# Notes on: Montavon, Gr\'egoire, Bach, S., Binder, A., Samek, W., & M\"uller, Klaus-Robert (2015): Explaining Nonlinear Classification Decisions With Deep Taylor Decomposition

## Overview

Propose an expansion for some input / activation about a root point using local Taylor decomposition, where the expansion is given by the recursion where

• denotes the relevance score of the j-th neuron in a given layer
• denotes the relevance score for the i-th neuron in the previous layer (i.e. before the layer which contains )

Intuitively, one can think of it as follows:

• Gradient measures sensitivity of some class-label to each pixel when the classifier is evaluated at the root point .
• A good root point is one that removes the object (e.g. as detected by the function ), but that minimally deviates from the original point .
• If is far away from , then is large, hence we assign large relevance to • If evaluted at the root point is large, then that means the input about this root point is very sensitive, hence we assign high releveance to nearby points which includes This method starts out by looking for a relevance score for the input which satisfies properties of

• conservation, i.e. sum assigned relevances in input space corresponds to total relevance detected by model (i.e. relevance i output) • positivity, i.e. ## Notation

• positive-valued function
• is an image
• quantifies the presence (or amount) of a certain type of boject in the image
• indicates absence of object in image
• assoicate to each pixel in the image a relevance score
• is the heatmap which contains the relevance score of each pixel
• denotes the point where on performs Taylor expansion, usually an infinitesimally small distance from the actual point , in direction of maximum descent, i.e. with small.

• denotes the set of neurons in the l-th layer (notation introduced by me)
• and denote the activations and relevances in some layer, i.e. • and denote the activations and relevances in the next layer (relative to layer containing all the ), i.e. • is a root point, which is the nearest point to such that ## Pixel-wise Decomposition of a function

One would like the heatmap to have the following properties

• Conservative , i.e. the sum of assigned relevances in pixel space corresponds to the total relevance detected by model • Positive: We say a heatmapping is consistent if and only if it is both conservative and positive.

## Taylor Decomposition

Taylor expansion of the function at some well-chosen root point , where which gives where we identify the summed elements as the relevances assigned to pixels in the image.

We can write this as the element-wise product between the gradient of the function at the root point and difference between the image and the root : • Gradient measures sensitivity of some class-label to each pixel when the classifier is evaluated at the root point .
• A good root point is one that removes the object (e.g. as detected by the function , but that minimally deviates from the original point .

### Deep Taylor Decomposition

• Don't consider whole NN function • Consider mapping of a set of neurons at a given layer to the relevance assignted to a neruon in the next layer, that is: • Assume two objects are related functionally by some function => would like to apply Taylor decomposition on this local function in order to redistribute relevance onto lower-layer relevances • Assume neighboring root point such that , we can write Taylor decomposition of at as that redistributes relevance from one layer to another below, where denotes the Taylor residual.

• If each local Taylor decomposition is conservative, then which is referred to as layer-wise relevance conservation.

• Positivity of relevance is also ensured
• If Taylor decompositions of local subfunctions are consistent, then the whole deep Taylor decomposition is also consistent

Only problem left is to identify the root-points, but this is indeed handled by the paper depending on whether or not the input is constrained, etc.