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''Let <math>A(\cdot)</math> be a self-adjoint, compact operator on an infinite dimensional Hilbert space <math>H</math>. Then, there exists in <math>H</math> a complete orthonormal system <math>\{e_1,e_2,\cdots \}</math> consisting of eigenvectors of <math>A(\cdot)</math>. Moreover, for every <math>x\in H</math>,<br /> | ''Let <math>A(\cdot)</math> be a self-adjoint, compact operator on an infinite dimensional Hilbert space <math>H</math>. Then, there exists in <math>H</math> a complete orthonormal system <math>\{e_1,e_2,\cdots \}</math> consisting of eigenvectors of <math>A(\cdot)</math>. Moreover, for every <math>x\in H</math>,<br /> | ||
<math> | <math> | ||
x = \sum_{i=1}^{\infty} | x = \sum_{i=1}^{\infty} <x, e_i> e_i, | ||
</math> | </math> | ||
where <math>\lambda_n</math> is the eigenvalue corresponding to <math>e_i</math>. Furthermore, if <math>A(\cdot)</math> has infinitely many distinct eigenvalues <math>\lambda_1, \lambda_2, \cdots</math>, then <math>\lambda_n \rightarrow 0</math> as <math>n \rightarrow 0</math>.'' | where <math>\lambda_n</math> is the eigenvalue corresponding to <math>e_i</math>. Furthermore, if <math>A(\cdot)</math> has infinitely many distinct eigenvalues <math>\lambda_1, \lambda_2, \cdots</math>, then <math>\lambda_n \rightarrow 0</math> as <math>n \rightarrow 0</math>.'' | ||
Revision as of 18:42, 3 June 2009
Maximum Variance Unfolding AKA Semidefinite Embedding
The main poposal of the technique is to lean a suitable kernel with several constraints when the data is given.
Here is the constraints for the kernel.
1. Semipositive definiteness
Kernel PCA is a kind of spectral decompostion in Hilber space. In functional analysis, we have
Spectral Theorem for Self-Adjoint Compact Operators
Let [math]\displaystyle{ A(\cdot) }[/math] be a self-adjoint, compact operator on an infinite dimensional Hilbert space [math]\displaystyle{ H }[/math]. Then, there exists in [math]\displaystyle{ H }[/math] a complete orthonormal system [math]\displaystyle{ \{e_1,e_2,\cdots \} }[/math] consisting of eigenvectors of [math]\displaystyle{ A(\cdot) }[/math]. Moreover, for every [math]\displaystyle{ x\in H }[/math],
[math]\displaystyle{
x = \sum_{i=1}^{\infty} \lt x, e_i\gt e_i,
}[/math]
where [math]\displaystyle{ \lambda_n }[/math] is the eigenvalue corresponding to [math]\displaystyle{ e_i }[/math]. Furthermore, if [math]\displaystyle{ A(\cdot) }[/math] has infinitely many distinct eigenvalues [math]\displaystyle{ \lambda_1, \lambda_2, \cdots }[/math], then [math]\displaystyle{ \lambda_n \rightarrow 0 }[/math] as [math]\displaystyle{ n \rightarrow 0 }[/math].