What does the (absolute) value of the Earth Mover Distance between two histograms mean?

Question

When using the Earth Mover Distance as a metric between two 2D histograms which represent normalized probability distributions, what does the value from the resulting optimization actually mean ?

For example say that I call (in algorithmic notation where $P$ and $Q$ are the two histograms in question):

emd(P,Q) = 1.56e-06

What does this absolute value actually mean ?

Answer 1

The Earth Mover's Distance is also known as the Wasserstein metric. The interpretation of the resulting value is a metric that defines you how similar two probability distributions (represented by your histograms) are.

Note that this only makes sense if your histograms are actually normed to have a sum of 1, to satisfy the condition on being a probability distribution.

Edit: A graphical introduction to this distance metric is given here.