Ery Arias-Castro, University of California, San Diego

Fukunaga and Hostetler proposed, in the 1970s, to cluster points in space according to the gradient flow of the underlying sampling density. Although their work has generated a good amount of interest, including in recent years, the theoretical underpinnings of this proposal have not been fully developed. Our contribution to the effort is multi-pronged. First, we describe how this definition of clustering coincides with the equally well-known definition of Hartigan based on the density level sets — also proposed in the 1970s. These two approaches to clustering may thus be considered together under the umbrella name of “modal clustering” — which has been in use for a while. We then present novel consistency results for the main modal clustering methods.