stat946f10

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{ \begin{align*} x = \sum_{i=1}^{\infty} \left\lt x, e_i\right\gt e_i, \end{align*} }[/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].