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the classification decision. A simple yes or no answer is sometimes of limited value in applications where questions like where something occurs or how it is structured are more relevant than a binary or real‐valued one‐dimensional assessment of mere presence or absence of a certain structure. In this section, we aim to explain in more detail the relation between classification and interpretability for multilayered neural networks discussed in the previous chapter.

      4.2.1 Pixel‐wise Decomposition

      We start with the concept of pixel‐wise image decomposition, which is designed to understand the contribution of a single pixel of an image x to the prediction f(x) made by a classifier f in an image classification task. We would like to find out, separately for each image x, which pixels contribute to what extent to a positive or negative classification result. In addition, we want to express this extent quantitatively by a measure. We assume that the classifier has real‐valued outputs with mapping f: ℝ^V → ℝ¹ such that f(x) > 0 denotes the presence of the learned structure. We are interested in finding out the contribution of each input pixel x_(d) of an input image x to a particular prediction f(x). The important constraint specific to classification consists in finding the differential contribution relative to the state of maximal uncertainty with respect to classification, which is then represented by the set of root points f(x₀) = 0. One possible way is to decompose the prediction f(x) as a sum of terms of the separate input dimensions x_d :

(4.1)

      Here, the qualitative interpretation is that R_d < 0 contributes evidence against the presence of a structure that is to be classified, whereas R_d > 0 contributes evidence for its presence. More generally, positive values should denote positive contributions and negative values, negative contributions.

      LRP: Returning to multilayer ANNs, we will introduce LRP as a concept defined by a set of constraints. In its general form, the concept assumes that the classifier can be decomposed into several layers of computation, which is a structure used in Deep NN. The first layer are the inputs, the pixels of the image; and the last layer is the real‐valued prediction output of the classifier f. The l‐th layer is modeled as a vector with dimensionality V(l). LRP assumes that we have a relevance score for each dimension of the vector z at layer l + 1. The idea is to find a relevance score for each dimension of the vector z at the next layer l which is closer to the input layer such that the following equation holds:

(4.2)

Iterating Eq. (4.2) from the last layer, which is the classifier output f(x), back to the input layer x consisting of image pixels then yields the desired Eq. (4.1). The relevance for the input layer will serve as the desired sum decomposition in Eq. (4.1). In the following, we will derive further constraints beyond Eqs. (4.1) and (4.2) and motivate them by examples. A decomposition satisfying Eq. (4.2) per se is neither unique, nor it is guaranteed that it yields a meaningful interpretation of the classifier prediction.

      As an example, suppose we have one layer. The inputs are x ∈ ℝ^V. We use a linear classifier with some arbitrary and dimension‐specific feature space mapping φ_d and a bias b:

      (4.3)

      Let us define the relevance for the second layer trivially as . Then, one possible LRP formula would be to define the relevance R⁽¹⁾ for the inputs x as

      (4.4)

This clearly satisfies Eqs. (4.1) and (4.2); however, the relevances R⁽¹⁾(x_d) of all input dimensions have the same sign as the prediction f(x). In terms of pixel‐wise decomposition interpretation, all inputs point toward the presence of a structure if f(x) > 0 and toward the absence of a structure if f(x) < 0. This is for many classification problems not a realistic interpretation. As a solution, for this example we define an alternative

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