samples/texts/2298363/page_4.md

Theorem 2 Let $h_g: \mathbb{R}^D \to \mathbb{R}^D$ be the explanation of classifier $g: \mathbb{R}^D \to \mathbb{R}$ with bounded derivatives $|\nabla_i g(x)| \le C \in \mathbb{R}_+$ for $i = 1, \dots, D$.

For a given target explanation $h^t: \mathbb{R}^D \to \mathbb{R}^D$, there exists another classifier $\tilde{g}: \mathbb{R}^D \to \mathbb{R}$ which completely agrees with the classifier $g$ on the data manifold $S$, i.e.

$$\tilde{g}{\mathrm{\mid }}_S=g{\mathrm{\mid }}_S\mathrm{.}(3)\backslash tilde\{g\}|\_S\; =\; g|\_S.\; \backslash quad\; (3)$$

In particular, both classifiers have the same train, validation, and test loss.

However, its explanation $h_{\tilde{g}}$ closely resembles the target $h^t$, i.e.

$$\text{MSE}(h_{\tilde{g}}(x),h^t(x))\le ϵ\mathrm{\forall }x\in S,(4)\backslash text\{MSE\}(h\_\{\backslash tilde\{g\}\}(x),\; h^t(x))\; \backslash le\; \backslash epsilon\; \backslash quad\; \backslash forall\; x\; \backslash in\; S,\; \backslash quad\; (4)$$

where $\text{MSE}(h, h') = \frac{1}{D} \sum_{i=1}^{D} (h_i - h'_i)^2$ denotes the mean-squared error and $\epsilon = \frac{d}{D}$.

Proof: By Theorem 1, we can find a function $G$ which agrees with $g$ on the data manifold $S$ but has the derivative

$$\mathrm{\nabla }G(x)=({\mathrm{\nabla }}_{1}g(x),\dots ,{\mathrm{\nabla }}_{d}g(x),h_{d+1}^t(x),\dots ,h_D^t(x))\backslash nabla\; G(x)\; =\; (\backslash nabla\_1\; g(x),\; \backslash dots,\; \backslash nabla\_d\; g(x),\; h\_\{d+1\}^t(x),\; \backslash dots,\; h\_D^t(x))$$

for all $x \in S$. By definition, this is its gradient explanation $h_G = \nabla G$.

As explained in Appendix A.2.1, we can assume without loss of generality that $|\nabla_i g(x)| \le 0.5$ for $i \in {1, \dots, D}$. We can furthermore rescale the target map such that $|h_i^t| \le 0.5$ for $i \in {1, \dots, D}$. This rescaling is merely conventional as it does not change the relative importance $h_i$ of any input component $x_i$ with respect to the others. It then follows that

$$\text{MSE}(h_G(x),h^t(x))=\frac{1}{D}\sum _{i=1}^D({\mathrm{\nabla }}_{i}G(x)-h_i^t(x){)}^2\mathrm{.}\backslash text\{MSE\}(h\_G(x),\; h^t(x))\; =\; \backslash frac\{1\}\{D\}\; \backslash sum\_\{i=1\}^\{D\}\; (\backslash nabla\_i\; G(x)\; -\; h\_i^t(x))^2\; .$$

This sum can be decomposed as

$$\frac{1}{D}\sum _{i=1}^d\underset{\le 1}{\underbrace{{({\mathrm{\nabla }}_{i}g(x)-h_i^t(x))}^2}}+\frac{1}{D}\sum _{i=d+1}^D\underset{=0}{\underbrace{{({\mathrm{\nabla }}_{i}G(x)-h_i^t(x))}^2}}\backslash frac\{1\}\{D\}\; \backslash sum\_\{i=1\}^\{d\}\; \backslash underbrace\{\backslash left(\backslash nabla\_i\; g(x)\; -\; h\_i^t(x)\backslash right)^2\}\_\{\backslash le\; 1\}\; +\; \backslash frac\{1\}\{D\}\; \backslash sum\_\{i=d+1\}^\{D\}\; \backslash underbrace\{\backslash left(\backslash nabla\_i\; G(x)\; -\; h\_i^t(x)\backslash right)^2\}\_\{=0\}$$

and from this, it follows that

$$\text{MSE}(h_G(x),h^t(x))\le \frac{d}{D}=ϵ,\backslash text\{MSE\}(h\_G(x),\; h^t(x))\; \backslash le\; \backslash frac\{d\}\{D\}\; =\; \backslash epsilon,$$

The proof then concludes by identifying $\tilde{g} = G$. □

Intuition: Somewhat roughly, this theorem can be understood as follows: two models, which behave identically on the data, need to only agree on the low-dimensional submanifold $S$. The gradients "orthogonal" to the submanifold $S$ are completely undetermined by this requirement. By the manifold assumption, there are however much more

"orthogonal" than "parallel" directions and therefore the explanation is largely controlled by these. We can use this fact to closely reproduce an arbitrary target while keeping the function's values on the data unchanged.

We stress however that there are a number of non-trivial differential geometric arguments needed in order to make these statements rigorous and quantitative. For example, it is entirely non-trivial that an extension to the embedding manifold exists for arbitrary choice of target explanation. This is shown by Theorem 1 whose proof is based on a differential geometric technique called partition of the unity subordinate to an open cover. See Appendix A.1 for details.

2.3. Explanation Manipulation: Methods

Flat Submanifolds and Logistic Regression: The previous theorem assumes that the data lies on an arbitrarily curved submanifold and therefore has to rely on relatively involved mathematical concepts of differential geometry. We will now illustrate the basic ideas in a much simpler context: we will assume that the data lies on a $d$-dimensional flat hyperplane $S \subset \mathbb{R}^D$.⁶ The points on the hyperplane $S$ obey the relation

$$\mathrm{\forall }x\in S:({\widehat{w}}^{(i)}{)}^{T}x=b_i,i\in \{1,\dots ,D-d\},(5)\backslash forall\; x\; \backslash in\; S\; :\; (\backslash hat\{w\}^\{(i)\})^T\; x\; =\; b\_i,\; \backslash quad\; i\; \backslash in\; \backslash \{1,\; \backslash dots,\; D-d\backslash \},\; \backslash quad\; (5)$$

where ${\hat{w}^{(i)} \in \mathbb{R}^D | i = 1, \dots, D-d}$ are a set of normal vectors to the hyperplane $S$ and $b_i \in \mathbb{R}$ are the affine translations. We furthermore assume that we use logistic regression as the classification algorithm, i.e.

$$g(x)=\sigma (w^{T}x+c),(6)g(x)\; =\; \backslash sigma(w^T\; x\; +\; c),\; \backslash quad\; (6)$$

where $w \in \mathbb{R}^D$, $c \in \mathbb{R}$ are the weights and the bias respectively and $\sigma(x) = \frac{1}{1+\exp(-x)}$ is the sigmoid function. This classifier has the gradient explanation⁷

$$h_{\text{grad}}(x)=w,(7)h\_\{\backslash text\{grad\}\}(x)\; =\; w,\; \backslash quad\; (7)$$

We can now define a modified classifier by

$$\tilde{g}(x)=\sigma (w^{T}x+\sum _i{\lambda }_i({\widehat{w}}^{(i)}{)}^{T}x-b_i+c),(8)\backslash tilde\{g\}(x)\; =\; \backslash sigma\backslash left(w^T\; x\; +\; \backslash sum\_i\; \backslash lambda\_i\; (\backslash hat\{w\}^\{(i)\})^T\; x\; -\; b\_i\; +\; c\backslash right),\; \backslash quad\; (8)$$

for arbitrary $\lambda_i \in \mathbb{R}$. By (5), it follows that both classifiers agree on the data manifold $S$, i.e.

$$\mathrm{\forall }x\in S:g(x)=\tilde{g}(x),(9)\backslash forall\; x\; \backslash in\; S\; :\; g(x)\; =\; \backslash tilde\{g\}(x),\; \backslash quad\; (9)$$

and therefore have the same train, validation, and test error. However, the gradient explanations are now given by

$$h_{\text{grad}}(x)=w+\sum _i{\lambda }_i{\widehat{w}}^{(i)}\mathrm{.}(10)h\_\{\backslash text\{grad\}\}(x)\; =\; w\; +\; \backslash sum\_\{i\}\; \backslash lambda\_\{i\}\; \backslash hat\{w\}^\{(i)\}.\; \backslash quad\; (10)$$

⁶In mathematics, these submanifolds are usually referred to as $d$-flats and only the case $d = D - 1$ is called hyperplane. We refrain from this terminology.

⁷We recall that in calculating the explanation map, we take the derivative before applying the final activation function.