I am currently studying a paper on Wasserstein Fair Classification. Several places they mention the Wasserstein barycenter, weighted barycenter distribution or the Wasserstein barycenter distribution. What is it? How is it defined? I can only find very deep measure-theoretic explanations, which are both a bit out of my league as a statistician, who never enjoyed analysis very much, but they simply also lack an intuition behind.
1
-
2$\begingroup$ You can have a look at this book An Invitation to Statistics in Wasserstein Space. $\endgroup$– AkiraCommented Nov 25, 2023 at 23:12
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
You can think of it as a kind of weighted average of probability measures. It's difficult not to fall into measure theoretic language for a thing like this, but its often useful to assume your measures are absolutely continuous, i.e. are defined by densities w.r.t the Lebesgue measure. In any case, the Wasserstein barycenter is the solution to the minimization problem:
$$\inf \left\{J(\nu) = \sum\limits_{i = 1}^p \frac{\lambda_i}{2}W_2^2(\nu_i, \nu) \right\}$$
where the $\nu_i$ are given "reference" measures and the $\lambda_i$ satisfy $\lambda_i > 0, \sum \lambda_i = 1$. In the case of $p = 2$ this corresponds to the McCann interpolation of $\nu_1, \nu_2$, which is the geodesic in the space of probability measures between them, with distance given by the $W_2$ metric.
To bring us back down to earth from measure theory, it's useful to recall that the Euclidean barycenter (weighted center of mass) is obtained by an analgous minimzation problem
$$\inf \left\{J(y) = \sum\limits_{i=1}^{p} \frac{\lambda_i}{2}|x_i - y|^2\right\}$$
