EMD

Earth mover's distance, Optimal Transport Problem

The minimum cost required to transform one probability distribution into another by moving probability mass, where cost is defined as the total distance that mass needs to be moved

It does not diverges even if

Support does not duplicate.

\text{EMD}(P, Q) = \min_{\gamma \in \Gamma(P,Q)} \sum_{x,y} \gamma(x,y) d(x,y)

Probability Density Function where

\sum_{y} \gamma(x, y) = P(x), \quad \sum_{x} \gamma(x, y) = Q(y), \quad \sum_{x, y} \gamma(x, y) = 1

Flow

Algorithm

Hungarian Matching

Earth mover's distance

In computer science, the earth mover's distance (EMD)[1] is a measure of dissimilarity between two frequency distributions, densities, or measures, over a metric space D. Informally, if the distributions are interpreted as two different ways of piling up earth (dirt) over D, the EMD captures the minimum cost of building the smaller pile using dirt taken from the larger, where cost is defined as the amount of dirt moved multiplied by the distance over which it is moved.

https://en.wikipedia.org/wiki/Earth_mover%27s_distance

EMD

Earth mover's distance, Optimal Transport Problem

Flow

Algorithm

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