Optimal transport is the general problem of moving one distribution of mass to another as efficiently as possible.
This is a VERY old problem.
Solving optimal transport
https://pythonot.github.io/quickstart.html?highlight=geomloss The optimal transport problem between discrete distributions is often expressed as
- is the metric cost matrix defining the cost to move mass from bin to bin .
- and are histograms on the simplex (positive, sum to 1) that represent the weights of each samples in the source and target distributions.
This Python library seems to explain:
- A memory efficient implement of Sinkhorn is found in Geomloss
For example, think of using a pile of dirt to fill a hole of the same volume, so as to minimize the average distance moved.
The Earth Mover’s Distance computes the optimal transport.
Also see Sinkhorn Distance
I also see it as Gradient Flow, but I definitely don’t think it’s the same thing.