Softmax Regression
From Ufldl
(→Properties of softmax regression parameterization) |
(→Properties of softmax regression parameterization) |
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p(y^{(i)} = j | x^{(i)} ; \theta) | p(y^{(i)} = j | x^{(i)} ; \theta) | ||
&= \frac{e^{(\theta_j-\psi)^T x^{(i)}}}{\sum_{l=1}^k e^{ (\theta_l-\psi)^T x^{(i)}}} \\ | &= \frac{e^{(\theta_j-\psi)^T x^{(i)}}}{\sum_{l=1}^k e^{ (\theta_l-\psi)^T x^{(i)}}} \\ | ||
- | &= \frac{e^{\theta_j^T x^{(i)}} e^{-\psi^Tx^{(i)}}}{\sum_{l=1}^k e^{\theta_l^T x^{(i)}} e^{-\psi^Tx^{(i)}}} | + | &= \frac{e^{\theta_j^T x^{(i)}} e^{-\psi^Tx^{(i)}}}{\sum_{l=1}^k e^{\theta_l^T x^{(i)}} e^{-\psi^Tx^{(i)}}} \\ |
&= \frac{e^{\theta_j^T x^{(i)}}}{\sum_{l=1}^k e^{ \theta_l^T} x^{(i)}} | &= \frac{e^{\theta_j^T x^{(i)}}}{\sum_{l=1}^k e^{ \theta_l^T} x^{(i)}} | ||
\end{align} | \end{align} |