# a question about kernelized locality preserving projections

kernel LPP is of form: $$\min_{\alpha} \ \alpha^{T}KLK\alpha \\ s.t. \ \alpha^{T}KDK\alpha = 1$$ and it eventually results in solving generalized eigenvalue problem below: $$KLK \alpha= \lambda KDK \alpha$$ It has same eigenvalues as: $$Ly=\lambda Dy$$ so here I solve the above eigen system knowing that: $$y=K \alpha$$ Notice the objective value of original kernel LPP is: $$\alpha^{T}KLK\alpha=\lambda\alpha^{T}KDK\alpha=\lambda y^{T}Dy$$ Does it mean that changing K won't affect the objective value of KLPP? Because here I get eigenvalues $\lambda$ and eigenvectors $y$ without using K.

Yes, you are correct. The minimum of the objective function is the minimal eigenvalue of the generalized eigenproblem $Ly = \lambda Dy$, regardless of $K$, at least as long as $K$ has full column rank.