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2. if w7x = 0 is perpendicular to the axis, then the for will also be a constant, k2, along these paths;
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3. for other weights, the influence of the position in feature space will be unbounded along these paths.
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Bounds on the values of the constants kl and can be readily determined by considering the modulus of the expression in (8.4): exp(w7z)
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Hence the magnitude of the influence of the position in feature space is bounded, in cases 1 and 2 above, by lx I This means that the y!~function for the bias term is 4 bounded by kl 5 1/4 the bias input is always 1, while for case 2, the Icz constant depends on the value of zpr which is unbounded. Hence we conclude that4
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for some constants k l and kz which depend on the path e. For the single perceptron = 1 while for MLP models it will take other values (see below). We see that the IC for a simple perceptron displays a variety of behaviors and, except in the case = 1, the single perceptron is not robust as the IC is unbounded. 8.5.2
Adding a hidden layer
We now look a t an MLP with a hidden layer as discussed in Section 2.1 (p. 9). Without loss of generality, we still restrict the discussion to MLPs with one output. We can readily extend these results to an MLP with more output units by simply
&J considering each output unit separately. Now considering - (equation (2.6)),
we can see that the last term, - p ,is the df(Yh)z
function for a single perceptron,
and that this term is multiplied by a number of terms, all of which are bounded by 1 except for the W p , h + l term. Hence the asymptotic results of the previous section (equations (8.5) and (8.6)) all apply to the R matrix of weights for an MLP when we choose the constant
lPll =
These results (equations 8.5 and 8.6) are shown diagrammatically in Figure 8.4. We need now only consider the behavior of the weights in the matrix. MLP of size 1.1.1 We consider the simplest case, an MLP of size 1.1.1, used t o separate two classes. The I" matrix consists of two weights, and 01, and the R matrix also consists of two weights, and w1. The fitted model is then mlp(z) = 1+exp 1 v1 exP{-(wo
+ w12))
If we then take the limits of the product w l z (so that we do not have to worry about the sign of q ) , see that we
mlp(z) =
+ exp(
- 01)
and lim
mlp(z) =
+ exp(-vo)
The first thing that we note is that, unlike the single perceptron, the fitted values may approach any target values in the range ( 0 , l ) by varying the choice of and u l . Hence, for the multi-layer perceptron, i t does not appear unreasonable to have target values other than 0 and 1, and the common practice of using target values of 0.1 and 0.9 appears to have some vindication. In 9 (p. 143) we show that using target values of { O . l , 0.9) may give a smoother separating boundary; however, it also has the undesirable effect that the outputs at the qth output unit may no = klz) (see Section 4.2, p. 35). longer be simply interpreted as an estimate of More importantly for our discussion here, unless the values of the limits exactly
decision hyperplane
rnfluence curves
normal to ..>. decision hyperplane
Figure 8.4 The shape the ICs for the R weights for an MLP with a hidden layer. The IC, given as dotted lines, should be interpreted as a surface sitting over the page. In a direction not parallel to the decision hyperplane, the IC redescends to 0. In a direction parallel to the decision hyperplane, it is unbounded. There are two exceptions to this. One is the IC for the bias term, which is constant along paths parallel to the decision boundary. The ot,her is the IC,,,, when the decision boundary is perpendicular to the axis. In this case also the IC takes a constant value along paths parallel to the decision boundary.
equal the target values, the of will approach a constant in the range l), rather t h a n going to 0. T h e sigmoid function output by a single perceptron (Figure 8.3) is, after a n ap= -f(z). However, propriate affine transformation, an odd function; t h a t is, the output of an MLP (with a hidden layer) does not have a comparable symmetry property. This can be easily seen from the fact t h a t its limits as z 4 *rn may take any value, shown in (8.8) and (8.9). This means t h a t the of (the - mlp(z) term), will likewise not have this symmetry, and the $ function for class 0 need not be symmetric with the function for class 1. For the second term of (2.5),
the of we consider the asymptotic behavior. Taking limits and once again considering the variable we see that: