Softmax Regression
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(→Weight Regularization) |
m (→Parameterization) |
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\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)} } \\ | ||
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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)} } \\ | ||
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&= | &= | ||
- | \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)} } \\ | ||
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&= | &= | ||
- | \frac{1}{ e^{ -\Theta_1 x^{(i)} } + 1 } | + | \frac{1}{ e^{ -\Theta_1^T x^{(i)} } + 1 } |
\begin{bmatrix} | \begin{bmatrix} | ||
1 \\ | 1 \\ |