Neural Networks - Algorithms, Applications,and ... - Csbdu.in

I 1.2From Neurons to ANS 27Figure 1.16This figure shows the x\,x 2 plane with the four points,(0,0), (1,0), (0, 1), and (1. 1), which make up the four inputvectors for the XOR problem. The line 9 — w\x\ + 1112x2divides the plane into two regions but cannot successfullyisolate the set of points (0,0) and (1, 1) from the points (0.1)and (1,0).This equation is the equation of a line in the x\,xi plane. That plane is illustratedin Figure 1.16, along with the four points that are the possible inputs to thenetwork. We can think of the problem as one of subdividing this space into regionsand then attaching labels to the regions that correspond to the right answerfor points in that region. We plot Eq. (1.5) for some values of 0, w\, and w 2 , asin Figure 1.16. The line can separate the plane into at most two distinct regions.We can then classify points in one region as belonging to the class having anoutput of 1, and those in the other region as belonging to the class having anoutput of 0; however, there is no way to arrange the position of the line so thatthe correct two points for each class both lie in the same region. (Try it.) Thesimple linear threshold unit cannot correctly perform the XOR function.Exercise 1.5: A linear node is one whose output is equal to its activation. Showthat a network such as the one in Figure 1.15, but with a linear output node,also is incapable of solving the XOR problem.Before showing a way to overcome this difficulty, we digress for a momentto introduce the concept of hyperplanes. This idea shows up occasionally in theliterature and can be useful in the evaluation of the performance of certain neuralnetworks. We have already used the concept to analyze the XOR problem.