By Nilsson N.J.

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14: A Piecewise Linear Machine The PWL machine groups its weighted summing units into R banks corresponding to the R categories. An input vector X is assigned to that category corresponding to the bank with the largest weighted sum. We can Introduction to Machine Learning c 1996 Nils J. Nilsson. All rights reserved. 3. NETWORKS OF TLUS 57 use an error-correction training algorithm similar to that used for a linear machine. If a pattern is classi ed incorrectly, we subtract (a constant times) the pattern vector from the weight vector producing the largest dot product (it was incorrectly the largest) and add (a constant times) the pattern vector to that weight vector in the correct bank of weight vectors whose dot product is locally largest in that bank.

The weights in the rst layer (except for the \threshold weights") can all have values of 1, ;1, or 0. Later, we will Introduction to Machine Learning c 1996 Nils J. Nilsson. All rights reserved. CHAPTER 4. 13: Three Planes Implemented by the Hidden Units Discuss half-space intersections, half-space unions, NP-hardness of optimal versions, single-sideerrorhypeplane methods, relation to \AQ" methods. Add diagrams showing the non-linear transformation performed by a layered network. present a training procedure for this rst layer of weights.

The example is a 2-clause expression in 3-CNF. If f is written in DNF, an Introduction to Machine Learning c 1996 Nils J. Nilsson. All rights reserved. 2. CLASSES OF BOOLEAN FUNCTIONS 25 application of De Morgan's law renders f in CNF, and vice versa. Because CNF and DNF are duals, there are also 2O(nk) functions in k-CNF. 4 Decision Lists Rivest has proposed a class of Boolean functions called decision lists Rivest, 1987]. A decision list is written as an ordered list of pairs: (tq vq ) (tq;1 vq;1 ) (ti vi ) (t2 v2) (T v1 ) where the vi are either 0 or 1, the ti are terms in (x1 : : : xn), and T is a term whose value is 1 (regardless of the values of the xi).