The Busemann function has recently found much interest in a variety of geometric machine learning problems, as it naturally defines projections onto geodesic rays of Riemannian manifolds and generalizes the notion of hyperplanes. As several sources of data can be conveniently modeled as probability distributions, it is natural to study this function in the Wasserstein space, which carries a rich formal Riemannian structure induced by Optimal Transport metrics. In this work, we investigate the existence and computation of Busemann functions in Wasserstein space, which admits geodesic rays. We establish closed-form expressions in two important cases: one-dimensional distributions and Gaussian measures. These results enable explicit projection schemes for probability distributions on $\mathbb{R}$, which in turn allow us to define novel Sliced-Wasserstein distances over Gaussian mixtures and labeled datasets. We demonstrate the efficiency of those original schemes on synthetic datasets as well as transfer learning problems.

The core strength of evolutionary learning is the wild, creative, generalpurpose generativity of the evolutionary process. The core weakness of evolutionary learning is its tendency to spend a lot of time exploring dead ends, even in cases where a bit of analytical or problem-specific reasoning would be able to identify the dead-end as such. Given this situation, it is natural that researchers have explored ways of injecting analytical (in particular, probabilistic) inference into the core of evolutionary algorithms - yielding a class of algorithms known as EDAs or Estimation of Distribution Algorithms [PGL02]. EDAs have proved successful for many types of problems. However, there is not yet a truly convincing EDA for optimizing problems centrally involving floating-point (rather than discrete) variables. And attempts to use EDAs for automated program learning, while interesting, have also failed to yield dramatically successful results.