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2, 10, and 11; it is used for applications to scheduling problems (5) and the knapsack problem (12). Other Related Networks High-order Boltzmann machines (HOBM) allow for terms of higher order in the consensus function than those for the standard Boltzmann machine (in which the consensus function has only first- and second-order terms). The theoretical results, such as uniqueness of the learned solution, which have been established for these HOBM do not hold for the Boltzmann machine with hidden units.

N, j n (xn ) 1 Nn j n =1 . . 2 N2 j 2 =1 N1 j 1 =1 n i=1 µi, j i (xi ) Lemma 1 The multidimensional receptive field functions selected in Eq. (5) satisfy three significant properties: a. Positivity, Rj1, j2, . , jn(x) Ͼ 0 for all x ʦ (x1, j1Ϫ1, x1, j1ϩ1) ϫ . . ϫ (xn, jnϪ1,, xn, jnϩ1) 1 xi, 2 xi, 3 xi, 4 gk (x) = Nn j n =1 ... , j n (x) : Rn → R (7) (8) In fact, according to the normalization property of Lemma 1(c), Eq. (8) is a convex combination of the weights w. (5) It is easy to see that the receptive fields so defined are normalized n-dimensional second-order splines.

The mapping generated, however, is dependent on the actual implementation of the CMAC. Let Ap be the number of association cells physically implemented by CMAC and A* be the number of maximally active elements of A for any given pattern. In practice, Ap is chosen to be at least 100 times A*. Then it can be shown that a x2 (4, 7) (5, 7) (6, 7) (7, 7) (4, 6) (5, 6) (6, 6) (7, 6) (4, 5) (5, 5) (6, 5) (7, 5) 2r 7 (x2) 2r 6 (x2) 2r 5 (x2) 2r 4 (x2) (4, 4) (5, 4) (6, 4) (7, 4) x1 1r 4 (x1) 1r 5 (x1) 1r 6 (x1) 1r 7 (x1) Figure 3.

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39.Neural Networks by John G. Webster (Editor)

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