Earth mover's distance

Imagine you have a bunch of toys on one side of your room and you want to move them to the other side of the room. To do that, you need to pick them up and move them one by one.

Earth mover's distance is kind of like that. But instead of toys, we have piles of dirt or sand (mathematicians call these "distributions"). And instead of just moving them from one side of the room to the other, we're trying to figure out how different they are from each other.

To measure this difference, we imagine that we're a giant excavator, and we're trying to "move" the dirt from one pile to another so that they match up perfectly. It's like trying to build a sandcastle by filling up a bucket of sand and pouring it into a different spot - we keep doing this until the two piles of sand look identical.

The amount of sand we have to move to get the two piles to match up perfectly is the earth mover's distance between them. The more sand we have to move, the farther apart the two piles are.

So, in math terms, the earth mover's distance measures the minimum cost of turning one distribution (pile of sand) into another, where the "cost" is the amount of dirt (sand) that needs to be moved.

Scientists and mathematicians use the earth mover's distance to compare things like images, sounds, or other types of data. By figuring out how much "movement" is needed to match up different distributions, they can learn more about how those distributions are similar or different.