The Earth Mover's Distance (EMD) is a method to evaluate dissimilarity between two multi-dimensional distributions in some feature space where a distance measure between single features, which we call the ground distance is given. The EMD ``lifts'' this distance from individual features to full distributions.
Intuitively, given two distributions, one can be seen as a mass of earth properly spread in space, the other as a collection of holes in that same space. Then, the EMD measures the least amount of work needed to fill the holes with earth. Here, a unit of work corresponds to transporting a unit of earth by a unit of ground distance.
A distribution can be represented by a set of clusters where each cluster is represented by its mean (or mode), and by the fraction of the distribution that belongs to that cluster. We call such a representation the signature of the distribution. The two signatures can have different sizes, for example, simple distributions have shorter signatures than complex ones.
Computing the EMD is based on a solution to the well-known transportation problem . Suppose that several suppliers, each with a given amount of goods, are required to supply several consumers, each with a given limited capacity. For each supplier-consumer pair, the cost of transporting a single unit of goods is given. The transportation problem is then to find a least-expensive flow of goods from the suppliers to the consumers that satisfies the consumers' demand. Matching signatures can be naturally cast as a transportation problem by defining one signature as the supplier and the other as the consumer, and by setting the cost for a supplier-consumer pair to equal the ground distance between an element in the first signature and an element in the second. Intuitively, the solution is then the minimum amount of ``work'' required to transform one signature into the other.
This can be formalized as the following linear programming problem: Let be the first signature with m clusters, where pi is the cluster representative and wpi is the weight of the cluster; the second signature with n clusters; and the ground distance matrix where dij is the ground distance between clusters pi and qj.
We want to find a flow
with fij the flow
between pi and qj, that minimizes the overall cost
The EMD has the following advantages
More details on the EMD can be found in .