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The term pruning is taken from its use in classification trees (Breiman et 1984, 510) and is used here to indicate that we are removing unwanted structure in order to leave a more parsimonious model. See Section 5.3 (p. 57) for a brief discussion of the literature on pruning MLP models. A problem with both the single-layer and the task-based approaches is that the resulting hidden layer may be of large size. For the task-based MLP, the hidden
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layer grows at O ( Q z ) where Q is the number of classes. The nature of this layer suggests some pruning ideas which are illustrated by way of the following simple example. Consider four classes, each a t the corner of a square in a twwdimensional feature space (Figure 6.2). This will need six discriminant functions for a full pairwise separation, and these are also illustrated. Note that while each unit has been selected to perform one task of separating a particular pair of classes, it may also separate other pairs of classes. A subset of units, that performs essentially the the full same tasks as the full set, should give as good a separation of the data set of units.
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Figure 6.2 Four classes in a two-dimensional feature space and a represent.at,ion of t,he six linear discriminant functions to pairwiseseparate them. T h e discriminant, functions have been given an arbitrary numbering which is used to refer to them in the following argument,.
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Using the hidden-layer output, we can calculate a diagnostic matrix of size Qx where T q h is the number of results > 0.5 so that w 7 z > 0) for the qth class given by the hidden-layer unit. An inspection of this matrix, given in Table 6.1, indicates which classes are being separated by the hyperplanes in feature space. An analysis of this matrix can deduce which tasks are being performed by which unit or discriminant function. This information is summarized in Table 6.2, which gives the task performed by each unit, an arbitrary numbering of that task, and a list of other tasks performed by that same unit. It is clear that many of the hidden-layer units duplicate tasks assigned to other units. As the matrix (and the matrix which is a compressed version of describe the separation achieved by the hyperplanes, an analysis of these
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PRUNING ALGORITHMS
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matrices should indicate which units can be removed. We now consider three different techniques that could be applied to this task:
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1. the
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2. hierarchical cluster analysis; and
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Table The matrix for the example illustrated in Figure 6.2. T k 2 is the results for the k t h class given by the j t h hidden-layer unit. The number of number of the discriminant function is shown at the top of the table and the class in t,he first column; is the number in class q. From this matrix, a simple analysis can deduce which tasks are being performed by which unit or discriminant function. This illformation is summarized in Table 6.2, which gives the task performed by each unit.
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A listing of the discriminant functions, with the same numbering as i n Figwe 6.2, together with (column 2) the classes which the discriminant function is designed t o separate, and (column 3) t.he numbers of any discriminant funct.ions it. performs the same task as.
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Table 6.2
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discriminant tion number
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which was designed to separate classes
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The first technique is to calculate the QR decomposition (Noble and Daniel, 1988, 55.9) of the matrix The absolute value of the diagonal elements of the matrix give the distance of the column of from the subspace spanned by the 1 , . . . , 1 columns (see Problem 6 . 2 , p. 90). These distances can be normalized by dividing by the norm of the hth column of f ( X 0 7 ) and forming the vector