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This is problem 12.10 in "The Elements of Statistical Learning":

Suppose you wish to carry out a linear discriminant analysis (two classes) using a vector of transformations of the input variables $h(x)$. Since $h(x)$ is high-dimensional, you will use a regularized within-class covariance matrix $W_h + \gamma I$. Show that the model can be estimated using only the inner products $K(x_i, x_{i'}) = \left < h(x_i), h(x_{i'}) \right >$.

How can I go about showing that regularized linear discriminant analysis can be estimated using only inner products, as in the "kernel trick" that is often used with SVM's?

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See the papers by Sebastian Mika and co-authors (I think this was the subject of Mika's PhD thesis). The original paper is here and for free here.

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