Local isomorphism to solve the pre-image problem in kernel methods

Kernel methods have been popular over the last decade to solve many computer vision, statistics and machine learning problems. An important, both theoretically and practically, open problem in kernel methods is the pre-image problem. The pre-image problem consists of finding a vector in the input space whose mapping is known in the feature space induced by a kernel. To solve the pre-image problem, this paper proposes a framework that computes an isomorphism between local Gram matrices in the input and feature space. Unlike existing methods that rely on analytic properties of kernels, our framework derives closed-form solutions to the pre-image problem in the case of non-differentiable and application-specific kernels. Experiments on the pre-image problem for visualizing cluster centers computed by kernel k-means and denoising high-dimensional images show that our algorithm outperforms state-of-the-art methods.