From 63dac245d8395985c70076a1a511e07b3dc2c776 Mon Sep 17 00:00:00 2001 From: Fangzhou Li Date: Mon, 20 May 2024 16:44:27 -0700 Subject: [PATCH] Update 07.3-concepts.Rmd Typos found: dot product sign switched. --- manuscript/07.3-concepts.Rmd | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/manuscript/07.3-concepts.Rmd b/manuscript/07.3-concepts.Rmd index 264922a53..8244ac0d5 100644 --- a/manuscript/07.3-concepts.Rmd +++ b/manuscript/07.3-concepts.Rmd @@ -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 $k$.