Statistics
Wasserstein Distance
The least work needed to reshape one pile of probability into another: mass times distance moved. Unlike overlap, it still knows two distant bumps are far apart.
PQarea between the CDFs = W₁
Drag the top plot to move Q
Wasserstein distance W1—
—
Distribution P
W1—
overlap—
What to observe
- Drag Q away from P and W₁ grows steadily with the gap, because it counts how far the mass has to travel.
- Once the bumps stop touching, the overlap is zero however far apart they are. W₁ still tells near from far, which is why Wasserstein GANs train on it.
- Centre Q on P but make it wider. W₁ is not zero: in 1D it is the area between the CDFs, and spreading mass out costs work too.
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