Earth Mover's Distance and Equivalent Metrics for Spaces with Semigroups
نویسنده
چکیده
introduce a multi-scale metric on a space equipped with a diffusion semigroup. We prove, under some technical conditions, that the norm dual to the space of Lipschitz functions with respect to this metric is equivalent to two other norms, one of which is a weighted sum of the averages at each scale, and one of which is a weighted sum of the difference of averages across scales. The notion of 'scale' is defined by the semigroup. For both equivalent norms, bigger scales have greater contribution. When the function is a difference of two probability distributions, the dual norm is equal to the Earth Mover's Distance with ground distance equal to the multi-scale metric induced by the semigroup. Approved for public release: distribution is unlimited.
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