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Recent research concentrated on the development of more efficient pruning techniques to solve the architecture selection problem. Several different approaches to pruning have been developed. This chapter groups these approaches in the following general classes: intuitive methods, evolutionary methods, information matrix methods, hypothesis testing methods and sensitivity analysis methods. Intuitive pruning techniques: Simple intuitive methods based on weight values and unit activation values have been proposed by Hagiwara [Hagiwara 1993]. The goodness factor Gi of unit i in layer l, Gi = Y^pY^j(wji i)2^ where the first sum is over all patterns, and oi is the output of the unit, assumes that an important unit is one which excites frequently and has large weights to other units. The consuming energy, Ei = Y^pY^jwji lj^l ji additionally assumes that unit i excites the units in the next layer. Both methods suffer from the flaw that when an unit's output is more frequently 0 than 1, that unit might be considered as being unimportant, while this is not necessarily the case. Magnitude- based pruning assumes that small weights are irrelevant [Hagiwara 1993, Lim and Ho 1994]. However, small weights may be of importance, especially compared to very large weights which cause saturation in hidden and output units. Also, large weights (in terms of their absolute value) may cancel each other out. Evolutionary pruning techniques: The use of genetic algorithms (GA) to prune NNs provides a biologically plausible approach to pruning [Kuscu and Thornton 1994, Reed 1993, Whitley and Bogart 1990, White and Ligomenides 1993]. Using GA terminology, the population consists of several pruned versions of the original network, each needed to be trained. Differently pruned networks are created by the application of mutation, reproduction and cross-over operators. These pruned networks "compete" for survival, being awarded for using fewer parameters and for improving generalization. GA NN pruning is thus a time-consuming process. Information matrix pruning techniques: Several researchers have used approximations to the Fisher information matrix to determine the optimal number of hidden units and weights. Based on the assumption that outputs are linearly activated, and that least squares estimators satisfy asymptotic normality, Cottrell et al. compute the relevance of a weight as a function of the information matrix, approximated by [Cottrell et al. 1994]
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P=\ Weights with a low relevance are removed. Hayashi [Hayashi 1993], Tamura et al. [Tamura et al. 1993], Xue et al. [Xue et al 1990] and Fletcher et al. [Fletcher et al. 1998] use Singular Value
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Decomposition (SVD) to analyze the hidden unit activation covariance matrix to determine the optimal number of hidden units. Based on the assumption that outputs are linearly activated, the rank of the covariance matrix is the optimal number of hidden units (also see [Fujita 1992]). SVD of this information matrix results in an eigenvalue and eigenvector decomposition where low eigenvalues correspond to irrelevant hidden units. The rank is the number of non-zero eigenvalues. Fletcher, Katkovnik, Steffens and Engelbrecht use the SVD of the conditional Fisher information matrix, as given in equation (7.22), together with likelihood-ratio tests to determine irrelevant hidden units [Fletcher et al. 1998]. In this case the conditional Fisher information matrix is restricted to weights between the hidden and output layer only, whereas previous techniques are based on all network weights. Each iteration of the pruning algorithm identifies exactly which hidden units to prune. Principal Component Analysis (PCA) pruning techniques have been developed that use the SVD of the Fisher information matrix to find the principal components (relevant parameters) [Levin et al. 1994, Kamimura 1993, Schittenkopf et al. 1997, Takahashi 1993]. These principal components are linear transformations of the original parameters, computed from the eigenvectors obtained from a SVD of the information matrix. The result of PCA is the orthogonal vectors on which variance in the data is maximally projected. Nonprincipal components/parameters (parameters that do not account for data variance) are pruned. Pruning using PCA is thus achieved through projection of the original w-dimensional space onto a w -dimensional linear subspace (w < w) spanned by the eigenvectors of the data's correlation or covariance matrix corresponding to the largest eigenvalues. Hypothesis testing techniques: Formal statistical hypothesis tests can be used to test the statistical significance of a subset of weights, or a subset of hidden units. Steppe, Bauer and Rogers [Steppe et al. 1996] and Fletcher, Katkovnik, Steffens and Engelbrecht [Fletcher et al. 1998] use the likelihoodratio test statistic to test the null hypothesis that a subset of weights is zero. Weights associated with a hidden unit are tested to see if they are statistically different from zero. If these weights are not statistically different from zero, the corresponding hidden unit is pruned. Belue and Bauer propose a method that injects a noisy input parameter into the NN model, and then use statistical tests to decide if the significances of the original NN parameters are higher than that of the injected noisy parameter [Belue and Bauer 1995]. Parameters with lower significances than the noisy parameter are pruned. Similarly, Prechelt [Prechelt 1995] and Finnoff et al. [Finnoff et al. 1993] test the assumption that a weight becomes zero during the training process. This approach is based on the observation that the distribution of weight values is roughly normal. Weights located in the left tail of this distribution are
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