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Update 07.3-concepts.Rmd #396

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May 22, 2024
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Update 07.3-concepts.Rmd #396

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2 changes: 1 addition & 1 deletion manuscript/07.3-concepts.Rmd
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Expand Up @@ -48,7 +48,7 @@ $$S_{C,k,l}(x)=\nabla h_{l,k}(\hat{f}_l(x))\cdot v_l^C$$
where $\hat{f}_l$ maps the input $x$ to the activation vector of the layer $l$ and $h_{l,k}$ maps the activation vector to the logit output of class $k$.

Mathematically, the sign of $S_{C,k,l}(x)$ only depends on the angle between the gradient of $h_{l,k}(\hat{f}_l(x))$ and $v_l^C$.
If the angle is greater than 90 degrees, $S_{C,k,l}(x)$ will be positive, and if the angle is less than 90 degrees, $S_{C,k,l}(x)$ will be negative.
If the angle is less than 90 degrees, $S_{C,k,l}(x)$ will be positive, and if the angle is greater than 90 degrees, $S_{C,k,l}(x)$ will be negative.
Since the gradient $\nabla h_{l,k}$ points to the direction that maximizes the output the most rapidly, conceptual sensitivity $S_{C,k,l}$, intuitively, indicates whether $v_l^C$ points to the similar direction that maximizes $h_{l,k}$.
Thus, $S_{C,k,l}(x)>0$ can be interpreted as concept $C$ encouraging the model to classify $x$ into class $k$.

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