Title: Understanding the shape of biological data: looking through the lens of optimal transport

Abstract: Modern advances in technology have led to the generation of ever-increasing amounts of quantitative data from biological systems, ranging from gene-expression snapshots of cell populations in a tissue to geometric arrangements of atoms within a molecule.

Datasets have inherent underlying structure that reflect the nature of the system being observed and the means of measurement. This structure can often be understood geometrically (e.g. in terms of manifolds, geodesics or clusters), and analysis methods should be adapted to the natural geometry of the data.

Optimal transport is a mathematical and computational framework that explicitly works with geometry in mind, and has deep connections to mathematical geometry, dynamical systems and statistical physics.

Intuitively, it compares distributions of points in a geometric space by leveraging the natural (geodesic) distance on that space.

We will discuss two applications of optimal transport to address biological data analyses. First, we develop both a theoretical and computational framework for inferring cellular dynamics based on optimal transport, and demonstrate its potential to extract the genetic logic that drives biological processes.

Second we consider datasets of hypergraphs (like graphs, but capturing higher-order relationships). Hypergraphs live in a non-Euclidean space and cannot be naturally vectorised. Using a generalised notion of optimal transport, we propose a method for learning linear representations of hypergraphs and show some applications to hypergraph ensembles in both simulated and chemical datasets.