Optimal Transport (OT) is a mathematical gem at the interface between probability,
analysis and optimization. The goal of that theory is to define geometric tools that
are useful to compare probability distributions. Let us briefly sketch some key ideas
using a vocabulary that was first introduced by Monge two centuries ago: a probability
distribution can be thought of as a pile of sand. Peaks indicate where likely observations
are to appear. Given a pair of probability distributions—two different piles of sand—
there are, in general, multiple ways to morph, transport or reshape the first pile so that
it matches the second. To every such transport we associate an a “global” cost, using
the “local” consideration of how much it costs to move a single grain of sand from one
location to another. The goal of optimal transport is to find the least costly transport,
and use it to derive an entire geometric toolbox for probability distributions.
2019-12-21 20:51:25
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Optimal
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