What is Optimal Transport?
What is optimal transport (OT)? Optimal transport is the general problem of moving one distribution of mass to another as efficiently as possible. 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. It is also the infinite-dimensional extension of the discrete problem of matching. Imagine two sets of data X and Y, each having the same number of data points N. A matching problem involves associating with each data point x in X, exactly one data point y in Y. The cost incurred for this match to occur is c(x,y). There are N! such possible matches. The cost of a particular match (called transport) is defined to be the average of c over all data points. The discrete OT problem is to find that transport that minimizes the average cost.
This basic problem has a wealth of applications within mathematics (in the theory of partial differential equations, geometry, functional analysis, optimization, probability and so on) as well as in other fields (image processing, data science, economics, chemical physics etc.) and is currently an extremely active research area in both theory and applications.
The Kanotorovich Initiative is dedicated towards research and dissemination of modern OT mathematics towards a wide audience of researchers, students, industry, policy makers and the general public.