fastGCVM: A Fast Algorithm for the Computation of the Discrete Generalized Cramér-von Mises Distance

Abstract

Comparing two random vectors by calculating a distance measure between the underlying probability density functions is a key ingredient in many applications, especially in the domain of image processing. For this purpose, the recently introduced generalized Cramér-von Mises distance is an interesting choice, since it is well defined even for the multivariate and discrete case. Unfortunately, the naive way of computing this distance, e.g., for two discrete two-dimensional random vectors ~x; ~y 2 [0; : : : ;n-1]2;n 2 N has a computational complexity of O(n5) that is impractical for most applications. This paper introduces fastGCVM, an algorithm that makes use of the well known concept of summed area tables and that allows to compute the generalized Cramér-von Mises distance with a computational complexity of O(n3) for the mentioned case. Two experiments demonstrate the achievable speed up and give an example for a practical application employing fastGCVM.