PCA
From Ufldl
(→Reducing the Data Dimension) |
(→Recovering an Approximation of the Data) |
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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 | + | the original value of <math>\textstyle x</math>? From an [[#Rotating the Data|earlier 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 | ||
