PIMS-UBC Distinguished Colloquium: Khanh Dao Duc

How different are two shapes? If each shape is represented as a point cloud, optimal transport (OT) gives a natural answer: find the matching between points that minimizes the average displacement. But OT depends on how the shapes are positioned. Rotate one of them, and the answer changes. The Gromov–Wasserstein (GW) problem avoids this by looking for the matching that best preserves internal distances, which yields a comparison invariant to rigid motions. This comes at a price: GW is a hard non-convex problem, and its formulation does not always fit the questions raised by modern biological imaging data. In this talk, I will present three ways to simplify or extend GW, from theory to algorithms. First, a stratified Wasserstein kernel embeds shapes into Euclidean space through ranked local distance profiles. This gives an isometry-invariant distance and a positive-definite kernel that scale to large datasets. Second, a Joint Gromov–Wasserstein objective extends metric measure spaces to collections of objects, which makes it possible to fit individual components, such as protein subunits, into a larger assembly. Third, a multilinear GW distance built on a multilinear form m defines a metric on shapes modulo the Lie group preserving m, which makes it possible, for instance, to detect chirality. Computing this distance is NP-hard in general, but its low-dimensional structure allows a polynomial-time approximation of the global solution. Throughout, I will illustrate these ideas with applications across scales and systems, including structural and cell biology, genomics, neuroscience, and animal behaviour.