Two photographers shoot the same mountain. One takes a hundred shots from the summit and only a couple from the base; the other does the opposite. If you try to align their images, the algorithm might decide that the summit in one photo matches the summit in the other simply because there are many similar points there, not because it's the same spot. Sampling density throws it off.
The usual method, entropic optimal transport, builds a transport plan that accounts for both point similarity and point counts. But when densities differ a lot, it can produce geometrically wrong pairings. The authors propose a weighted rescaling: you can weaken the influence of density as much as you like, even down to ignoring it entirely. It's like giving the algorithm glasses that cut the glare from uneven sampling.
On model data, the method recovers geometrically accurate correspondences and outperforms existing approaches when point densities differ sharply. This is useful in medicine, biology, astronomy — anywhere you need to compare shapes, not observation counts.
Optimal transport was invented back in the 18th century to figure out the cheapest way to move dirt from a ditch to a mound. Today it helps compare images, texts, and even cells.