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4/21/2011
1
Introduction to
Dimension Reduction
Starting from PCA, Diffusion maps, …
Need of Dimension Reduction
•
Realworld data (speech signals, digital
photographs, or fMRI scans) usually has a high
dimensionality.
•
In order to handle such realworld data
adequately, its dimensionality needs to be
reduced.
•
Dimensionality reduction is the transformation of
highdimensional data into a
meaningful
representation of reduced dimensionality.
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2
Meaningful?
•
Ideally, the reduced representation should have a
dimensionality that corresponds to the intrinsic
dimensionality of the data.
•
The intrinsic dimensionality of data is the
minimum number of parameters
needed to
account for the observed properties of the data.
•
In mathematical terms, intrinsic dimensionality
means that the points in dataset X are lying on or
near a manifold with dimensionality d that is
embedded in the Ddimensional space.
Examples of manifolds
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 Spring '11
 Li

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