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It is clear that in ill-posed problems, such as the given example, the need for a penalization term is paramount. Fixed-slope training appears to work well a penalization method. While its implementation is more complex than that of weight decay, for example, it has an intuitive geometric interpretation in terms of the slopes of the activation functions, and of the fitted surface. Fixing the slope of the activation function makes the MLP model more resistant to the influence of isolated points.
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Figure We consider the data set from Example 2 in Section 9.3.2, p. 146. As in the decision 9, a single point has been moved and the resulting effects noted boundaries, which are calculated for the linear discriminant function (LDF), a standard MLP niodel and a fixed-slope MLP. An inspection of the figure shows that neither t h e LDF nor t.he standard MLP are resistant t o an aberrant d a t a point, although their non-resistant a point in class "+" by virtue of how behavior is quite different. T h e LDF is influenced far that. point is (in Mahalanobis distance) from the mean of class whereas the standard MLP has a non linear influence funct,ion t h a t has a pronounced ridge along the lines of linear The fixed slope MLP is separability. See particularly the difference between plots 4 and resistant to the moving point.
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A comparison of several classification techniques on the simulated scene. For each classifier the error rate on the image is given.
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II classifier 11 Linear discriminant functions
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Quadratic discriminant functions Nearest neighbor MLP (of size 4.2.3), a cross-entropy penalty function, softmax outputs MLP, as above, with weight decay X = 0.1
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error rate 0.2536 0.2552 0.2320 0.2600 0.2216 0.2776 0.2208 0.2192 0.2224
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MLP, as above, with fixed slope, unconstrained MLP, as above, with fixed slope y = 6 MLP, as above, with fixed slope y = 5 MLP. above. with fixed slope = 4 ILILP, as above, with fixed slope y = 3
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Adby, P. R. and Dempster, M. A. H. (1974). Chapman and Hall, London. Ahrned, S. and Tesauro, G. (1988). Scaling and generalization in neural networks: a case study. In Touretzky kt al. (1988), pp. 3-9. Aires, F., Prigent, C., and Rossow, W. B. (2004). Neural network uncertainty assessment using bayesian statistics: A remote sensing application. 16(11):2415-2458. Aitchison, J. and Dunsmore, I. R. (1975). bridge University Press, Cambridge. Almeida, L. and Wellekens, C., editors (1990). 4 12, Berlin. Cam-
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Alpaydin, E. (1991). Gal: Networks that grow when they learn and shrink when they forget. Preprint, International Computer Science Institute, Berkeley, CA. Aniari, S., Murata, N., Muller, K.-R., Finke, M., and Yang, H. (1997). Asymptotic statistical theory of overtraining and cross-validation. on 8(5). Previously, University of Tokyo Techreport: METR 9506, 1995. Anderson, J. A . and Rosenfeld, E., editors (1988). M I T Press, Cambridge, MA.
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A Statistical Approach to Neural Networks for Pattern Recognition by Robert A . Dunne Copyright @ 2007 John Wiley & Sons, Inc.
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Anderson, T . W. (1984). ley, New York, second edition.
Anderson, T. W. and Bahadur, R. R. (1962). Classification into two multivariate of normal distributions with different covariance matrices. 33:420-431. Apostol, T . (1967).
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Baldi, P. (1991). Computing with arrays of bell-shaped and sigmoid functions. In Lippmann et al. pp. 735-742. Baldi, P. and Hornik, K. (1995). Learning in linear neural networks: A survey. on 6(4):837-858. Barnard, E. and Cole, R. A. (1989). A neural-net training program based on conjugate-gradient optimization. Technical Report CSE 89-014, Oregon Graduate Institute of Science and Technology. Rarron, A. R. (1984). Predicted squared error: A criterion for automatic model selection. In Farlow, S. J., editor, Marcel Decker, New York. Bartlett, P. L. (1998). The sample complexity of pattern classification with neural networks: The size of the weights is more important than the size of the network. on 44(2):525- 536. Barton, S. A. (1991). A matrix method for optimizing a neural network. 3:450-459. Battiti, R. (1992). First- and second-xder methods for learning: between steepest descent and Newton s method. 4:141-166. Baum, E. and Haussler, D. (1989). What size net gives valid generalization 1:151-160. Baum, E. B. (1990). When are k-nearest neighbor and back propagation accurate for feasible sized sets of examples In Almeida and Wellekens (1990), pp. 2-25. Becker, R. A., Chambers, J. M., and Wilks, A. R. (1988). Wadsworth and Brooks, Monteray, CA.
Becker, S . and le Cun, Y. (1989). Improving the convergence of back-propagation learning with second order methods. In Touretzky et al. (1988), pp. 29-37. Benediktsson, J. A,, Sveinsson, J. feature extraction of AVIRIS data. 33(5):1194-1205. and Arnason, K. (1995). Classification and on
Besag, J . (1974). Spatial interaction and the statistical analysis of lattice systems (with discussion). of 36(2):192-236. Besag, J. (1986). On the statistical analysis of dirty pictures (with discussion). of 48(3):259-302.