C++ Neural Networks and Fuzzy Logic: Preface

Figure 1.6

  Corrupted “minus” pattern.

We will call the corresponding bipolar vector A = (1, −1, −1, 1, 1, 1, −1, −1, −1). You get the activation

vector (−12, −2, −8, 4, 4, 4, −8, −2, −8) giving the output vector, C− = (−1, −1, −1, 1, 1, 1, −1, −1, −1). In

other words, the character −, corrupted slightly, is recalled as the character − by the Hopfield network. The

intended pattern is recognized.

We now input a bipolar vector that is different from the vectors corresponding to the exemplars, and see

whether the network can store the corresponding pattern. The vector we choose is B = (1, −1, 1, −1, −1, −1, 1,

−1, 1). The corresponding neuron activations are given by the vector (12, −2, 12, −4, −4, −4, 12, −2, 12)

which causes the output to be the vector (1, −1, 1, −1, −1, −1, 1, −1, 1), same as B. An additional pattern,

which is a 3x3 grid with only the corner pixels black, as shown in Figure 1.7, is also recalled since it is

autoassociated, by this Hopfield network.

Figure 1.7

  Pattern result.

If we omit part of the pattern in Figure 1.7, leaving only the top corners black, as in Figure 1.8, we get the

bipolar vector D = (1, −1, 1, −1, −1, −1, −1, −1, −1). You can consider this also as an incomplete or corrupted

version of the pattern in Figure 1.7. The network activations turn out to be (4, −2, 4, −4, −4, −4, 8, −2, 8) and

C++ Neural Networks and Fuzzy Logic:Preface

Binary and Bipolar Inputs

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give the output (1, −1, 1, −1, −1, −1, 1, −1, 1), which is B.