Hamilton-Jacobi equations on graphs with applications to semi-supervised learning and data depth

Jeff Calder, Mahmood Ettehad

Abstract: Shortest path graph distances are widely used in data science and machine learning, since they can approximate the underlying geodesic distance on the data manifold. However, the shortest path distance is highly sensitive to the addition of corrupted edges in the graph, either through noise or an adversarial perturbation. In this talk we present a family of Hamilton-Jacobi equations on graphs that we call the p-eikonal equation. We show that the p-eikonal equation with p=1 is a provably robust distance-type function on a graph, and the limiting case in which p goes to infinity recovers shortest path distances. While the p-eikonal equation does not correspond to a shortest-path graph distance, we nonetheless show that the continuum limit of the p-eikonal equation on a random geometric graph recovers a geodesic density weighted distance in the continuum. We show the results of applications to data depth and semi-supervised learning.

Submission Type: Abstract