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Show that the minimizer is obtained by a generalized eigenvalue problem.

$$\alpha=\underset{1^TK\alpha=0; \ \alpha^TK^2\alpha=1}{\text {arg min}} \gamma ||f||_{K}^2+f^TLf$$

Details: $K$ is a constant reproducing Kernel matrix with real entries, and is positive semi-definite and the entries in the matrix are obtained through a Mercer Kernel $k:X\times X$ and has a reproducing kernel Hilbert space of functions $\mathcal{H}_K$ defined from $X\rightarrow \mathbb{R}$ associated with a corrsponding norm being $||.||_K$. Also, $f$ is real-valued, $\gamma$ is positive and $L$ is a constant p.s.d matrix with real entries.

Reference: Eq 13 in the paper- http://people.cs.uchicago.edu/~niyogi/papersps/BNSaistats.pdf The minimization problem posted above is posed in the paper and the authors give the solution in terms of a generalized eigen value problem. I am having trouble trying to pose the minimization as a generalized eigen value problem.

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Can you provide some more detail? What is $K$? And what is $L$? Is $\gamma$ positive? Is $f$ real valued? –  bartgol Oct 29 '12 at 13:08
    
@bartgol Sure! Posted these updates in the question. –  qlinck Oct 29 '12 at 14:45

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