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      The choice of a Taylor base point x₀ is a free parameter in this setup. As stated above, in the case of classification, we are interested in finding out the contribution of each pixel relative to the state of maximal uncertainty of the prediction given by the set of points f(x₀) = 0, since f(x) > 0 denotes the presence and f(x) < 0 denotes the absence of the learned structure. Thus, x₀ should be chosen to be a root of the predictor f. Thus, the above equation simplifies to

      (4.14)

      The pixel‐wise decomposition contains a nonlinear dependence on the prediction point x beyond the Taylor series, as a close root point x₀ needs to be found. Thus, the whole pixel‐wise decomposition is not a linear, but a locally linear algorithm, as the root point x₀ depends on the prediction point x.

      4.2.2 Pixel‐wise Decomposition for Multilayer NN

      Pixel‐wise decomposition for multilayer networks: In the previous chapter, we discussed NN networks built as a set of interconnected neurons organized in a layered structure. They define a mathematical function when combined with each other that maps the first‐layer neurons (input) to the last‐layer neurons (output). In this section, we denote each neuron by x_i , where i is an index for the neuron. By convention, we associate different indices for each layer of the network. We denote by ∑_i the summation over all neurons of a given layer, and by ∑_j the summation over all neurons of another layer. We denote by x_(d) the neurons corresponding to the pixel activations (i.e., with which we would like to obtain a decomposition of the classification decision). A common mapping from one layer to the next one consists of a linear projection followed by a nonlinear function: z_ij = x_j w_ij , z_j = ∑_i z_ij + b_j , x_j = g(z_j), where w_ij is a weight connecting neuron x_i to neuron x_j, b_j is a bias term, and g is a nonlinear activation function. Multilayer networks stack several of these layers, each of them being composed of a large number of neurons. Common nonlinear functions are the hyperbolic tangent g(t) = tanh (t) or the rectification function g(t) = max (0, t)

      Taylor‐type decomposition: Denoting by f : ℝ^M ↦ ℝ^N the vector‐valued multivariate function implementing the mapping between input and output of the network, a first possible explanation of the classification decision x ↦ fx) can be obtained by Taylor expansion at a near root point x₀ of the decision function f:

      (4.15)

      The derivative ∂fx)/∂x_(d) required for pixel‐wise decomposition can be computed efficiently by reusing the network topology using the backpropagation algorithm discussed in the previous chapter. Having backpropagated the derivatives up to a certain layer j, we can compute the derivative of the previous layer i using the chain rule:

      (4.16)

Figure 4.2 Relevance propagation.

Layer‐wise relevance backpropagation: As an alternative to Taylor‐type decomposition, it is possible to compute relevances at each layer in a backward pass, that is, express relevances as a function of upper‐layer relevances , and backpropagating relevances until we reach the input (pixels). Figure 4.2 depicts a graphic example. The method works as follows: Knowing the relevance of a certain neuron for the classification decision f(x), one would like to obtain a decomposition of this relevance in terms of the messages sent to the neurons of the previous layers. We call these messages R_{i ← j}. As before, the conservation property

      (4.17)

must hold. In the case of a linear network f(x) = ∑_i z_ij where the relevance R_j = f(x), such a decomposition is immediately given by R_{i ← j} = z_ij . However, in the general case, the neuron activation x_j is a nonlinear function of z_j . Nevertheless, for the hyperbolic tangent and the rectifying function – two simple monotonically increasing functions satisfying g(0) = 0‐the pre‐activations z_ij still provide a sensible way to measure the relative contribution of each neuron x_i to R_j. A first possible choice of relevance decomposition is based on the ratio of local and global pre‐activations and is given by

(4.18)

      These relevances R_{i ← j} are easily shown to approximate the conservation properties, in particular:

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