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## Competition between two neurons

We shall start the derivation of the competition mechanism by considering a network with only one neuron. We shall adopt the reconstruction error minimisation approach, and start by defining the reconstruction mapping (the mapping in section 2.1), which has parameters :

The mapping is linear since the reconstruction is linearly dependent on the parameters and the output y. Without any loss of generality we can assume that the weight vector is normalised to unity: . In order to obtain a similar coding as in figure 3.1 B, we shall also require the output y to be non-negative. By assuming a quadratic reconstruction error and solving equation 2.1 (derivation is given in appendix A) we obtain

We have thus derived a neuron, which is most active when the direction of the input vector is the same as the direction of the weight vector.

Now suppose we have two neurons with reconstruction mapping

Again we can assume that the weight vectors are normalised to unity and that the outputs are non-negative. We expect the neurons to compete for the output. All competition should be inhibitory, and therefore we shall require the outputs to be at most what they would be if the neurons were be alone, that is, . This time the solution to equation 2.1 turns out to be (derivation is given in appendix A)

where and . We have assumed that the degenerate case , where c = 1, can be omitted. The solution for is similar with indices 1 and 2 interchanged. The value is the projection of the input on the weight vector . The value c gives a measure for the similarity of the weight vectors. If c = 1, the weight vectors are exactly the same. The smaller the c is the more dissimilar are the weight vectors. We shall call the value c a correlation, because in some cases it can be interpreted as the measure of correlation between and .

It is useful to define a new variable and examine its solutions:

We shall call the new variable a winning strength, because it can be interpreted as the result of competition between the neurons. Figure 3.2 shows the outputs and the winning strengths as a function of the direction of input (equations 3.4 and 3.5). When only one neuron is active, its winning strength is one. We can interpret this so that the neuron has won the competition and is a full winner. When both neurons participate in the representation of the input, the winning strengths take values between zero and one. Both neurons are winners to some extent, but neither is a full winner. When a neuron has a zero output, also its winning strength is zero, and we can say that the neuron has completely lost the competition.

By examining the equation 3.4 we notice that if the correlation c between neurons is negative or zero, the winning strength is one whenever the projection of the input on the weight vector is positive: . This means that neurons with dissimilar weight vectors have no mutual competition. The closer c comes to one, the smaller become the outputs of the neurons. This means that neurons with similar weight vectors compete most. Another notable property of the winning strengths is that they depend only on the direction of the input, not the magnitude. If the input vector is multiplied with a positive constant, the winning strengths remain unchanged.

Figure 3.2: The outputs (at the left) and the winning strengths (at the right) of a network with two neurons are plotted as a function of the angle of the input. The directions of the weight vectors are denoted by `*'.

Next: Combining the competitions Up: Competition mechanism Previous: Competition mechanism

Harri Lappalainen
Thu May 9 14:06:29 DST 1996