  Optimal transport (OT) theory can be informally described using the words of
the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in
hand has to move a large pile of sand lying on a construction site. The goal of
the worker is to erect with all that sand a target pile with a prescribed shape
(for example, that of a giant sand castle). Naturally, the worker wishes to
minimize her total effort, quantified for instance as the total distance or
time spent carrying shovelfuls of sand. Mathematicians interested in OT cast
that problem as that of comparing two probability distributions, two different
piles of sand of the same volume. They consider all of the many possible ways
to morph, transport or reshape the first pile into the second, and associate a
"global" cost to every such transport, using the "local" consideration of how
much it costs to move a grain of sand from one place to another. Recent years
have witnessed the spread of OT in several fields, thanks to the emergence of
approximate solvers that can scale to sizes and dimensions that are relevant to
data sciences. Thanks to this newfound scalability, OT is being increasingly
used to unlock various problems in imaging sciences (such as color or texture
processing), computer vision and graphics (for shape manipulation) or machine
learning (for regression, classification and density fitting). This short book
reviews OT with a bias toward numerical methods and their applications in data
sciences, and sheds lights on the theoretical properties of OT that make it
particularly useful for some of these applications.
