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We present a new optimization procedure which is particularly suited for the solution of second-order kernel methods like e.g. Kernel-PCA. Common to these methods is that there is a cost function to be optimized, under a positive definite quadratic constraint, which bounds the solution. For example, in kernel-PCA the constraint provides unit length and orthogonal (in feature space) principal components. The cost function is often quadratic which allows to solve the problem as a generalized eigenvalue problem. However, in contrast to support vector machines, which employ box constraints, quadratic constraints usually do not lead to sparse solutions. Here we give up the structure of the generalized eigenvalue problem in favor of a non-quadratic regularization term added to the cost function, which enforces sparse solutions. To optimize this more 'complicated' cost function, we introduce a modified conjugate gradient descent method. Starting from an admissible point, all iterations are carried out inside the subspace of admissible solutions, which is defined by the hyper-ellipsoidal constraint surface.
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