Department of Mathematics

Speaker: Facundo Mémoli
Title: A Spectral Notion of Gromov-Wasserstein Distances
Affiliation: University of Adelaide, Australia
Date: Monday, February 4, 2013
Place and Time: Room 101, Love Building, 3:35-4:30 pm
Refreshments: Room 204, Love Building, 3:00 pm

Abstract. We introduce a spectral notion of distance between shapes (closed Riemannian manifolds) and study its theoretical properties. We show that our distance satisfies the properties of a metric on the class of isometric shapes, which means, in particular, that two shapes are at 0 distance if and only if they are isometric. Our construction is similar to the Gromov-Wasserstein distance, but rather than viewing shapes merely as metric spaces, we define our distance via the comparison of heat kernels.

This allows us to relate this distance to previously proposed spectral invariants used for shape comparison, such as the spectrum of the Laplace-Beltrami operator and statistics of diffusion distances. In addition, the heat kernel provides a natural notion of scale, which is useful for multi-scale shape comparison. We also prove a hierarchy of lower bounds for our distance, which provide increasing discriminative power at the cost of an increase in computational complexity.