optimal transport

With Suroor Gandhi (NYU) and Adrian Price-Whelan (Flatiron) we have been able to formulate (we think) some questions about unseen gravitational matter (dark matter and unmapped stars and gas) in the Milky Way into questions about transformations that map one set of points onto another set of points. How, you might ask? By thinking about dynamical processes that set up point distributions in phase space.

Being physicists, we figured that we can do this all ourselves! And being Bayesians, we reached for probabilistic methods. Like: Build a kernel density estimate on one set of points and maximize the likelihood given the other set of points and the transformation. That's great! But it has high computational complexity, and it is slow to compute. But for our purposes, we don't need this to be a likelihood, so we found out (through Soledad Villar, NYU) about optimal transport

Despite its name, optimal transport is about solving problems of this type (find transformations that match point sets) with fast, good algorithms. The optimal-transport setting brings a clever objective function (that looks like earth-mover distance) and a high-performance tailored algorithm to match (that looks like linear programming). I don't understand any of this yet, but Math may have just saved our day. I hope I have said here recently how valuable it is to talk out problems with applied mathematicians!