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

## Table of Contents

## 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

- If is far away from , then is large, hence we assign large

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 asthat

*redistributes*relevance from one layer to another*below*, where denotes the Taylor residual.If each

*local*Taylor decomposition is**conservative**, thenwhich 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.*