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zzmanzz
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Homework Statement
Suppose I wish to fit a plane
[tex] z = w_1 + w_2x +w_3y [/tex]
to a data set [tex] (x_1,y_1,z_1), ... ,(x_n,y_n,z_n) [/tex]
Using gradient descent
Homework Equations
http://en.wikipedia.org/wiki/Stochastic_gradient_descent
The Attempt at a Solution
I'm basically trying to figure out the 3-dimensional version of the example on wiki.
The objective function to e minimized is:
[tex] Q(w) = \sum_{i = 1}^n Q_i(w) = \sum_{i = 1}^n (w_1 + w_2x_i + w_3y_i - z_i)^2 [/tex]
I want to find the parameters of [tex]w_1,w_2,w_3 [/tex]
The iterative method updates the parameters [tex]w^{(0)}_1,w^{(0)}_2,w^{(0)}_3 [/tex]
1-step in the iteration
[tex]
\left( \begin{array}{ccc}
w^{(1)}_1 \\
w^{(1)}_2\\
w^{(1)}_3 \end{array} \right) = \left( \begin{array}{ccc}
w^{(0)}_1 \\
w^{(0)}_2 \\
w^{(0)}_3 \end{array} \right) + \alpha \times \left( \begin{array}{ccc}
2(w^{(0)}_1 + w^{(0)}_2x_i + w^{(0)}_3 y_i - z_i) \\
2x_i(w^{(0)}_1 + w^{(0)}_2x_i + w^{(0)}_3 y_i - z_i) \\
2y_i(w^{(0)}_1+ w^{(0)}_2x_i + w^{(0)}_3 y_i - z_i) \end{array} \right) [/tex]
[tex]\alpha [\tex] is the step size.
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