Section: New Results

Gromov-Wasserstein methods in graphics and machine learning

G. Peyré, J. Solomon, M. Cuturi

A bottleneck of optimal transport (OT) methods for some applications in graphics and machine learning is that it requires the knowledge of an a priori fixed ground cost. This cost is often chosen as some power of a distance, which in turn requires that the data to compare or modify are pre-registered in a common embedding metric space (e.g. the 3-D or 2-D Euclidean space for shapes matching). For many applications (such a shape matching in vision or molecule comparison in quantum chemistry), this is simply not the case. We thus propose in [18], [21] to extend the computational machinery of OT to cope with an unknown cost by using the so-called Gromov-Wasserstein distance. This distance allows to compare probability distributions living in different and un-registered metric spaces, by coupling together pairs of points instead of single points. This allows to formulate a non-convex energy minimization, which is similar to the graph matching problem. We propose to use the entropic regularization scheme to solve it numerically, and we showed that it leads to a very effective Sinkhorn-like algorithm. In [18] (published in SIGGRAPH, the best computer graphics conference) we explore various application in computer graphics (such as shape matching or organization of collections of surfaces and images), while [21] (published in ICML, one of the two best machine learning conference) we extend this machinery to compute interpolation and barycenter of several metric space, with application to shape interpolation and supervised learning for quantum chemistry.

Figure 13. Example of matching induced between an input 3-D shape (on the left) and 3-D or 2-D shapes using the transport coupling computed using our entropy-regularized Gromov-Wasserstein problem. From [18].