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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, the number of samples in the sample cloud. | + | * '''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_Old Kiwi]] | * [[MDS_Old Kiwi]] | ||
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| + | == Nonlinear Methods: When Data Embedded in Low-Dimensional Non-Linear Manifold == | ||
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| + | Let's consider the case when the data is embedded in a low-dimensional non-linear manifold. Because of the non-linear structure of the manifold, the traditional [[MDS_Old Kiwi]] won't capture the true structure of the data. Recently several methods have been developed to deal with such a non-linar configuration. Here we briefly review two of them: | ||
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| + | * [[Isomaps_Old Kiwi]] | ||
| + | * [[Laplacian Eigenmaps_Old Kiwi]] | ||
Latest revision as of 09:41, 28 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.
Outer Characteristics of the point cloud Methods
Inner characteristics of the point cloud Methods
Nonlinear Methods: When Data Embedded in Low-Dimensional Non-Linear Manifold
Let's consider the case when the data is embedded in a low-dimensional non-linear manifold. Because of the non-linear structure of the manifold, the traditional MDS_Old Kiwi won't capture the true structure of the data. Recently several methods have been developed to deal with such a non-linar configuration. Here we briefly review two of them:
