PCA

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(PCA on Images)
(Rotating the Data)
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This is the training set rotated into the <math>\textstyle u_1</math>,<math>\textstyle u_2</math> basis. In the general
This is the training set rotated into the <math>\textstyle u_1</math>,<math>\textstyle u_2</math> basis. In the general
case, <math>\textstyle U^Tx</math> will be the training set rotated into the basis  
case, <math>\textstyle U^Tx</math> will be the training set rotated into the basis  
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<math>\textstyle u_1</math>,<math>\textstyle u_2</math>,\ldots,<math>\textstyle u_n</math>.  
+
<math>\textstyle u_1</math>,<math>\textstyle u_2</math>, ...,<math>\textstyle u_n</math>.  
-
One of the properties of <math>\textstyle U</math> is that it is an orthogonal basis, and thus <math>\textstyle U^TU = UU^T = I</math>.
+
One of the properties of <math>\textstyle U</math> is that it satisfies <math>\textstyle U^TU = UU^T = I</math>;
 +
another way of saying this is that <math>U</math> is an "orthogonal" matrix.  
So if you ever need to go back from the rotated vectors <math>\textstyle x_{\rm rot}</math> back to the  
So if you ever need to go back from the rotated vectors <math>\textstyle x_{\rm rot}</math> back to the  
original data <math>\textstyle x</math>, you can compute  
original data <math>\textstyle x</math>, you can compute  

Revision as of 23:13, 29 April 2011

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