Softmax Regression

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(Weight Regularization)
m (Parameterization)
Line 158: Line 158:
\begin{align}
\begin{align}
-
h(x^{(i)}) &= \frac{1}{ 1 + \sum_{j=1}^{n-1}{e^{ \Theta_j x^{(i)} }} }
+
h(x^{(i)}) &= \frac{1}{ 1 + \sum_{j=1}^{n-1}{e^{ \Theta_j^T x^{(i)} }} }
\begin{bmatrix}  
\begin{bmatrix}  
e^{ \Theta_1^T x^{(i)} } \\
e^{ \Theta_1^T x^{(i)} } \\
Line 182: Line 182:
h(x^{(i)}) &=  
h(x^{(i)}) &=  
-
\frac{1}{ 1 + e^{ \Theta_1 x^{(i)} } }
+
\frac{1}{ 1 + e^{ \Theta_1^T x^{(i)} } }
\begin{bmatrix}  
\begin{bmatrix}  
e^{ \Theta_1^T x^{(i)} } \\
e^{ \Theta_1^T x^{(i)} } \\
Line 190: Line 190:
&=  
&=  
-
\frac{e^{ \Theta_1 x^{(1)} } }{ 1 + e^{ \Theta_1 x^{(i)} } }
+
\frac{e^{ \Theta_1^T x^{(1)} } }{ 1 + e^{ \Theta_1^T x^{(i)} } }
\cdot
\cdot
-
\frac{1}{e^{ \Theta_1 x^{(1)} } }
+
\frac{1}{e^{ \Theta_1^T x^{(1)} } }
\begin{bmatrix}  
\begin{bmatrix}  
e^{ \Theta_1^T x^{(i)} } \\
e^{ \Theta_1^T x^{(i)} } \\
Line 200: Line 200:
&=  
&=  
-
\frac{1}{ e^{ -\Theta_1 x^{(i)} } + 1 }
+
\frac{1}{ e^{ -\Theta_1^T x^{(i)} } + 1 }
\begin{bmatrix}  
\begin{bmatrix}  
1 \\
1 \\

Revision as of 23:24, 7 May 2011

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