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Then, the relevance of a feature dimension x_d depends on the sign of the term in Eq. (4.5). This is for many classification problems a more plausible interpretation. This second example shows that LRP is able to deal with nonlinearities such as the feature space mapping φ_d to some extent and how an example of LRP satisfying Eq. (4.2) may look like in practice.

The above example gives an intuition about what relevance R is, namely, the local contribution to the prediction function f(x). In that sense, the relevance of the output layer is the prediction itself: f(x). This first example shows what one could expect as a decomposition for the linear case. The linear case is not a novelty; however, it provides a first intuition. A more graphic and nonlinear example is given in Figure 4.1. The upper part of the figure shows a neural‐network‐shaped classifier with neurons and weights w_ij on connections between neurons. Each neuron i has an output a_i from an activation function.The top layer consists of one output neuron, indexed by 7. For each neuron i we would like to compute a relevance R_i . We initialize the top layer relevance as the function value; thus, . LRP in Eq. (4.2) requires now to hold

      (4.6)

      (4.7)

      We will make two assumptions for this example. First, we express the layer‐wise relevance in terms of messages between neurons i and j which can be sent along each connection. The messages are, however, directed from a neuron toward its input neurons, in contrast to what happens at prediction time, as shown in the lower part of Figure 4.1. Second, we define the relevance of any neuron except neuron 7 as the sum of incoming messages (k : i is input for neuron k):

(4.8)

Figure 4.1 (a) Neural network (NN) as a classifier, (b) NN during the relevance computation.

For example, Note that neuron 7 has no incoming messages anyway. Instead, its relevance is defined as . In Eq. (4.8) and the following text, the terms input and source have the meaning of being an input to another neuron in the direction defined during the time of classification, not during the time of computation of LRP. For example in Figure 4.1, neurons 1 and 2 are the inputs and source for neuron 4, while neuron 6 is the sink for neurons 2 and 3. Given the two assumptions encoded in Eq. (4.8), the LRP by Eq. (4.2) can be satisfied by the following sufficient condition:

       and . In general, this condition can be expressed as

(4.9)

The difference between condition (4.9) and definition (4.8) is that in condition Скачать книгу