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. The semipositive definiteness interprets the kernel matrix as storing the inner products of vectors in a Hilber space. Furthermore, the semipositive definiteness also means all eigenvalues are non-negative.

2. Centering
Considering the centering process in Kernel PCA, it is also required here. The condition is given by
[math]\displaystyle{ \sum_i \Phi(x_i) =0 . }[/math]
Equivalently,
[math]\displaystyle{ 0 = |\sum_i \Phi(x_i)|^2 = \sum_{ij}\Phi(x_i)\Phi(x_j)=\sum_{ij}K_{ij}. }[/math]