Learning across Incomparable Spaces (in Biomedical Applications)

Traditional machine learning tasks require a direct comparison between samples of different distributions and solely operate on data from identical metric spaces. However, one might be interested in modeling only topological or relational aspects of a distribution, either because the absolute location of the data manifold is irrelevant, e.g. distributions over spaces of learned representations, or not available, e.g. if the data is accessible only as a weighted graph among sample points. Other applications might require the use of different data sources for an analysis, e.g. different patient information need to be integrated into the algorithmic decision process. In these settings, the general principle of traditional machine learning algorithms is not satisfied and methods and distance metrics operating on incomparable spaces are required.

In this talk we discuss optimal transport, a mathematical framework for comparing probability distributions while respecting the underlying geometry and its recent success in biomedical applications. We discuss important limitations that curtail its broader applicability and introduce the Gromov-Wasserstein distance, a generalization of classic optimal transport distances to incomparable ground spaces.