| ### Chapter 4 - MLP Representational Capabilities
Neural Smithing: Supervised Learning in Feedforward Artificial Neural Networks
Russell D. Reed and Robert J. Marks II
Copyright © 1999 Massachusetts Institute of Technology

## 4.1 Representational Capability

The representational capability of a network can be defined as the range of mappings it can implement when the weights are varied. A particular mapping is within the representational capability of a net if there is a set of weights for which the net performs the mapping. Theories about representational capability answer questions like: Given a set of inputs and associated target outputs, is there a set of weights that allow the network to perform the desired mapping? Is the network capable of generating the desired outputs from the given inputs? In some cases, an exact solution may be required while in others, an approximation may be allowed. Results say whether a particular problem might be solvable by a particular network without necessarily producing the weights that generate the solution. If the answer is negative, we know the network cannot solve the problem and this saves us the expense of a futile search. If the answer is positive, we know a solution exists but these results don't guarantee that it will be easy to find or that a chosen optimization method will be able to find it. Results of this kind provide broad guidelines in selecting an architecture for a problem. One well-known representational result, for example, is that single-layer networks are capable of representing only linearly separable functions.

Figure 4.1: MLP structure. A multilayer perceptron is a cascade of single-layer perceptrons. There is a layer of input nodes and a layer of output nodes with one or more hidden layers in between.

Two Hidden Layers Are Sufficient When designing a layered network, an obvious first question is how many layers to use. Lippmann [247] shows that two hidden layers are sufficient to create classification regions of any desired shape. In figure 4.2, linear threshold units in the first hidden layer divide the input space into half-spaces with hyperplanes, units in the second hidden layer AND (form intersections of) these half-spaces to produce convex regions, and the output units OR (form unions of) the convex regions into arbitrary, possibly unconnected, shapes. Given a sufficient number of units, a network can be formed that divides the input space in any way desired, producing 0 when the input is in one region and 1 when it is in the other. The boundaries are piecewise linear, but any smooth boundary can be approximated with enough units.

Figure 4.2: Three layers of linear threshold units are sufficient to define arbitrary regions. Units in the first hidden layer divide the input space with hyperplanes, units in the second hidden layer can form convex regions bounded by these hyperplanes, and output units can combine the regions defined by the second layer into arbitrarily shaped, possibly unconnected, regions. Here, the boundaries are piecewise linear, but any smooth boundary can be approximated with enough units (based on [247].)

To approximate continuous functions, one can add and subtract (rather than logically OR) convex regions with appropriate weighting factors so two hidden layers are also sufficient to approximate any desired bounded continuous function [234].

One-hidden-layer Nets Can Represent Nonconvex, Disjoint Regions Units in the second hidden layer of figure 4.2 respond to convex regions in the input space. Because these would be the outputs in a single-hidden-layer network, it is sometimes mistakenly assumed that single-hidden-layer networks can recognize only convex decision regions.

Figure 4.3: Single-hidden-layer networks can recognize nonconvex decision regions. Here, eight hidden units in a single layer create a nonconvex "square donut." Each dashed line is the decision boundary of one of the hidden units. The hidden unit is active (1) on the side of the line indicated by the pointer and connects to the output unit with the weight indicated next to the pointer. The other numbers show the summed input to the output unit in different regions. Thresholding at 8.5 creates the square donut.
Figure 4.4: Weiland and Leighton [407] provide this example of a nonconvex region recognized by a single-hidden-layer network. Thresholding at 8.7 creates the 'C' shaped region: (a) a nonconvex decision region, and (b) the network. There are 10 hidden nodes. Each has one connection with weight w = ± 1 from either the x or y input. Numbers inside the circles are the node thresholds.
Figure 4.5: Disjoint regions recognized by single-hidden-layer networks. In addition to nonconvex regions, single-hidden-layer networks can also recognize disjoint (unconnected) regions.

Counter-examples are shown in figures 4.3 through 4.5 Figures 4.3and 4.4 illustrate nonconvex regions recognized by three-layer nets. (Figure 4.4 is derived from Weiland and Leighton [407].) Figure 4.5 shows examples of disjoint (unconnected) regions recognizable by three-layer networks. Other examples can be found in [180], [255], [249].

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