Is functional analysis actually what we want? Since this is going to be something of an incendiary post, I feel the need to begin with a disclaimer. I am a very early career scientist. My background is mainly in mathematics and theoretical computer science, and I am completely an autodidact when it comes to the physical sciences. So what I’m about to say should be read with such context in mind.
Estimating Monge Maps Kantorovich relaxes Monge If you look in a modern treatment of optimal transportation, you will likely find the problem defined as something like $$ W_c(\mu,\nu) = \inf_{\pi \in \Pi(\mu,\nu)} \int_{X \times Y} c(x,y) d\pi $$ This is the modern definition of the problem, where we model the objective as finding the minimum cost coupling between $\mu$ and $\nu$. But this is not the original formulation of the problem.
Prelude In this series I’m going to detail some interesting ideas I tried in my research that didn’t pan out. By doing so I hope to help other people avoid wasting time on the same things I did. Perhaps they will even realize that something here is not a waste of time and do something cool with it.
Definitions One of my areas of interest is Optimal Transportation. The curious reader can no doubt find a better expose of these ideas on Wikipedia or in one of many books, but the gist is …