# Introduction to Optimal Transport and OTT

This lesson will introduce Optimal Transport and the OTT library.

We talked about Wasserstein distance in the last lesson. It’s quite important to get a background of the underlying theory of **optimal transport (OT)** first.

Note:This lesson is a basic introduction to optimal transport (OT). It can be skipped if needed.

## Optimal transport

In 1781, French mathematician Gaspard Monge presented the following problem:

_“A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The worker’s goal is to erect a target pile with a prescribed shape (for example, a giant sandcastle). Naturally, the worker wishes to minimize her total effort, quantified, for instance, as the total distance or time spent carrying shovelfuls of sand.”

[G. Monge, Mémoire sur la théorie des déblais et des remblais (De l’Imprimerie Royale, 1781)]

So the problem is quite clear:

- We have a distribution $\mathcal{X}$ (the raw pile of sand in this case).
- We want to
**transport**it to another distribution $\mathcal{Y}$. - We want to reduce the cost (be it
*time*or*cost*) of the transportation, $T:\mathcal{X} \to \mathcal{Y}$.

This problem applies in almost any area of daily life whenever we have to move a distribution rather than a single item. Common areas include computer vision, bioinformatics (especially in **single-cell sequencing**), and machine learning.

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