PCA

From Ufldl

Jump to: navigation, search
m (Removed broken equation links)
Line 164: Line 164:
Now, <math>\textstyle \tilde{x} \in \Re^k</math> is a lower-dimensional, "compressed" representation
Now, <math>\textstyle \tilde{x} \in \Re^k</math> is a lower-dimensional, "compressed" representation
of the original <math>\textstyle x \in \Re^n</math>.  Given <math>\textstyle \tilde{x}</math>, how can we recover an approximation <math>\textstyle \hat{x}</math> to  
of the original <math>\textstyle x \in \Re^n</math>.  Given <math>\textstyle \tilde{x}</math>, how can we recover an approximation <math>\textstyle \hat{x}</math> to  
-
the original value of <math>\textstyle x</math>?  From Equation~(\ref{eqn-unrotate}), we know that <math>\textstyle x = U x_{\rm rot}</math>.  Further,  
+
the original value of <math>\textstyle x</math>?  From the [[#Rotating the Data|previous section]], we know that <math>\textstyle x = U x_{\rm rot}</math>.  Further,  
we can think of <math>\textstyle \tilde{x}</math> as an approximation to <math>\textstyle x_{\rm rot}</math>, where we have
we can think of <math>\textstyle \tilde{x}</math> as an approximation to <math>\textstyle x_{\rm rot}</math>, where we have
set the last <math>\textstyle n-k</math> components to zeros.  Thus, given <math>\textstyle \tilde{x} \in \Re^k</math>, we can  
set the last <math>\textstyle n-k</math> components to zeros.  Thus, given <math>\textstyle \tilde{x} \in \Re^k</math>, we can  
Line 173: Line 173:
= \sum_{i=1}^k u_i \tilde{x}_i.
= \sum_{i=1}^k u_i \tilde{x}_i.
\end{align}</math>
\end{align}</math>
-
The final equality above comes from the definition of <math>\textstyle U</math> given in Equation~(\ref{eqn-udefinition}).
+
The final equality above comes from the definition of <math>\textstyle U</math> [#Example and Mathematical Background|given earlier].
(In a practical implementation, we wouldn't actually zero pad <math>\textstyle \tilde{x}</math> and then multiply
(In a practical implementation, we wouldn't actually zero pad <math>\textstyle \tilde{x}</math> and then multiply
by <math>\textstyle U</math>, since that would mean multiplying a lot of things by zeros; instead, we'd just  
by <math>\textstyle U</math>, since that would mean multiplying a lot of things by zeros; instead, we'd just  

Revision as of 01:10, 23 April 2011

Personal tools