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Doraemonzzz · Doraemonzzz · commit 933e3b9dbb29 · 2018-10-06T09:08:44.000+08:00
diff --git a/Chapter8/md/Chapter8 Support Vector Machines.md b/Chapter8/md/Chapter8 Support Vector Machines.md
@@ -820,35 +820,29 @@ $$
 $$
 所以$Φ(x) = (Φ_1(x), Φ_2(x))​$的kernel为$K_1+K_2​$
 
-(b)$\Phi(x)$是$Φ_1(x)Φ_2(x)^T$每一行拼接而成的向量，设$Φ_1(x),Φ_2(x)\in R^n$，给出以下记号
+(b)$\Phi(x)$是$Φ_1(x)Φ_2^T(x)$每一行拼接而成的向量，设$Φ_1(x),Φ_2(x)\in R^n$，给出以下记号
 $$
-\Phi^i(x)=Φ_1^{i}(x)Φ_2(x)^T\in R^n\\
+\Phi^i(x)=Φ_1^{i}(x)Φ^T_2(x)\in R^n\\
 Φ^{i}_1(x)为Φ_1(x)的第i个分量
 $$
 那么
 $$
 \Phi(x) =(Φ^i(x),...,Φ^n(x))^T \in R^{n^2}\\
 $$
 
-$$
-\Phi^i(x)=Φ_1^{i}(x)Φ_2(x)^T\in R^n\\
-\Phi(x) =(Φ^i(x),...,Φ^n(x))^T \in R^{n^2}\\
-Φ^{i}_1(x)为Φ_1(x)的第i个分量
-$$
-
-接着计算$\Phi(x) \Phi^T(x) $，注意$\Phi^i(x)$为行向量
+接着计算$\Phi(x) \Phi^T(x') $，注意$\Phi^i(x)$为行向量
 $$
 \begin{aligned}
-(Φ(x) Φ(x^{'})^T)
+(Φ(x) Φ^T(x^{'}))
 &=\sum_{i=1}^n  (Φ^i(x))Φ^i(x^{'})^T\\
 &=\sum_{i=1}^n (Φ_1^{i}(x)Φ_2(x)^T)(Φ_1^{i}(x^{'})Φ_2(x^{'})^T)^T\\
-&=\sum_{i=1}^nΦ_1^{i}(x)Φ_1^{i}(x^{'})Φ_2(x)^TΦ_2(x^{'})\\
+&=\sum_{i=1}^nΦ_1^{i}(x)Φ_1^{i}(x^{'})Φ^T_2(x)Φ_2(x^{'})\\
 &=\sum_{i=1}^nΦ_1^{i}(x)Φ_1^{i}(x^{'})K_2(x,x^{'})\\
 &=K_2(x,x^{'})\sum_{i=1}^nΦ_1^{i}(x)Φ_1^{i}(x^{'})\\
 &=K_2(x,x^{'})K_1(x,x^{'})
 \end{aligned}
 $$
-所以$Φ(x)$的kernel为$K_1(x,x^{'})K_2(x,x^{'})$
+所以$Φ(x)$对应的kernel为$K_1(x,x^{'})K_2(x,x^{'})$
 
 (c)由(a),(b)可以直接推出
 
