Sanity checks for saliency maps, Equation sheets, [XAI-6 (1)]

논문 제목 : Sanity checks for saliency maps

논문 주소 : arxiv.org/abs/1810.03292

Sanity Checks for Saliency Maps

주요 수식 정리:

0) Definition

input : $x \in \mathbb{R}^d $

model : $ S : \mathbb{R}^d -> \mathbb{R}^C$, C : the number of classes

1) Gradient with respect to input 

$$E_{grad} (x) = \frac{\partial S}{\partial x}$$

2) Gradient $\odot$ Input (Gradient element-wise product with the input)

$$E_{Grad\odot input}(x)=x\odot \frac{\partial S}{\partial x}$$

3) Guided Backpropagation (GBP)

Feature maps derived during the forward pass : $\{f^l, f^{l-1},...,f^0\}$

Intermediate representations obtained during the backward pass : $\{R^l,R^{l-1},...,R^0\}$

$$f^l=relu(f^{l-1})$$

$$ R^{l+1} = \frac{\partial f^{out}}{\partial f^{l+1}}$$

GBP aims to zero out negative gradients during computation of R.

$$R^l=1_{R^{l+1}>0}1_{f^l>0}R^{l+1}$$

위 식의 $1_{R^{l+1}>0}$ 는 positive gradient만 전달, $1_{f^l>0}$은 positive activation만 전달을 의미한다.

4) Integrated Gradients (IG)

$$E_{IG}(x)=(x-\bar x) \times \ \int_0^1 \frac{\partial S(\bar x+\alpha (x-\bar x)}{\partial x}d\alpha$$

$\bar x$는 baseline input으로 주로 zero로 set이 된다.

5) SmoothGrad

$$E_{sg}(x)=\frac{1}{N}\sum_{i=1}^N E(x+g_i), \quad g_i \sim N(0,\sigma^2)$$

6) VarGrad

$V$ : the variance.

$$E_{vg}(x)=V(E(x+g_i)),\quad g_i \sim N(0,\sigma^2)$$

7) GradCAM and Guided GradCAM

$A^k$ : last convolutional layer에서 추출된 feature map

$$\alpha_c^k = \frac{1}{Z}\sum_i \sum_j \frac{\partial S}{\partial A_{ij}^k} $$

$$E_grad = ReLU(\sum_k \alpha_c^k A^k)$$

$$E_{guided-gradcam}(x)=E_{grad}\odot E_{gbp}$$

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나중에 쉽게 보려고 정리해놨다. (언제 논문 켜서 확인하니..)