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'''1. Semipositive definiteness'''<br />
'''1. Semipositive definiteness'''<br />
Kernel PCA is a kind of spectral decompostion in Hilber space. In functional analysis, we have<br />
Kernel PCA is a kind of spectral decompostion in Hilber space. The semipositive definiteness interprets the kernel matrix as storing the inner products of vectors in a Hilber space.
'' '''Spectral Theorem for Self-Adjoint Compact Operators''' ''<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 />
'''2. Centering  '''<br />
<math>
x = \sum_{i=1}^{\infty} <x, e_i> e_i,
</math><br />
''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 18:48, 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. The semipositive definiteness interprets the kernel matrix as storing the inner products of vectors in a Hilber space.

2. Centering

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