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Dimension reduction2 - Dimension Reduction 2 Some more...

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4/21/2011 1 Dimension Reduction 2 Some more detailed discussion Dealing with high dimensional data is time consuming, even when dealing with cameras and images, the images are highly compressed so that they can be used easier. They are compressed down to parameters that define the original picture. The parameters can be used to retrieve the image when needing to be viewed again. How do we perform this compression of data, or dimension reduction? Suppose we have some data points ? ? ∈ ℝ 𝐷 , 𝑖 = 1,2, … , 𝑛 where D is very large. We want ? ? ? ? ? ∈ ℝ 𝑑 where M is a map and d<<D (much much less than). The best case would be that we can perfectly go to ? ? from ? ? and vice versa but in reality we do lose some information. We saw this with PCA ? ? = 𝑎 ?1 𝑎 ?2 𝑎 ?𝑑 ∈ ℝ 𝐷
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4/21/2011 2 Our situation: ? = ? 1 , ? 2 , … , ? 𝑛 ∈ ℝ 𝐷 D is very large Come up with a mapping to get ? = ? ? , ? 2 , … , ? 𝑛 ∈ ℝ 𝑑 d << D How do we pull this off? With a dimension reduction method: Direct PCA Dual PCA Kernel PCA
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