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Spectral methods are widely used to reduce data dimensionality in order to enable a more effective use of several pattern recognition techniques such as clustering algorithms. Here we review the most popular spectral methods. | Spectral methods are widely used to reduce data dimensionality in order to enable a more effective use of several pattern recognition techniques such as clustering algorithms. Here we review the most popular spectral methods. | ||
| − | Consider a collection of sample points <math>\{x_1,x_2,\cdots,x_n\}</math> where <math> x_i \in R^m</math>. We divide the methods in two categories: | + | Consider a collection of sample points <math>\{\vec{x_1},\vec{x_2},\cdots,\vec{x_n}\}</math> where <math> \vec{x_i} \in R^m</math>. We divide the methods in two categories: |
* '''Outer Characteristics of the point cloud:''' These methods require the spectral analysis of a positive definite kernel of dimension m, the extrinsic dimensionality of the data. | * '''Outer Characteristics of the point cloud:''' These methods require the spectral analysis of a positive definite kernel of dimension m, the extrinsic dimensionality of the data. | ||
* '''Inner characteristics of the point cloud:''' These methods require the spectral analysis of a positive definite kernel of dimension n, or the number of samples in the sample cloud. This happens because this set of methods explore the structure of pairwise similarities among all the data samples. | * '''Inner characteristics of the point cloud:''' These methods require the spectral analysis of a positive definite kernel of dimension n, or the number of samples in the sample cloud. This happens because this set of methods explore the structure of pairwise similarities among all the data samples. | ||
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== Outer Characteristics of the point cloud Methods == | == Outer Characteristics of the point cloud Methods == | ||
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* [[MDS_OldKiwi]] | * [[MDS_OldKiwi]] | ||
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| + | == Nonlinear Methods == | ||
Revision as of 10:28, 18 April 2008
Spectral methods are widely used to reduce data dimensionality in order to enable a more effective use of several pattern recognition techniques such as clustering algorithms. Here we review the most popular spectral methods.
Consider a collection of sample points $ \{\vec{x_1},\vec{x_2},\cdots,\vec{x_n}\} $ where $ \vec{x_i} \in R^m $. We divide the methods in two categories:
- Outer Characteristics of the point cloud: These methods require the spectral analysis of a positive definite kernel of dimension m, the extrinsic dimensionality of the data.
- Inner characteristics of the point cloud: These methods require the spectral analysis of a positive definite kernel of dimension n, or the number of samples in the sample cloud. This happens because this set of methods explore the structure of pairwise similarities among all the data samples.
