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I want to solve the problem argmax f(p), where f is a black-box function, and P is a discrete structure. You can think of p as a string, tree, etc. With just this, I know that there is nothing other than exhaustive search for us. But, suppose I define a one-one feature transform phi, such that phi(p) belongs to the real space. Now, we would be able to use standard numerical optimization techniques to optimize in phi's range space, and then use the inverse of phi to get p. But, suppose phi(p) is an infinite dimensional feature vector, and thus, we only have the kernel K(p1, p2) defined. How, then, can I solve this optimization problem efficiently? |
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I think you're looking for the pre-image problem in kernel methods, don't you? See reference 28 here. |
