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I am trying to implement a 2D version of the EMD/Wasserstein Distance to measure the distance of sets of 2D weighted points.

Let $A = \{a_{1}, a_{2}, ..., a_{m}\}$, be a weighted point set such that $a_{i} = \{(x_{i}, w_{i})\}$ for $i = 1,...,m$, where $x_{i} \in R^{2}$ or in general $R^{k}$ with $w_{i} \in R^{+} \cup \{0\}$ being its corresponding weights.

Let $B$ be an other set of points.

How would I go about computing the distance between these 2 sets. Moreover how do I implement something like this (Python or pseudocode)?

Mostly I don't know how to treat the 2D histograms (surrogates for the 2 distributions) and do I use the euclidean distance between the points?

Thanks.

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