Fast Evaluation of Kernel Distances

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2021-04-01

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In this work we are concerned with the analysis and implementation of fast computational methods for the evaluation of kernel distances. The application we are considering is shape matching and shape classification. The shape matching problem is as follows: We are given two shape representations of an object of interest (in our case, mitral valves of the human heart). We seek a diffeomorphism that establishes a spatial correspondence between these two shapes. We consider an optimal control formulation in which the sought-after diffeomorphism is parameterized by a time-dependent smooth velocity field. To make the problem computational tractable, we consider a self-reproducing kernel Hilbert space formulation. This formulation necessitates a repeated evaluation of sums of Gaussian kernels to evaluate several mathematical operators that arise in our formulation, which in turn constitutes massive computational costs and by that the main bottleneck of our current implementation. In the present work, we investigate the performance of existing algorithms that belong to a class of methods referred to as Fast Gauss Transforms to address these massive computational costs.

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