ( j + k ) if j 6= k. Repeat this procedure until all patterns are stable. It appears that, in practice, this algorithm usually converges. There exist cases, however, where the algorithm remains oscillatory (try to nd one)! The second problem stated above can be alleviated by applying the Hebb rule in reverse to the spurious stable state, but with a low learning factor (Hop eld, Feinstein, & Palmer, 1983).

32) When eq. 31) holds for two vectors ui and ui+1 they are said to be conjugate. Now, starting at some point p0 , the rst minimisation direction u0 is taken equal to g0 = ;rf (p0 ), resulting in a new point p1. 33) k pk gTi gi Next, calculate pi+2 = pi+1 + i+1 ui+1 where i+1 is chosen so as to minimise f (pi+2 )3 . , see (Stoer i= & 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).

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